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• 1. Likelihood-free Methods Likelihood-free Methods Christian P. Robert Universit´ Paris-Dauphine & CREST e http://www.ceremade.dauphine.fr/~xian February 16, 2012
• 2. Likelihood-free MethodsOutlineIntroductionThe Metropolis-Hastings Algorithmsimulation-based methods in EconometricsGenetics of ABCApproximate Bayesian computationABC for model choiceABC model choice consistency
• 3. Likelihood-free Methods IntroductionApproximate Bayesian computation Introduction Monte Carlo basics Population Monte Carlo AMIS The Metropolis-Hastings Algorithm simulation-based methods in Econometrics Genetics of ABC Approximate Bayesian computation
• 4. Likelihood-free Methods Introduction Monte Carlo basicsGeneral purpose Given a density π known up to a normalizing constant, and an integrable function h, compute h(x)˜ (x)µ(dx) π Π(h) = h(x)π(x)µ(dx) = π (x)µ(dx) ˜ when h(x)˜ (x)µ(dx) is intractable. π
• 5. Likelihood-free Methods Introduction Monte Carlo basicsMonte Carlo basics Generate an iid sample x1 , . . . , xN from π and estimate Π(h) by N ΠMC (h) = N −1 ˆ N h(xi ). i=1 ˆ as LLN: ΠMC (h) −→ Π(h) N If Π(h2 ) = h2 (x)π(x)µ(dx) < ∞, √ L CLT: ˆ N ΠMC (h) − Π(h) N N 0, Π [h − Π(h)]2 .
• 6. Likelihood-free Methods Introduction Monte Carlo basicsMonte Carlo basics Generate an iid sample x1 , . . . , xN from π and estimate Π(h) by N ΠMC (h) = N −1 ˆ N h(xi ). i=1 ˆ as LLN: ΠMC (h) −→ Π(h) N If Π(h2 ) = h2 (x)π(x)µ(dx) < ∞, √ L CLT: ˆ N ΠMC (h) − Π(h) N N 0, Π [h − Π(h)]2 . Caveat Often impossible or ineﬃcient to simulate directly from Π
• 7. Likelihood-free Methods Introduction Monte Carlo basicsImportance Sampling For Q proposal distribution such that Q(dx) = q(x)µ(dx), alternative representation Π(h) = h(x){π/q}(x)q(x)µ(dx).
• 8. Likelihood-free Methods Introduction Monte Carlo basicsImportance Sampling For Q proposal distribution such that Q(dx) = q(x)µ(dx), alternative representation Π(h) = h(x){π/q}(x)q(x)µ(dx). Principle Generate an iid sample x1 , . . . , xN ∼ Q and estimate Π(h) by N ΠIS (h) = N −1 ˆ Q,N h(xi ){π/q}(xi ). i=1
• 9. Likelihood-free Methods Introduction Monte Carlo basicsImportance Sampling (convergence) Then ˆ as LLN: ΠIS (h) −→ Π(h) Q,N and if Q((hπ/q)2 ) < ∞, √ L CLT: ˆ N(ΠIS (h) − Π(h)) Q,N N 0, Q{(hπ/q − Π(h))2 } .
• 10. Likelihood-free Methods Introduction Monte Carlo basicsImportance Sampling (convergence) Then ˆ as LLN: ΠIS (h) −→ Π(h) Q,N and if Q((hπ/q)2 ) < ∞, √ L CLT: ˆ N(ΠIS (h) − Π(h)) Q,N N 0, Q{(hπ/q − Π(h))2 } . Caveat ˆ If normalizing constant unknown, impossible to use ΠIS Q,N Generic problem in Bayesian Statistics: π(θ|x) ∝ f (x|θ)π(θ).
• 11. Likelihood-free Methods Introduction Monte Carlo basicsSelf-Normalised Importance Sampling Self normalized version N −1 N ˆ ΠSNIS (h) = {π/q}(xi ) h(xi ){π/q}(xi ). Q,N i=1 i=1
• 12. Likelihood-free Methods Introduction Monte Carlo basicsSelf-Normalised Importance Sampling Self normalized version N −1 N ˆ ΠSNIS (h) = {π/q}(xi ) h(xi ){π/q}(xi ). Q,N i=1 i=1 ˆ as LLN : ΠSNIS (h) −→ Π(h) Q,N and if Π((1 + h2 )(π/q)) < ∞, √ L CLT : ˆ N(ΠSNIS (h) − Π(h)) Q,N N 0, π {(π/q)(h − Π(h)}2 ) .
• 13. Likelihood-free Methods Introduction Monte Carlo basicsSelf-Normalised Importance Sampling Self normalized version N −1 N ˆ ΠSNIS (h) = {π/q}(xi ) h(xi ){π/q}(xi ). Q,N i=1 i=1 ˆ as LLN : ΠSNIS (h) −→ Π(h) Q,N and if Π((1 + h2 )(π/q)) < ∞, √ L CLT : ˆ N(ΠSNIS (h) − Π(h)) Q,N N 0, π {(π/q)(h − Π(h)}2 ) . c The quality of the SNIS approximation depends on the choice of Q
• 14. Likelihood-free Methods Introduction Monte Carlo basicsIterated importance sampling Introduction of an algorithmic temporal dimension : (t) (t−1) xi ∼ qt (x|xi ) i = 1, . . . , n, t = 1, . . . and n ˆ 1 (t) (t) It = i h(xi ) n i=1 is still unbiased for (t) (t) πt (xi ) i = (t) (t−1) , i = 1, . . . , n qt (xi |xi )
• 15. Likelihood-free Methods Introduction Population Monte Carlo PMC: Population Monte Carlo Algorithm At time t = 0 iid Generate (xi,0 )1≤i≤N ∼ Q0 Set ωi,0 = {π/q0 }(xi,0 ) iid Generate (Ji,0 )1≤i≤N ∼ M(1, (¯ i,0 )1≤i≤N ) ω Set xi,0 = xJi ,0 ˜
• 16. Likelihood-free Methods Introduction Population Monte Carlo PMC: Population Monte Carlo Algorithm At time t = 0 iid Generate (xi,0 )1≤i≤N ∼ Q0 Set ωi,0 = {π/q0 }(xi,0 ) iid Generate (Ji,0 )1≤i≤N ∼ M(1, (¯ i,0 )1≤i≤N ) ω Set xi,0 = xJi ,0 ˜ At time t (t = 1, . . . , T ), ind Generate xi,t ∼ Qi,t (˜i,t−1 , ·) x Set ωi,t = {π(xi,t )/qi,t (˜i,t−1 , xi,t )} x iid Generate (Ji,t )1≤i≤N ∼ M(1, (¯ i,t )1≤i≤N ) ω Set xi,t = xJi,t ,t . ˜ [Capp´, Douc, Guillin, Marin, & CPR, 2009, Stat.& Comput.] e
• 17. Likelihood-free Methods Introduction Population Monte CarloNotes on PMC After T iterations of PMC, PMC estimator of Π(h) given by T N ¯ 1 ΠPMC (h) = N,T ¯ ωi,t h(xi,t ). T t=1 i=1
• 18. Likelihood-free Methods Introduction Population Monte CarloNotes on PMC After T iterations of PMC, PMC estimator of Π(h) given by T N ¯ 1 ΠPMC (h) = N,T ¯ ωi,t h(xi,t ). T t=1 i=1 1. ωi,t means normalising over whole sequence of simulations ¯ 2. Qi,t ’s chosen arbitrarily under support constraint 3. Qi,t ’s may depend on whole sequence of simulations 4. natural solution for ABC
• 19. Likelihood-free Methods Introduction Population Monte CarloImproving quality The eﬃciency of the SNIS approximation depends on the choice of Q, ranging from optimal q(x) ∝ |h(x) − Π(h)|π(x) to useless ˆ var ΠSNIS (h) = +∞ Q,N
• 20. Likelihood-free Methods Introduction Population Monte CarloImproving quality The eﬃciency of the SNIS approximation depends on the choice of Q, ranging from optimal q(x) ∝ |h(x) − Π(h)|π(x) to useless ˆ var ΠSNIS (h) = +∞ Q,N Example (PMC=adaptive importance sampling) Population Monte Carlo is producing a sequence of proposals Qt aiming at improving eﬃciency ˆ ˆ Kull(π, qt ) ≤ Kull(π, qt−1 ) or var ΠSNIS (h) ≤ var ΠSNIS ,∞ (h) Qt ,∞ Qt−1 [Capp´, Douc, Guillin, Marin, Robert, 04, 07a, 07b, 08] e
• 21. Likelihood-free Methods Introduction AMISMultiple Importance Sampling Reycling: given several proposals Q1 , . . . , QT , for 1 ≤ t ≤ T generate an iid sample t t x1 , . . . , xN ∼ Qt and estimate Π(h) by T N ΠMIS (h) = T −1 ˆ Q,N N −1 h(xit )ωit t=1 i=1 where π(xit ) ωit = correct... qt (xit )
• 22. Likelihood-free Methods Introduction AMISMultiple Importance Sampling Reycling: given several proposals Q1 , . . . , QT , for 1 ≤ t ≤ T generate an iid sample t t x1 , . . . , xN ∼ Qt and estimate Π(h) by T N ΠMIS (h) = T −1 ˆ Q,N N −1 h(xit )ωit t=1 i=1 where π(xit ) ωit = T still correct! T −1 =1 q (xit )
• 23. Likelihood-free Methods Introduction AMISMixture representation Deterministic mixture correction of the weights proposed by Owen and Zhou (JASA, 2000) The corresponding estimator is still unbiased [if not self-normalised] All particles are on the same weighting scale rather than their own Large variance proposals Qt do not take over Variance reduction thanks to weight stabilization & recycling [K.o.] removes the randomness in the component choice [=Rao-Blackwellisation]
• 24. Likelihood-free Methods Introduction AMISGlobal adaptation Global Adaptation At iteration t = 1, · · · , T , ˆ 1. For 1 ≤ i ≤ N1 , generate xit ∼ T3 (ˆt−1 , Σt−1 ) µ 2. Calculate the mixture importance weight of particle xit ωit = π xit δit where t−1 δit = ˆ ˆ qT (3) xit ; µl , Σl l=0
• 25. Likelihood-free Methods Introduction AMIS Backward reweighting 3. If t ≥ 2, actualize the weights of all past particles, xil 1≤l ≤t −1 ωil = π xit δil where ˆ δil = δil + qT (3) xil ; µt−1 , Σt−1 ˆ ˆ 4. Compute IS estimates of target mean and variance µt and Σt , ˆ where t N1 t N1 µt ˆj = ωil (xj )li ωil . . . l=1 i=1 l=1 i=1
• 26. Likelihood-free Methods Introduction AMISA toy example 10 0 −10 −20 −30 −40 −20 0 20 40 Banana shape benchmark: marginal distribution of (x1 , x2 ) for the parameters 2 σ1 = 100 and b = 0.03. Contours represent 60% (red), 90% (black) and 99.9% (blue) conﬁdence regions in the marginal space.
• 27. Likelihood-free Methods Introduction AMISA toy example p=5 p=10 p=20 q 1e+05 20000 40000 60000 80000 60000 8e+04 6e+04 20000 q 4e+04 q q q q 0 Banana shape example: boxplots of 10 replicate ESSs for the AMIS scheme (left) and the NOT-AMIS scheme (right) for p = 5, 10, 20.
• 28. Likelihood-free Methods The Metropolis-Hastings AlgorithmThe Metropolis-Hastings Algorithm Introduction The Metropolis-Hastings Algorithm Monte Carlo Methods based on Markov Chains The Metropolis–Hastings algorithm The random walk Metropolis-Hastings algorithm simulation-based methods in Econometrics Genetics of ABC Approximate Bayesian computation
• 29. Likelihood-free Methods The Metropolis-Hastings Algorithm Monte Carlo Methods based on Markov ChainsRunning Monte Carlo via Markov Chains Epiphany! It is not necessary to use a sample from the distribution f to approximate the integral I= h(x)f (x)dx ,
• 30. Likelihood-free Methods The Metropolis-Hastings Algorithm Monte Carlo Methods based on Markov ChainsRunning Monte Carlo via Markov Chains Epiphany! It is not necessary to use a sample from the distribution f to approximate the integral I= h(x)f (x)dx , Principle: Obtain X1 , . . . , Xn ∼ f (approx) without directly simulating from f , using an ergodic Markov chain with stationary distribution f [Metropolis et al., 1953]
• 31. Likelihood-free Methods The Metropolis-Hastings Algorithm Monte Carlo Methods based on Markov ChainsRunning Monte Carlo via Markov Chains (2) Idea For an arbitrary starting value x (0) , an ergodic chain (X (t) ) is generated using a transition kernel with stationary distribution f
• 32. Likelihood-free Methods The Metropolis-Hastings Algorithm Monte Carlo Methods based on Markov ChainsRunning Monte Carlo via Markov Chains (2) Idea For an arbitrary starting value x (0) , an ergodic chain (X (t) ) is generated using a transition kernel with stationary distribution f Insures the convergence in distribution of (X (t) ) to a random variable from f . For a “large enough” T0 , X (T0 ) can be considered as distributed from f Produce a dependent sample X (T0 ) , X (T0 +1) , . . ., which is generated from f , suﬃcient for most approximation purposes.
• 33. Likelihood-free Methods The Metropolis-Hastings Algorithm Monte Carlo Methods based on Markov ChainsRunning Monte Carlo via Markov Chains (2) Idea For an arbitrary starting value x (0) , an ergodic chain (X (t) ) is generated using a transition kernel with stationary distribution f Insures the convergence in distribution of (X (t) ) to a random variable from f . For a “large enough” T0 , X (T0 ) can be considered as distributed from f Produce a dependent sample X (T0 ) , X (T0 +1) , . . ., which is generated from f , suﬃcient for most approximation purposes. Problem: How can one build a Markov chain with a given stationary distribution?
• 34. Likelihood-free Methods The Metropolis-Hastings Algorithm The Metropolis–Hastings algorithmThe Metropolis–Hastings algorithm Basics The algorithm uses the objective (target) density f and a conditional density q(y |x) called the instrumental (or proposal) distribution
• 35. Likelihood-free Methods The Metropolis-Hastings Algorithm The Metropolis–Hastings algorithmThe MH algorithm Algorithm (Metropolis–Hastings) Given x (t) , 1. Generate Yt ∼ q(y |x (t) ). 2. Take Yt with prob. ρ(x (t) , Yt ), X (t+1) = x (t) with prob. 1 − ρ(x (t) , Yt ), where f (y ) q(x|y ) ρ(x, y ) = min ,1 . f (x) q(y |x)
• 36. Likelihood-free Methods The Metropolis-Hastings Algorithm The Metropolis–Hastings algorithmFeatures Independent of normalizing constants for both f and q(·|x) (ie, those constants independent of x) Never move to values with f (y ) = 0 The chain (x (t) )t may take the same value several times in a row, even though f is a density wrt Lebesgue measure The sequence (yt )t is usually not a Markov chain
• 37. Likelihood-free Methods The Metropolis-Hastings Algorithm The Metropolis–Hastings algorithmConvergence properties The M-H Markov chain is reversible, with invariant/stationary density f since it satisﬁes the detailed balance condition f (y ) K (y , x) = f (x) K (x, y )
• 38. Likelihood-free Methods The Metropolis-Hastings Algorithm The Metropolis–Hastings algorithmConvergence properties The M-H Markov chain is reversible, with invariant/stationary density f since it satisﬁes the detailed balance condition f (y ) K (y , x) = f (x) K (x, y ) If q(y |x) > 0 for every (x, y ), the chain is Harris recurrent and T 1 lim h(X (t) ) = h(x)df (x) a.e. f . T →∞ T t=1 lim K n (x, ·)µ(dx) − f =0 n→∞ TV for every initial distribution µ
• 39. Likelihood-free Methods The Metropolis-Hastings Algorithm The random walk Metropolis-Hastings algorithmRandom walk Metropolis–Hastings Use of a local perturbation as proposal Yt = X (t) + εt , where εt ∼ g , independent of X (t) . The instrumental density is now of the form g (y − x) and the Markov chain is a random walk if we take g to be symmetric g (x) = g (−x)
• 40. Likelihood-free Methods The Metropolis-Hastings Algorithm The random walk Metropolis-Hastings algorithm Algorithm (Random walk Metropolis) Given x (t) 1. Generate Yt ∼ g (y − x (t) ) 2. Take f (Yt )  Y with prob. min 1, , (t+1) t X = f (x (t) )  (t) x otherwise.
• 41. Likelihood-free Methods The Metropolis-Hastings Algorithm The random walk Metropolis-Hastings algorithmRW-MH on mixture posterior distribution 3 2 µ2 1 0 X −1 −1 0 1 2 3 µ1 Random walk MCMC output for .7N (µ1 , 1) + .3N (µ2 , 1)
• 42. Likelihood-free Methods The Metropolis-Hastings Algorithm The random walk Metropolis-Hastings algorithmAcceptance rate A high acceptance rate is not indication of eﬃciency since the random walk may be moving “too slowly” on the target surface
• 43. Likelihood-free Methods The Metropolis-Hastings Algorithm The random walk Metropolis-Hastings algorithmAcceptance rate A high acceptance rate is not indication of eﬃciency since the random walk may be moving “too slowly” on the target surface If x (t) and yt are “too close”, i.e. f (x (t) ) f (yt ), yt is accepted with probability f (yt ) min ,1 1. f (x (t) ) and acceptance rate high
• 44. Likelihood-free Methods The Metropolis-Hastings Algorithm The random walk Metropolis-Hastings algorithmAcceptance rate A high acceptance rate is not indication of eﬃciency since the random walk may be moving “too slowly” on the target surface If average acceptance rate low, the proposed values f (yt ) tend to be small wrt f (x (t) ), i.e. the random walk [not the algorithm!] moves quickly on the target surface often reaching its boundaries
• 45. Likelihood-free Methods The Metropolis-Hastings Algorithm The random walk Metropolis-Hastings algorithmRule of thumb In small dimensions, aim at an average acceptance rate of 50%. In large dimensions, at an average acceptance rate of 25%. [Gelman,Gilks and Roberts, 1995]
• 46. Likelihood-free Methods The Metropolis-Hastings Algorithm The random walk Metropolis-Hastings algorithmNoisy AR(1) Target distribution of x given x1 , x2 and y is −1 τ2 exp (x − ϕx1 )2 + (x2 − ϕx)2 + (y − x 2 )2 . 2τ 2 σ2 For a Gaussian random walk with scale ω small enough, the random walk never jumps to the other mode. But if the scale ω is suﬃciently large, the Markov chain explores both modes and give a satisfactory approximation of the target distribution.
• 47. Likelihood-free Methods The Metropolis-Hastings Algorithm The random walk Metropolis-Hastings algorithmNoisy AR(2) Markov chain based on a random walk with scale ω = .1.
• 48. Likelihood-free Methods The Metropolis-Hastings Algorithm The random walk Metropolis-Hastings algorithmNoisy AR(3) Markov chain based on a random walk with scale ω = .5.
• 49. Likelihood-free Methods simulation-based methods in EconometricsMeanwhile, in a Gaulish village... Similar exploration of simulation-based techniques in Econometrics Simulated method of moments Method of simulated moments Simulated pseudo-maximum-likelihood Indirect inference [Gouri´roux & Monfort, 1996] e
• 50. Likelihood-free Methods simulation-based methods in EconometricsSimulated method of moments o Given observations y1:n from a model yt = r (y1:(t−1) , t , θ) , t ∼ g (·) simulate 1:n , derive yt (θ) = r (y1:(t−1) , t , θ) and estimate θ by n arg min (yto − yt (θ))2 θ t=1
• 51. Likelihood-free Methods simulation-based methods in EconometricsSimulated method of moments o Given observations y1:n from a model yt = r (y1:(t−1) , t , θ) , t ∼ g (·) simulate 1:n , derive yt (θ) = r (y1:(t−1) , t , θ) and estimate θ by n n 2 arg min yto − yt (θ) θ t=1 t=1
• 52. Likelihood-free Methods simulation-based methods in EconometricsMethod of simulated moments Given a statistic vector K (y ) with Eθ [K (Yt )|y1:(t−1) ] = k(y1:(t−1) ; θ) ﬁnd an unbiased estimator of k(y1:(t−1) ; θ), ˜ k( t , y1:(t−1) ; θ) Estimate θ by n S arg min K (yt ) − ˜ t k( s , y1:(t−1) ; θ)/S θ t=1 s=1 [Pakes & Pollard, 1989]
• 53. Likelihood-free Methods simulation-based methods in EconometricsIndirect inference ˆ Minimise (in θ) the distance between estimators β based on pseudo-models for genuine observations and for observations simulated under the true model and the parameter θ. [Gouri´roux, Monfort, & Renault, 1993; e Smith, 1993; Gallant & Tauchen, 1996]
• 54. Likelihood-free Methods simulation-based methods in EconometricsIndirect inference (PML vs. PSE) Example of the pseudo-maximum-likelihood (PML) ˆ β(y) = arg max log f (yt |β, y1:(t−1) ) β t leading to ˆ ˆ arg min ||β(yo ) − β(y1 (θ), . . . , yS (θ))||2 θ when ys (θ) ∼ f (y|θ) s = 1, . . . , S
• 55. Likelihood-free Methods simulation-based methods in EconometricsIndirect inference (PML vs. PSE) Example of the pseudo-score-estimator (PSE) 2 ˆ ∂ log f β(y) = arg min (yt |β, y1:(t−1) ) β t ∂β leading to ˆ ˆ arg min ||β(yo ) − β(y1 (θ), . . . , yS (θ))||2 θ when ys (θ) ∼ f (y|θ) s = 1, . . . , S
• 56. Likelihood-free Methods simulation-based methods in EconometricsConsistent indirect inference ...in order to get a unique solution the dimension of the auxiliary parameter β must be larger than or equal to the dimension of the initial parameter θ. If the problem is just identiﬁed the diﬀerent methods become easier...
• 57. Likelihood-free Methods simulation-based methods in EconometricsConsistent indirect inference ...in order to get a unique solution the dimension of the auxiliary parameter β must be larger than or equal to the dimension of the initial parameter θ. If the problem is just identiﬁed the diﬀerent methods become easier... Consistency depending on the criterion and on the asymptotic identiﬁability of θ [Gouri´roux, Monfort, 1996, p. 66] e
• 58. Likelihood-free Methods simulation-based methods in EconometricsAR(2) vs. MA(1) example true (AR) model yt = t −θ t−1 and [wrong!] auxiliary (MA) model yt = β1 yt−1 + β2 yt−2 + ut R code x=eps=rnorm(250) x[2:250]=x[2:250]-0.5*x[1:249] simeps=rnorm(250) propeta=seq(-.99,.99,le=199) dist=rep(0,199) bethat=as.vector(arima(x,c(2,0,0),incl=FALSE)\$coef) for (t in 1:199) dist[t]=sum((as.vector(arima(c(simeps[1],simeps[2:250]-propeta[t]* simeps[1:249]),c(2,0,0),incl=FALSE)\$coef)-bethat)^2)
• 59. Likelihood-free Methods simulation-based methods in EconometricsAR(2) vs. MA(1) example One sample: 0.8 0.6 distance 0.4 0.2 0.0 −1.0 −0.5 0.0 0.5 1.0
• 60. Likelihood-free Methods simulation-based methods in EconometricsAR(2) vs. MA(1) example Many samples: 6 5 4 3 2 1 0 0.2 0.4 0.6 0.8 1.0
• 61. Likelihood-free Methods simulation-based methods in EconometricsChoice of pseudo-model Pick model such that ˆ 1. β(θ) not ﬂat (i.e. sensitive to changes in θ) ˆ 2. β(θ) not dispersed (i.e. robust agains changes in ys (θ)) [Frigessi & Heggland, 2004]
• 62. Likelihood-free Methods simulation-based methods in EconometricsABC using indirect inference We present a novel approach for developing summary statistics for use in approximate Bayesian computation (ABC) algorithms by using indirect inference(...) In the indirect inference approach to ABC the parameters of an auxiliary model ﬁtted to the data become the summary statistics. Although applicable to any ABC technique, we embed this approach within a sequential Monte Carlo algorithm that is completely adaptive and requires very little tuning(...) [Drovandi, Pettitt & Faddy, 2011] c Indirect inference provides summary statistics for ABC...
• 63. Likelihood-free Methods simulation-based methods in EconometricsABC using indirect inference ...the above result shows that, in the limit as h → 0, ABC will be more accurate than an indirect inference method whose auxiliary statistics are the same as the summary statistic that is used for ABC(...) Initial analysis showed that which method is more accurate depends on the true value of θ. [Fearnhead and Prangle, 2012] c Indirect inference provides estimates rather than global inference...
• 64. Likelihood-free Methods Genetics of ABCGenetic background of ABC ABC is a recent computational technique that only requires being able to sample from the likelihood f (·|θ) This technique stemmed from population genetics models, about 15 years ago, and population geneticists still contribute signiﬁcantly to methodological developments of ABC. [Griﬃth & al., 1997; Tavar´ & al., 1999] e
• 65. Likelihood-free Methods Genetics of ABCPopulation genetics [Part derived from the teaching material of Raphael Leblois, ENS Lyon, November 2010] Describe the genotypes, estimate the alleles frequencies, determine their distribution among individuals, populations and between populations; Predict and understand the evolution of gene frequencies in populations as a result of various factors. c Analyses the eﬀect of various evolutive forces (mutation, drift, migration, selection) on the evolution of gene frequencies in time and space.
• 66. Likelihood-free Methods Genetics of ABCA[B]ctors Les mécanismes de l’évolution Towards an explanation of the mechanisms of evolutionary changes, mathematical models ofl’évolution ont to be developed •! Les mathématiques de evolution started in the years 1920-1930. commencé à être développés dans les années 1920-1930 Ronald A Fisher (1890-1962) Sewall Wright (1889-1988) John BS Haldane (1892-1964)
• 67. Likelihood-free Methods Genetics of ABCWright-Fisher model de Le modèle Wright-Fisher •! En l’absence de mutation et de sélection,A population of constant les fréquences alléliquessize, in which individuals dérivent (augmentent et diminuent) inévitablement reproduce at the same time. jusqu’à la fixation d’un allèle Each gene in a generation is a copy of a gene of the •! La dérivepreviousdonc à la conduit generation. perte de variation génétique à l’intérieur In the absence of mutation des populations and selection, allele frequencies derive inevitably until the ﬁxation of an allele.
• 68. Likelihood-free Methods Genetics of ABCCoalescent theory [Kingman, 1982; Tajima, Tavar´, &tc] e !"#\$%&((")**+\$,-"."/010234%".5"*\$*%()23\$15"6" Coalescence theory interested in the genealogy of a sample of !!7**+\$,-",()5534%" common ancestor of the sample. " genes back in time to the " "!"7**+\$,-"8",\$)(5,1,"9"
• 69. Likelihood-free Methods Genetics of ABCCommon ancestor Les différentes lignées fusionnent (coalescent) au fur et à mesure que l’on remonte vers le Time of coalescence passé (T) Modélisation du processus de dérive génétique The diﬀerent lineages mergeen “remontant dans lein the past. when we go back temps” jusqu’à l’ancêtre commun d’un échantillon de gènes 6
• 70. Likelihood-free Methods Genetics of ABC Sous l’hypothèse de neutralité des marqueurs génétiques étudiés, les mutations sont indépendantes de la généalogieNeutral généalogie ne dépend que des processus démographiques i.e. la mutations On construit donc la généalogie selon les paramètres démographiques (ex. N), puis on ajoute aassumption les Under the posteriori of mutations sur les différentes neutrality, the mutations branches,independentau feuilles de are du MRCA of the l’arbregenealogy. On obtient ainsi des données de We construct the genealogy polymorphisme sous les modèles according to the démographiques et mutationnels demographic parameters, considérés then we add a posteriori the mutations. 20
• 71. Likelihood-free Methods Genetics of ABCDemo-genetic inference Each model is characterized by a set of parameters θ that cover historical (time divergence, admixture time ...), demographics (population sizes, admixture rates, migration rates, ...) and genetic (mutation rate, ...) factors The goal is to estimate these parameters from a dataset of polymorphism (DNA sample) y observed at the present time Problem: most of the time, we can not calculate the likelihood of the polymorphism data f (y|θ).
• 72. Likelihood-free Methods Genetics of ABCUntractable likelihood Missing (too missing!) data structure: f (y|θ) = f (y|G , θ)f (G |θ)dG G The genealogies are considered as nuisance parameters. This problematic thus diﬀers from the phylogenetic approach where the tree is the parameter of interesst.
• 73. Likelihood-free Methods Genetics of ABCA genuine !""#\$%&()*+,(-*.&(/+0\$"1)()&\$/+2!,03! example of application 1/+*%*"4*+56(""4&7()&\$/.+.1#+4*.+8-9:*.+ 94 Pygmies populations: do they have a common origin? Is there a lot of exchanges between pygmies and non-pygmies populations?
• 74. Likelihood-free Methods Genetics of ABC !""#\$%&()*+,(-*.&(/+0\$"1)()&\$/+2!,03! 1/+*%*"4*+56(""4&7()&\$/.+.1#+4*.+8-9:*.+Alternative scenarios Différents scénarios possibles, choix de scenario par ABC Verdu et al. 2009 96
• 75. 1/+*%*"4*+56(""4&7()&\$/.+.1#+4*.+Likelihood-free Methods !""#\$%&()*+,(-*.&(/+0\$"1)()&\$/+2!,03 Genetics of ABC 1/+*%*"4*+56(""4&7()&\$/.+.1#+4*.+8-9:*.Différentsresults Simulation scénarios possibles, choix de scenari Différents scénarios possibles, choix de scenario par ABC c Scenario 1A is chosen. largement soutenu par rapport aux Le scenario 1a est autres ! plaide pour une origine commune des Le scenario 1a est largement soutenu par populations pygmées d’Afrique de l’Ouest rap Verdu e
• 76. Likelihood-free Methods Genetics of ABC !""#\$%&()*+,(-*.&(/+0\$"1)()&\$/+2!,03!Most likely scenario 1/+*%*"4*+56(""4&7()&\$/.+.1#+4*.+8-9:*.+ Scénario évolutif : on « raconte » une histoire à partir de ces inférences Verdu et al. 2009 99
• 77. Likelihood-free Methods Approximate Bayesian computationApproximate Bayesian computation Introduction The Metropolis-Hastings Algorithm simulation-based methods in Econometrics Genetics of ABC Approximate Bayesian computation ABC basics Alphabet soup Automated summary statistic selection
• 78. Likelihood-free Methods Approximate Bayesian computation ABC basicsUntractable likelihoods Cases when the likelihood function f (y|θ) is unavailable and when the completion step f (y|θ) = f (y, z|θ) dz Z is impossible or too costly because of the dimension of z c MCMC cannot be implemented!
• 79. Likelihood-free Methods Approximate Bayesian computation ABC basicsUntractable likelihoods
• 80. Likelihood-free Methods Approximate Bayesian computation ABC basicsIllustrations Example Stochastic volatility model: for Highest weight trajectories t = 1, . . . , T , 0.4 0.2 yt = exp(zt ) t , zt = a+bzt−1 +σηt , 0.0 −0.2 T very large makes it diﬃcult to −0.4 include z within the simulated 0 200 400 t 600 800 1000 parameters
• 81. Likelihood-free Methods Approximate Bayesian computation ABC basicsIllustrations Example Potts model: if y takes values on a grid Y of size k n and f (y|θ) ∝ exp θ Iyl =yi l∼i where l∼i denotes a neighbourhood relation, n moderately large prohibits the computation of the normalising constant
• 82. Likelihood-free Methods Approximate Bayesian computation ABC basicsIllustrations Example Inference on CMB: in cosmology, study of the Cosmic Microwave Background via likelihoods immensely slow to computate (e.g WMAP, Plank), because of numerically costly spectral transforms [Data is a Fortran program] [Kilbinger et al., 2010, MNRAS]
• 83. Likelihood-free Methods Approximate Bayesian computation ABC basicsIllustrations Example Phylogenetic tree: in population genetics, reconstitution of a common ancestor from a sample of genes via a phylogenetic tree that is close to impossible to integrate out [100 processor days with 4 parameters] [Cornuet et al., 2009, Bioinformatics]
• 84. Likelihood-free Methods Approximate Bayesian computation ABC basicsThe ABC method Bayesian setting: target is π(θ)f (x|θ)
• 85. Likelihood-free Methods Approximate Bayesian computation ABC basicsThe ABC method Bayesian setting: target is π(θ)f (x|θ) When likelihood f (x|θ) not in closed form, likelihood-free rejection technique:
• 86. Likelihood-free Methods Approximate Bayesian computation ABC basicsThe ABC method Bayesian setting: target is π(θ)f (x|θ) When likelihood f (x|θ) not in closed form, likelihood-free rejection technique: ABC algorithm For an observation y ∼ f (y|θ), under the prior π(θ), keep jointly simulating θ ∼ π(θ) , z ∼ f (z|θ ) , until the auxiliary variable z is equal to the observed value, z = y. [Tavar´ et al., 1997] e
• 87. Likelihood-free Methods Approximate Bayesian computation ABC basicsWhy does it work?! The proof is trivial: f (θi ) ∝ π(θi )f (z|θi )Iy (z) z∈D ∝ π(θi )f (y|θi ) = π(θi |y) . [Accept–Reject 101]
• 88. Likelihood-free Methods Approximate Bayesian computation ABC basicsEarlier occurrence ‘Bayesian statistics and Monte Carlo methods are ideally suited to the task of passing many models over one dataset’ [Don Rubin, Annals of Statistics, 1984] Note Rubin (1984) does not promote this algorithm for likelihood-free simulation but frequentist intuition on posterior distributions: parameters from posteriors are more likely to be those that could have generated the data.
• 89. Likelihood-free Methods Approximate Bayesian computation ABC basicsA as A...pproximative When y is a continuous random variable, equality z = y is replaced with a tolerance condition, (y, z) ≤ where is a distance
• 90. Likelihood-free Methods Approximate Bayesian computation ABC basicsA as A...pproximative When y is a continuous random variable, equality z = y is replaced with a tolerance condition, (y, z) ≤ where is a distance Output distributed from π(θ) Pθ { (y, z) < } ∝ π(θ| (y, z) < ) [Pritchard et al., 1999]
• 91. Likelihood-free Methods Approximate Bayesian computation ABC basicsABC algorithm Algorithm 1 Likelihood-free rejection sampler 2 for i = 1 to N do repeat generate θ from the prior distribution π(·) generate z from the likelihood f (·|θ ) until ρ{η(z), η(y)} ≤ set θi = θ end for where η(y) deﬁnes a (not necessarily suﬃcient) statistic
• 92. Likelihood-free Methods Approximate Bayesian computation ABC basicsOutput The likelihood-free algorithm samples from the marginal in z of: π(θ)f (z|θ)IA ,y (z) π (θ, z|y) = , A ,y ×Θ π(θ)f (z|θ)dzdθ where A ,y = {z ∈ D|ρ(η(z), η(y)) < }.
• 93. Likelihood-free Methods Approximate Bayesian computation ABC basicsOutput The likelihood-free algorithm samples from the marginal in z of: π(θ)f (z|θ)IA ,y (z) π (θ, z|y) = , A ,y ×Θ π(θ)f (z|θ)dzdθ where A ,y = {z ∈ D|ρ(η(z), η(y)) < }. The idea behind ABC is that the summary statistics coupled with a small tolerance should provide a good approximation of the posterior distribution: π (θ|y) = π (θ, z|y)dz ≈ π(θ|y) .
• 94. Likelihood-free Methods Approximate Bayesian computation ABC basicsConvergence of ABC (ﬁrst attempt) What happens when → 0?
• 95. Likelihood-free Methods Approximate Bayesian computation ABC basicsConvergence of ABC (ﬁrst attempt) What happens when → 0? If f (·|θ) is continuous in y , uniformly in θ [!], given an arbitrary δ > 0, there exists 0 such that < 0 implies π(θ) f (z|θ)IA ,y (z) dz π(θ)f (y|θ)(1 δ)µ(B ) ∈ A ,y ×Θ π(θ)f (z|θ)dzdθ Θ π(θ)f (y|θ)dθ(1 ± δ)µ(B )
• 96. Likelihood-free Methods Approximate Bayesian computation ABC basicsConvergence of ABC (ﬁrst attempt) What happens when → 0? If f (·|θ) is continuous in y , uniformly in θ [!], given an arbitrary δ > 0, there exists 0 such that < 0 implies π(θ) f (z|θ)IA ,y (z) dz π(θ)f (y|θ)(1 δ) XX ) µ(BX ∈ A ,y ×Θ π(θ)f (z|θ)dzdθ Θ π(θ)f (y|θ)dθ(1 ± δ) µ(B ) XXX
• 97. Likelihood-free Methods Approximate Bayesian computation ABC basicsConvergence of ABC (ﬁrst attempt) What happens when → 0? If f (·|θ) is continuous in y , uniformly in θ [!], given an arbitrary δ > 0, there exists 0 such that < 0 implies π(θ) f (z|θ)IA ,y (z) dz π(θ)f (y|θ)(1 δ) XX ) µ(BX ∈ A ,y ×Θ π(θ)f (z|θ)dzdθ Θ π(θ)f (y|θ)dθ(1 ± δ) X µ(B ) X X [Proof extends to other continuous-in-0 kernels K ]
• 98. Likelihood-free Methods Approximate Bayesian computation ABC basicsConvergence of ABC (second attempt) What happens when → 0?
• 99. Likelihood-free Methods Approximate Bayesian computation ABC basicsConvergence of ABC (second attempt) What happens when → 0? For B ⊂ Θ, we have A f (z|θ)dz f (z|θ)π(θ)dθ ,y B π(θ)dθ = dz B A ,y ×Θ π(θ)f (z|θ)dzdθ A ,y A ,y ×Θ π(θ)f (z|θ)dzdθ B f (z|θ)π(θ)dθ m(z) = dz A ,y m(z) A ,y ×Θ π(θ)f (z|θ)dzdθ m(z) = π(B|z) dz A ,y A ,y ×Θ π(θ)f (z|θ)dzdθ which indicates convergence for a continuous π(B|z).
• 100. Likelihood-free Methods Approximate Bayesian computation ABC basicsProbit modelling on Pima Indian women Example (R benchmark) 200 Pima Indian women with observed variables plasma glucose concentration in oral glucose tolerance test diastolic blood pressure diabetes pedigree function presence/absence of diabetes
• 101. Likelihood-free Methods Approximate Bayesian computation ABC basicsProbit modelling on Pima Indian women Example (R benchmark) 200 Pima Indian women with observed variables plasma glucose concentration in oral glucose tolerance test diastolic blood pressure diabetes pedigree function presence/absence of diabetes Probability of diabetes function of above variables P(y = 1|x) = Φ(x1 β1 + x2 β2 + x3 β3 ) ,
• 102. Likelihood-free Methods Approximate Bayesian computation ABC basicsProbit modelling on Pima Indian women Example (R benchmark) 200 Pima Indian women with observed variables plasma glucose concentration in oral glucose tolerance test diastolic blood pressure diabetes pedigree function presence/absence of diabetes Probability of diabetes function of above variables P(y = 1|x) = Φ(x1 β1 + x2 β2 + x3 β3 ) , Test of H0 : β3 = 0 for 200 observations of Pima.tr based on a g -prior modelling: β ∼ N3 (0, n XT X)−1
• 103. Likelihood-free Methods Approximate Bayesian computation ABC basicsProbit modelling on Pima Indian women Example (R benchmark) 200 Pima Indian women with observed variables plasma glucose concentration in oral glucose tolerance test diastolic blood pressure diabetes pedigree function presence/absence of diabetes Probability of diabetes function of above variables P(y = 1|x) = Φ(x1 β1 + x2 β2 + x3 β3 ) , Test of H0 : β3 = 0 for 200 observations of Pima.tr based on a g -prior modelling: β ∼ N3 (0, n XT X)−1 Use of importance function inspired from the MLE estimate distribution ˆ ˆ β ∼ N (β, Σ)
• 104. Likelihood-free Methods Approximate Bayesian computation ABC basicsPima Indian benchmark 80 100 1.0 80 60 0.8 60 0.6 Density Density Density 40 40 0.4 20 20 0.2 0.0 0 0 −0.005 0.010 0.020 0.030 −0.05 −0.03 −0.01 −1.0 0.0 1.0 2.0 Figure: Comparison between density estimates of the marginals on β1 (left), β2 (center) and β3 (right) from ABC rejection samples (red) and MCMC samples (black) .
• 105. Likelihood-free Methods Approximate Bayesian computation ABC basicsMA example Back to the MA(q) model q xt = t + ϑi t−i i=1 Simple prior: uniform over the inverse [real and complex] roots in q Q(u) = 1 − ϑi u i i=1 under the identiﬁability conditions
• 106. Likelihood-free Methods Approximate Bayesian computation ABC basicsMA example Back to the MA(q) model q xt = t + ϑi t−i i=1 Simple prior: uniform prior over the identiﬁability zone, e.g. triangle for MA(2)
• 107. Likelihood-free Methods Approximate Bayesian computation ABC basicsMA example (2) ABC algorithm thus made of 1. picking a new value (ϑ1 , ϑ2 ) in the triangle 2. generating an iid sequence ( t )−q<t≤T 3. producing a simulated series (xt )1≤t≤T
• 108. Likelihood-free Methods Approximate Bayesian computation ABC basicsMA example (2) ABC algorithm thus made of 1. picking a new value (ϑ1 , ϑ2 ) in the triangle 2. generating an iid sequence ( t )−q<t≤T 3. producing a simulated series (xt )1≤t≤T Distance: basic distance between the series T ρ((xt )1≤t≤T , (xt )1≤t≤T ) = (xt − xt )2 t=1 or distance between summary statistics like the q autocorrelations T τj = xt xt−j t=j+1
• 109. Likelihood-free Methods Approximate Bayesian computation ABC basicsComparison of distance impact Evaluation of the tolerance on the ABC sample against both distances ( = 100%, 10%, 1%, 0.1%) for an MA(2) model
• 110. Likelihood-free Methods Approximate Bayesian computation ABC basicsComparison of distance impact 4 1.5 3 1.0 2 0.5 1 0.0 0 0.0 0.2 0.4 0.6 0.8 −2.0 −1.0 0.0 0.5 1.0 1.5 θ1 θ2 Evaluation of the tolerance on the ABC sample against both distances ( = 100%, 10%, 1%, 0.1%) for an MA(2) model
• 111. Likelihood-free Methods Approximate Bayesian computation ABC basicsComparison of distance impact 4 1.5 3 1.0 2 0.5 1 0.0 0 0.0 0.2 0.4 0.6 0.8 −2.0 −1.0 0.0 0.5 1.0 1.5 θ1 θ2 Evaluation of the tolerance on the ABC sample against both distances ( = 100%, 10%, 1%, 0.1%) for an MA(2) model
• 112. Likelihood-free Methods Approximate Bayesian computation ABC basicsHomonomy The ABC algorithm is not to be confused with the ABC algorithm The Artiﬁcial Bee Colony algorithm is a swarm based meta-heuristic algorithm that was introduced by Karaboga in 2005 for optimizing numerical problems. It was inspired by the intelligent foraging behavior of honey bees. The algorithm is speciﬁcally based on the model proposed by Tereshko and Loengarov (2005) for the foraging behaviour of honey bee colonies. The model consists of three essential components: employed and unemployed foraging bees, and food sources. The ﬁrst two components, employed and unemployed foraging bees, search for rich food sources (...) close to their hive. The model also deﬁnes two leading modes of behaviour (...): recruitment of foragers to rich food sources resulting in positive feedback and abandonment of poor sources by foragers causing negative feedback. [Karaboga, Scholarpedia]
• 113. Likelihood-free Methods Approximate Bayesian computation ABC basicsABC advances Simulating from the prior is often poor in eﬃciency
• 114. Likelihood-free Methods Approximate Bayesian computation ABC basicsABC advances Simulating from the prior is often poor in eﬃciency Either modify the proposal distribution on θ to increase the density of x’s within the vicinity of y ... [Marjoram et al, 2003; Bortot et al., 2007, Sisson et al., 2007]
• 115. Likelihood-free Methods Approximate Bayesian computation ABC basicsABC advances Simulating from the prior is often poor in eﬃciency Either modify the proposal distribution on θ to increase the density of x’s within the vicinity of y ... [Marjoram et al, 2003; Bortot et al., 2007, Sisson et al., 2007] ...or by viewing the problem as a conditional density estimation and by developing techniques to allow for larger [Beaumont et al., 2002]
• 116. Likelihood-free Methods Approximate Bayesian computation ABC basicsABC advances Simulating from the prior is often poor in eﬃciency Either modify the proposal distribution on θ to increase the density of x’s within the vicinity of y ... [Marjoram et al, 2003; Bortot et al., 2007, Sisson et al., 2007] ...or by viewing the problem as a conditional density estimation and by developing techniques to allow for larger [Beaumont et al., 2002] .....or even by including in the inferential framework [ABCµ ] [Ratmann et al., 2009]
• 117. Likelihood-free Methods Approximate Bayesian computation Alphabet soupABC-NP Better usage of [prior] simulations by adjustement: instead of throwing away θ such that ρ(η(z), η(y)) > , replace θ’s with locally regressed transforms (use with BIC) θ∗ = θ − {η(z) − η(y)}T β ˆ [Csill´ry et al., TEE, 2010] e ˆ where β is obtained by [NP] weighted least square regression on (η(z) − η(y)) with weights Kδ {ρ(η(z), η(y))} [Beaumont et al., 2002, Genetics]
• 118. Likelihood-free Methods Approximate Bayesian computation Alphabet soupABC-NP (regression) Also found in the subsequent literature, e.g. in Fearnhead-Prangle (2012) : weight directly simulation by Kδ {ρ(η(z(θ)), η(y))} or S 1 Kδ {ρ(η(zs (θ)), η(y))} S s=1 [consistent estimate of f (η|θ)]
• 119. Likelihood-free Methods Approximate Bayesian computation Alphabet soupABC-NP (regression) Also found in the subsequent literature, e.g. in Fearnhead-Prangle (2012) : weight directly simulation by Kδ {ρ(η(z(θ)), η(y))} or S 1 Kδ {ρ(η(zs (θ)), η(y))} S s=1 [consistent estimate of f (η|θ)] Curse of dimensionality: poor estimate when d = dim(η) is large...
• 120. Likelihood-free Methods Approximate Bayesian computation Alphabet soupABC-NP (density estimation) Use of the kernel weights Kδ {ρ(η(z(θ)), η(y))} leads to the NP estimate of the posterior expectation i θi Kδ {ρ(η(z(θi )), η(y))} i Kδ {ρ(η(z(θi )), η(y))} [Blum, JASA, 2010]
• 121. Likelihood-free Methods Approximate Bayesian computation Alphabet soupABC-NP (density estimation) Use of the kernel weights Kδ {ρ(η(z(θ)), η(y))} leads to the NP estimate of the posterior conditional density ˜ Kb (θi − θ)Kδ {ρ(η(z(θi )), η(y))} i i Kδ {ρ(η(z(θi )), η(y))} [Blum, JASA, 2010]
• 122. Likelihood-free Methods Approximate Bayesian computation Alphabet soupABC-NP (density estimations) Other versions incorporating regression adjustments ˜ Kb (θi∗ − θ)Kδ {ρ(η(z(θi )), η(y))} i i Kδ {ρ(η(z(θi )), η(y))}
• 123. Likelihood-free Methods Approximate Bayesian computation Alphabet soupABC-NP (density estimations) Other versions incorporating regression adjustments ˜ Kb (θi∗ − θ)Kδ {ρ(η(z(θi )), η(y))} i i Kδ {ρ(η(z(θi )), η(y))} In all cases, error E[ˆ (θ|y)] − g (θ|y) = cb 2 + cδ 2 + OP (b 2 + δ 2 ) + OP (1/nδ d ) g c var(ˆ (θ|y)) = g (1 + oP (1)) nbδ d [Blum, JASA, 2010]
• 124. Likelihood-free Methods Approximate Bayesian computation Alphabet soupABC-NP (density estimations) Other versions incorporating regression adjustments ˜ Kb (θi∗ − θ)Kδ {ρ(η(z(θi )), η(y))} i i Kδ {ρ(η(z(θi )), η(y))} In all cases, error E[ˆ (θ|y)] − g (θ|y) = cb 2 + cδ 2 + OP (b 2 + δ 2 ) + OP (1/nδ d ) g c var(ˆ (θ|y)) = g (1 + oP (1)) nbδ d [standard NP calculations]
• 125. Likelihood-free Methods Approximate Bayesian computation Alphabet soupABC-NCH Incorporating non-linearities and heterocedasticities: σ (η(y)) ˆ θ∗ = m(η(y)) + [θ − m(η(z))] ˆ ˆ σ (η(z)) ˆ
• 126. Likelihood-free Methods Approximate Bayesian computation Alphabet soupABC-NCH Incorporating non-linearities and heterocedasticities: σ (η(y)) ˆ θ∗ = m(η(y)) + [θ − m(η(z))] ˆ ˆ σ (η(z)) ˆ where m(η) estimated by non-linear regression (e.g., neural network) ˆ σ (η) estimated by non-linear regression on residuals ˆ log{θi − m(ηi )}2 = log σ 2 (ηi ) + ξi ˆ [Blum & Fran¸ois, 2009] c
• 127. Likelihood-free Methods Approximate Bayesian computation Alphabet soupABC-NCH (2) Why neural network?
• 128. Likelihood-free Methods Approximate Bayesian computation Alphabet soupABC-NCH (2) Why neural network? ﬁghts curse of dimensionality selects relevant summary statistics provides automated dimension reduction oﬀers a model choice capability improves upon multinomial logistic [Blum & Fran¸ois, 2009] c
• 129. Likelihood-free Methods Approximate Bayesian computation Alphabet soupABC-MCMC Markov chain (θ(t) ) created via the transition function  θ ∼ Kω (θ |θ(t) ) if x ∼ f (x|θ ) is such that x = y   π(θ )Kω (t) |θ ) θ (t+1) = and u ∼ U(0, 1) ≤ π(θ(t) )K (θ |θ(t) ) ,  ω (θ  (t) θ otherwise,
• 130. Likelihood-free Methods Approximate Bayesian computation Alphabet soupABC-MCMC Markov chain (θ(t) ) created via the transition function  θ ∼ Kω (θ |θ(t) ) if x ∼ f (x|θ ) is such that x = y   π(θ )Kω (t) |θ ) θ (t+1) = and u ∼ U(0, 1) ≤ π(θ(t) )K (θ |θ(t) ) ,  ω (θ  (t) θ otherwise, has the posterior π(θ|y ) as stationary distribution [Marjoram et al, 2003]
• 131. Likelihood-free Methods Approximate Bayesian computation Alphabet soupABC-MCMC (2) Algorithm 2 Likelihood-free MCMC sampler Use Algorithm 1 to get (θ(0) , z(0) ) for t = 1 to N do Generate θ from Kω ·|θ(t−1) , Generate z from the likelihood f (·|θ ), Generate u from U[0,1] , π(θ )Kω (θ(t−1) |θ ) if u ≤ I π(θ(t−1) Kω (θ |θ(t−1) ) A ,y (z ) then set (θ(t) , z(t) ) = (θ , z ) else (θ(t) , z(t) )) = (θ(t−1) , z(t−1) ), end if end for
• 132. Likelihood-free Methods Approximate Bayesian computation Alphabet soupWhy does it work? Acceptance probability does not involve calculating the likelihood and π (θ , z |y) q(θ (t−1) |θ )f (z(t−1) |θ (t−1) ) × π (θ (t−1) , z(t−1) |y) q(θ |θ (t−1) )f (z |θ ) π(θ ) XX|θ IA ,y (z )  ) f (z X X = π(θ (t−1) ) f (z(t−1) |θ (t−1) ) IA ,y (z(t−1) ) q(θ (t−1) |θ ) f (z(t−1) |θ (t−1) ) × q(θ |θ (t−1) ) X|θ X ) f (z XX
• 133. Likelihood-free Methods Approximate Bayesian computation Alphabet soupWhy does it work? Acceptance probability does not involve calculating the likelihood and π (θ , z |y) q(θ (t−1) |θ )f (z(t−1) |θ (t−1) ) × π (θ (t−1) , z(t−1) |y) q(θ |θ (t−1) )f (z |θ ) π(θ ) XX|θ IA ,y (z )  ) f (z X X = hh ( π(θ (t−1) ) ((hhhhh) IA ,y (z(t−1) ) f (z(t−1) |θ (t−1) ((( h (( hh ( q(θ (t−1) |θ ) ((hhhhh) f (z(t−1) |θ (t−1) (( ((( h × q(θ |θ (t−1) ) X|θ X ) f (z XX
• 134. Likelihood-free Methods Approximate Bayesian computation Alphabet soupWhy does it work? Acceptance probability does not involve calculating the likelihood and π (θ , z |y) q(θ (t−1) |θ )f (z(t−1) |θ (t−1) ) × π (θ (t−1) , z(t−1) |y) q(θ |θ (t−1) )f (z |θ ) π(θ ) X|θ IA ,y (z ) X ) f (z X X = hh ( π(θ (t−1) ) ((hhhhh) XX(t−1) ) f (z(t−1) |θ (t−1) IAX (( X ((( h ,y (zX  X hh ( q(θ (t−1) |θ ) ((hhhhh) f (z(t−1) |θ (t−1) (( ((( h × q(θ |θ (t−1) ) X|θ X ) f (z XX π(θ )q(θ (t−1) |θ ) = IA ,y (z ) π(θ (t−1) q(θ |θ (t−1) )
• 135. Likelihood-free Methods Approximate Bayesian computation Alphabet soupA toy example Case of 1 1 x ∼ N (θ, 1) + N (θ, 1) 2 2 under prior θ ∼ N (0, 10)
• 136. Likelihood-free Methods Approximate Bayesian computation Alphabet soupA toy example Case of 1 1 x ∼ N (θ, 1) + N (θ, 1) 2 2 under prior θ ∼ N (0, 10) ABC sampler thetas=rnorm(N,sd=10) zed=sample(c(1,-1),N,rep=TRUE)*thetas+rnorm(N,sd=1) eps=quantile(abs(zed-x),.01) abc=thetas[abs(zed-x)<eps]
• 137. Likelihood-free Methods Approximate Bayesian computation Alphabet soupA toy example Case of 1 1 x ∼ N (θ, 1) + N (θ, 1) 2 2 under prior θ ∼ N (0, 10) ABC-MCMC sampler metas=rep(0,N) metas[1]=rnorm(1,sd=10) zed[1]=x for (t in 2:N){ metas[t]=rnorm(1,mean=metas[t-1],sd=5) zed[t]=rnorm(1,mean=(1-2*(runif(1)<.5))*metas[t],sd=1) if ((abs(zed[t]-x)>eps)||(runif(1)>dnorm(metas[t],sd=10)/dnorm(metas[t-1],sd=10))){ metas[t]=metas[t-1] zed[t]=zed[t-1]} }
• 138. Likelihood-free Methods Approximate Bayesian computation Alphabet soupA toy example x =2
• 139. Likelihood-free Methods Approximate Bayesian computation Alphabet soupA toy example x =2 0.30 0.25 0.20 0.15 0.10 0.05 0.00 −4 −2 0 2 4 θ
• 140. Likelihood-free Methods Approximate Bayesian computation Alphabet soupA toy example x = 10
• 141. Likelihood-free Methods Approximate Bayesian computation Alphabet soupA toy example x = 10 0.25 0.20 0.15 0.10 0.05 0.00 −10 −5 0 5 10 θ
• 142. Likelihood-free Methods Approximate Bayesian computation Alphabet soupA toy example x = 50
• 143. Likelihood-free Methods Approximate Bayesian computation Alphabet soupA toy example x = 50 0.10 0.08 0.06 0.04 0.02 0.00 −40 −20 0 20 40 θ
• 144. Likelihood-free Methods Approximate Bayesian computation Alphabet soupABCµ [Ratmann, Andrieu, Wiuf and Richardson, 2009, PNAS] Use of a joint density f (θ, |y) ∝ ξ( |y, θ) × πθ (θ) × π ( ) where y is the data, and ξ( |y, θ) is the prior predictive density of ρ(η(z), η(y)) given θ and x when z ∼ f (z|θ)
• 145. Likelihood-free Methods Approximate Bayesian computation Alphabet soupABCµ [Ratmann, Andrieu, Wiuf and Richardson, 2009, PNAS] Use of a joint density f (θ, |y) ∝ ξ( |y, θ) × πθ (θ) × π ( ) where y is the data, and ξ( |y, θ) is the prior predictive density of ρ(η(z), η(y)) given θ and x when z ∼ f (z|θ) Warning! Replacement of ξ( |y, θ) with a non-parametric kernel approximation.
• 146. Likelihood-free Methods Approximate Bayesian computation Alphabet soupABCµ details Multidimensional distances ρk (k = 1, . . . , K ) and errors k = ρk (ηk (z), ηk (y)), with ˆ 1 k ∼ ξk ( |y, θ) ≈ ξk ( |y, θ) = K [{ k −ρk (ηk (zb ), ηk (y))}/hk ] Bhk b ˆ then used in replacing ξ( |y, θ) with mink ξk ( |y, θ)
• 147. Likelihood-free Methods Approximate Bayesian computation Alphabet soupABCµ details Multidimensional distances ρk (k = 1, . . . , K ) and errors k = ρk (ηk (z), ηk (y)), with ˆ 1 k ∼ ξk ( |y, θ) ≈ ξk ( |y, θ) = K [{ k −ρk (ηk (zb ), ηk (y))}/hk ] Bhk b ˆ then used in replacing ξ( |y, θ) with mink ξk ( |y, θ) ABCµ involves acceptance probability ˆ π(θ , ) q(θ , θ)q( , ) mink ξk ( |y, θ ) ˆ π(θ, ) q(θ, θ )q( , ) mink ξk ( |y, θ)
• 148. Likelihood-free Methods Approximate Bayesian computation Alphabet soupABCµ multiple errors [ c Ratmann et al., PNAS, 2009]
• 149. Likelihood-free Methods Approximate Bayesian computation Alphabet soupABCµ for model choice [ c Ratmann et al., PNAS, 2009]
• 150. Likelihood-free Methods Approximate Bayesian computation Alphabet soupQuestions about ABCµ For each model under comparison, marginal posterior on used to assess the ﬁt of the model (HPD includes 0 or not).
• 151. Likelihood-free Methods Approximate Bayesian computation Alphabet soupQuestions about ABCµ For each model under comparison, marginal posterior on used to assess the ﬁt of the model (HPD includes 0 or not). Is the data informative about ? [Identiﬁability] How is the prior π( ) impacting the comparison? How is using both ξ( |x0 , θ) and π ( ) compatible with a standard probability model? [remindful of Wilkinson ] Where is the penalisation for complexity in the model comparison? [X, Mengersen & Chen, 2010, PNAS]
• 152. Likelihood-free Methods Approximate Bayesian computation Alphabet soupABC-PRC Another sequential version producing a sequence of Markov (t) (t) transition kernels Kt and of samples (θ1 , . . . , θN ) (1 ≤ t ≤ T )
• 153. Likelihood-free Methods Approximate Bayesian computation Alphabet soupABC-PRC Another sequential version producing a sequence of Markov (t) (t) transition kernels Kt and of samples (θ1 , . . . , θN ) (1 ≤ t ≤ T ) ABC-PRC Algorithm (t−1) 1. Select θ at random from previous θi ’s with probabilities (t−1) ωi (1 ≤ i ≤ N). 2. Generate (t) (t) θi ∼ Kt (θ|θ ) , x ∼ f (x|θi ) , 3. Check that (x, y ) < , otherwise start again. [Sisson et al., 2007]
• 154. Likelihood-free Methods Approximate Bayesian computation Alphabet soupWhy PRC? Partial rejection control: Resample from a population of weighted particles by pruning away particles with weights below threshold C , replacing them by new particles obtained by propagating an existing particle by an SMC step and modifying the weights accordinly. [Liu, 2001]
• 155. Likelihood-free Methods Approximate Bayesian computation Alphabet soupWhy PRC? Partial rejection control: Resample from a population of weighted particles by pruning away particles with weights below threshold C , replacing them by new particles obtained by propagating an existing particle by an SMC step and modifying the weights accordinly. [Liu, 2001] PRC justiﬁcation in ABC-PRC: Suppose we then implement the PRC algorithm for some c > 0 such that only identically zero weights are smaller than c Trouble is, there is no such c...
• 156. Likelihood-free Methods Approximate Bayesian computation Alphabet soupABC-PRC weight (t) Probability ωi computed as (t) (t) (t) (t) ωi ∝ π(θi )Lt−1 (θ |θi ){π(θ )Kt (θi |θ )}−1 , where Lt−1 is an arbitrary transition kernel.
• 157. Likelihood-free Methods Approximate Bayesian computation Alphabet soupABC-PRC weight (t) Probability ωi computed as (t) (t) (t) (t) ωi ∝ π(θi )Lt−1 (θ |θi ){π(θ )Kt (θi |θ )}−1 , where Lt−1 is an arbitrary transition kernel. In case Lt−1 (θ |θ) = Kt (θ|θ ) , all weights are equal under a uniform prior.
• 158. Likelihood-free Methods Approximate Bayesian computation Alphabet soupABC-PRC weight (t) Probability ωi computed as (t) (t) (t) (t) ωi ∝ π(θi )Lt−1 (θ |θi ){π(θ )Kt (θi |θ )}−1 , where Lt−1 is an arbitrary transition kernel. In case Lt−1 (θ |θ) = Kt (θ|θ ) , all weights are equal under a uniform prior. Inspired from Del Moral et al. (2006), who use backward kernels Lt−1 in SMC to achieve unbiasedness
• 159. Likelihood-free Methods Approximate Bayesian computation Alphabet soupABC-PRC bias Lack of unbiasedness of the method
• 160. Likelihood-free Methods Approximate Bayesian computation Alphabet soupABC-PRC bias Lack of unbiasedness of the method Joint density of the accepted pair (θ(t−1) , θ(t) ) proportional to (t−1) (t) (t−1) (t) π(θ |y )Kt (θ |θ )f (y |θ ), For an arbitrary function h(θ), E [ωt h(θ(t) )] proportional to (t) π(θ (t) )Lt−1 (θ (t−1) |θ (t) ) (t−1) (t) (t−1) (t) (t−1) (t) h(θ ) π(θ |y )Kt (θ |θ )f (y |θ )dθ dθ π(θ (t−1) )Kt (θ (t) |θ (t−1) ) (t) π(θ (t) )Lt−1 (θ (t−1) |θ (t) ) (t−1) (t−1) ∝ h(θ ) π(θ )f (y |θ ) π(θ (t−1) )Kt (θ (t) |θ (t−1) ) (t) (t−1) (t) (t−1) (t) × Kt (θ |θ )f (y |θ )dθ dθ (t) (t) (t−1) (t) (t−1) (t−1) (t) ∝ h(θ )π(θ |y ) Lt−1 (θ |θ )f (y |θ )dθ dθ .
• 161. Likelihood-free Methods Approximate Bayesian computation Alphabet soupA mixture example (1) Toy model of Sisson et al. (2007): if θ ∼ U(−10, 10) , x|θ ∼ 0.5 N (θ, 1) + 0.5 N (θ, 1/100) , then the posterior distribution associated with y = 0 is the normal mixture θ|y = 0 ∼ 0.5 N (0, 1) + 0.5 N (0, 1/100) restricted to [−10, 10]. Furthermore, true target available as π(θ||x| < ) ∝ Φ( −θ)−Φ(− −θ)+Φ(10( −θ))−Φ(−10( +θ)) .
• 162. Likelihood-free Methods Approximate Bayesian computation Alphabet soup“Ugly, squalid graph...” 1.0 1.0 1.0 1.0 1.0 0.8 0.8 0.8 0.8 0.8 0.6 0.6 0.6 0.6 0.6 0.4 0.4 0.4 0.4 0.4 0.2 0.2 0.2 0.2 0.2 0.0 0.0 0.0 0.0 0.0 −3 −1 1 3 −3 −1 1 3 −3 −1 1 3 −3 −1 1 3 −3 −1 1 3 θ θ θ θ θ 1.0 1.0 1.0 1.0 1.0 0.8 0.8 0.8 0.8 0.8 0.6 0.6 0.6 0.6 0.6 0.4 0.4 0.4 0.4 0.4 0.2 0.2 0.2 0.2 0.2 0.0 0.0 0.0 0.0 0.0 −3 −1 1 3 −3 −1 1 3 −3 −1 1 3 −3 −1 1 3 −3 −1 1 3 θ θ θ θ θ Comparison of τ = 0.15 and τ = 1/0.15 in Kt
• 163. Likelihood-free Methods Approximate Bayesian computation Alphabet soupA PMC version Use of the same kernel idea as ABC-PRC but with IS correction ( check PMC ) Generate a sample at iteration t by N (t) (t−1) (t−1) πt (θ ˆ )∝ ωj Kt (θ(t) |θj ) j=1 modulo acceptance of the associated xt , and use an importance (t) weight associated with an accepted simulation θi (t) (t) (t) ωi ∝ π(θi ) πt (θi ) . ˆ c Still likelihood free [Beaumont et al., 2009]
• 164. Likelihood-free Methods Approximate Bayesian computation Alphabet soupOur ABC-PMC algorithm Given a decreasing sequence of approximation levels 1 ≥ ... ≥ T, 1. At iteration t = 1, For i = 1, ..., N (1) (1) Simulate θi ∼ π(θ) and x ∼ f (x|θi ) until (x, y ) < 1 (1) Set ωi = 1/N (1) Take τ 2 as twice the empirical variance of the θi ’s 2. At iteration 2 ≤ t ≤ T , For i = 1, ..., N, repeat (t−1) (t−1) Pick θi from the θj ’s with probabilities ωj (t) 2 (t) generate θi |θi ∼ N (θi , σt ) and x ∼ f (x|θi ) until (x, y ) < t (t) (t) N (t−1) −1 (t) (t−1) Set ωi ∝ π(θi )/ j=1 ωj ϕ σt θi − θj ) 2 (t) Take τt+1 as twice the weighted empirical variance of the θi ’s
• 165. Likelihood-free Methods Approximate Bayesian computation Alphabet soupSequential Monte Carlo SMC is a simulation technique to approximate a sequence of related probability distributions πn with π0 “easy” and πT as target. Iterated IS as PMC: particles moved from time n to time n via kernel Kn and use of a sequence of extended targets πn˜ n πn (z0:n ) = πn (zn ) ˜ Lj (zj+1 , zj ) j=0 where the Lj ’s are backward Markov kernels [check that πn (zn ) is a marginal] [Del Moral, Doucet & Jasra, Series B, 2006]
• 166. Likelihood-free Methods Approximate Bayesian computation Alphabet soupSequential Monte Carlo (2) Algorithm 3 SMC sampler (0) sample zi ∼ γ0 (x) (i = 1, . . . , N) (0) (0) (0) compute weights wi = π0 (zi )/γ0 (zi ) for t = 1 to N do if ESS(w (t−1) ) < NT then resample N particles z (t−1) and set weights to 1 end if (t−1) (t−1) generate zi ∼ Kt (zi , ·) and set weights to (t) (t) (t−1) (t) (t−1) πt (zi ))Lt−1 (zi ), zi )) wi = wi−1 (t−1) (t−1) (t) πt−1 (zi ))Kt (zi ), zi )) end for
• 167. Likelihood-free Methods Approximate Bayesian computation Alphabet soupABC-SMC [Del Moral, Doucet & Jasra, 2009] True derivation of an SMC-ABC algorithm Use of a kernel Kn associated with target π n and derivation of the backward kernel π n (z )Kn (z , z) Ln−1 (z, z ) = πn (z) Update of the weights M m m=1 IA n (xin ) win ∝ wi(n−1) M m (xi(n−1) ) m=1 IA n−1 m when xin ∼ K (xi(n−1) , ·)
• 168. Likelihood-free Methods Approximate Bayesian computation Alphabet soupABC-SMCM Modiﬁcation: Makes M repeated simulations of the pseudo-data z given the parameter, rather than using a single [M = 1] simulation, leading to weight that is proportional to the number of accepted zi s M 1 ω(θ) = Iρ(η(y),η(zi ))< M i=1 [limit in M means exact simulation from (tempered) target]
• 169. Likelihood-free Methods Approximate Bayesian computation Alphabet soupProperties of ABC-SMC The ABC-SMC method properly uses a backward kernel L(z, z ) to simplify the importance weight and to remove the dependence on the unknown likelihood from this weight. Update of importance weights is reduced to the ratio of the proportions of surviving particles Major assumption: the forward kernel K is supposed to be invariant against the true target [tempered version of the true posterior]
• 170. Likelihood-free Methods Approximate Bayesian computation Alphabet soupProperties of ABC-SMC The ABC-SMC method properly uses a backward kernel L(z, z ) to simplify the importance weight and to remove the dependence on the unknown likelihood from this weight. Update of importance weights is reduced to the ratio of the proportions of surviving particles Major assumption: the forward kernel K is supposed to be invariant against the true target [tempered version of the true posterior] Adaptivity in ABC-SMC algorithm only found in on-line construction of the thresholds t , slowly enough to keep a large number of accepted transitions
• 171. Likelihood-free Methods Approximate Bayesian computation Alphabet soupA mixture example (2) Recovery of the target, whether using a ﬁxed standard deviation of τ = 0.15 or τ = 1/0.15, or a sequence of adaptive τt ’s. 1.0 1.0 1.0 1.0 1.0 0.8 0.8 0.8 0.8 0.8 0.6 0.6 0.6 0.6 0.6 0.4 0.4 0.4 0.4 0.4 0.2 0.2 0.2 0.2 0.2 0.0 0.0 0.0 0.0 0.0 −3 −2 −1 0 1 2 3 −3 −2 −1 0 1 2 3 −3 −2 −1 0 1 2 3 −3 −2 −1 0 1 2 3 −3 −2 −1 0 1 2 3 θ θ θ θ θ
• 172. Likelihood-free Methods Approximate Bayesian computation Alphabet soupYet another ABC-PRC Another version of ABC-PRC called PRC-ABC with a proposal distribution Mt , S replications of the pseudo-data, and a PRC step: PRC-ABC Algorithm (t−1) (t−1) 1. Select θ at random from the θi ’s with probabilities ωi (t) iid (t−1) 2. Generate θi ∼ Kt , x1 , . . . , xS ∼ f (x|θi ), set (t) (t) ωi = Kt {η(xs ) − η(y)} Kt (θi ) , s (t) and accept with probability p (i) = ωi /ct,N (t) (t) 3. Set ωi ∝ ωi /p (i) (1 ≤ i ≤ N) [Peters, Sisson & Fan, Stat & Computing, 2012]
• 173. Likelihood-free Methods Approximate Bayesian computation Alphabet soupYet another ABC-PRC Another version of ABC-PRC called PRC-ABC with a proposal distribution Mt , S replications of the pseudo-data, and a PRC step: (t−1) (t−1) Use of a NP estimate Mt (θ) = ωi ψ(θ|θi (with a ﬁxed variance?!) S = 1 recommended for “greatest gains in sampler performance” ct,N upper quantile of non-zero weights at time t
• 174. Likelihood-free Methods Approximate Bayesian computation Alphabet soupWilkinson’s exact BC ABC approximation error (i.e. non-zero tolerance) replaced with exact simulation from a controlled approximation to the target, convolution of true posterior with kernel function π(θ)f (z|θ)K (y − z) π (θ, z|y) = , π(θ)f (z|θ)K (y − z)dzdθ with K kernel parameterised by bandwidth . [Wilkinson, 2008]
• 175. Likelihood-free Methods Approximate Bayesian computation Alphabet soupWilkinson’s exact BC ABC approximation error (i.e. non-zero tolerance) replaced with exact simulation from a controlled approximation to the target, convolution of true posterior with kernel function π(θ)f (z|θ)K (y − z) π (θ, z|y) = , π(θ)f (z|θ)K (y − z)dzdθ with K kernel parameterised by bandwidth . [Wilkinson, 2008] Theorem The ABC algorithm based on the assumption of a randomised observation y = y + ξ, ξ ∼ K , and an acceptance probability of ˜ K (y − z)/M gives draws from the posterior distribution π(θ|y).
• 176. Likelihood-free Methods Approximate Bayesian computation Alphabet soupHow exact a BC? “Using to represent measurement error is straightforward, whereas using to model the model discrepancy is harder to conceptualize and not as commonly used” [Richard Wilkinson, 2008]
• 177. Likelihood-free Methods Approximate Bayesian computation Alphabet soupHow exact a BC? Pros Pseudo-data from true model and observed data from noisy model Interesting perspective in that outcome is completely controlled Link with ABCµ and assuming y is observed with a measurement error with density K Relates to the theory of model approximation [Kennedy & O’Hagan, 2001] Cons Requires K to be bounded by M True approximation error never assessed Requires a modiﬁcation of the standard ABC algorithm
• 178. Likelihood-free Methods Approximate Bayesian computation Alphabet soupABC for HMMs Speciﬁc case of a hidden Markov model Xt+1 ∼ Qθ (Xt , ·) Yt+1 ∼ gθ (·|xt ) 0 where only y1:n is observed. [Dean, Singh, Jasra, & Peters, 2011]
• 179. Likelihood-free Methods Approximate Bayesian computation Alphabet soupABC for HMMs Speciﬁc case of a hidden Markov model Xt+1 ∼ Qθ (Xt , ·) Yt+1 ∼ gθ (·|xt ) 0 where only y1:n is observed. [Dean, Singh, Jasra, & Peters, 2011] Use of speciﬁc constraints, adapted to the Markov structure: 0 0 y1 ∈ B(y1 , ) × · · · × yn ∈ B(yn , )
• 180. Likelihood-free Methods Approximate Bayesian computation Alphabet soupABC-MLE for HMMs ABC-MLE deﬁned by ˆ 0 0 θn = arg max Pθ Y1 ∈ B(y1 , ), . . . , Yn ∈ B(yn , ) θ
• 181. Likelihood-free Methods Approximate Bayesian computation Alphabet soupABC-MLE for HMMs ABC-MLE deﬁned by ˆ 0 0 θn = arg max Pθ Y1 ∈ B(y1 , ), . . . , Yn ∈ B(yn , ) θ Exact MLE for the likelihood same basis as Wilkinson! 0 pθ (y1 , . . . , yn ) corresponding to the perturbed process (xt , yt + zt )1≤t≤n zt ∼ U(B(0, 1) [Dean, Singh, Jasra, & Peters, 2011]
• 182. Likelihood-free Methods Approximate Bayesian computation Alphabet soupABC-MLE is biased ABC-MLE is asymptotically (in n) biased with target l (θ) = Eθ∗ [log pθ (Y1 |Y−∞:0 )]
• 183. Likelihood-free Methods Approximate Bayesian computation Alphabet soupABC-MLE is biased ABC-MLE is asymptotically (in n) biased with target l (θ) = Eθ∗ [log pθ (Y1 |Y−∞:0 )] but ABD-MLE converges to true value in the sense l n (θn ) → l (θ) for all sequences (θn ) converging to θ and n
• 184. Likelihood-free Methods Approximate Bayesian computation Alphabet soupNoisy ABC-MLE Idea: Modify instead the data from the start 0 (y1 + ζ1 , . . . , yn + ζn ) [ see Fearnhead-Prangle ] noisy ABC-MLE estimate 0 0 arg max Pθ Y1 ∈ B(y1 + ζ1 , ), . . . , Yn ∈ B(yn + ζn , ) θ [Dean, Singh, Jasra, & Peters, 2011]
• 185. Likelihood-free Methods Approximate Bayesian computation Alphabet soupConsistent noisy ABC-MLE Degrading the data improves the estimation performances: Noisy ABC-MLE is asymptotically (in n) consistent under further assumptions, the noisy ABC-MLE is asymptotically normal increase in variance of order −2 likely degradation in precision or computing time due to the lack of summary statistic
• 186. Likelihood-free Methods Approximate Bayesian computation Alphabet soupSMC for ABC likelihood Algorithm 4 SMC ABC for HMMs Given θ for k = 1, . . . , n do 1 1 N N generate proposals (xk , yk ), . . . , (xk , yk ) from the model weigh each proposal with ωk l =I l B(yk + ζk , ) (yk ) 0 l renormalise the weights and sample the xk ’s accordingly end for approximate the likelihood by n N l ωk N k=1 l=1 [Jasra, Singh, Martin, & McCoy, 2010]
• 187. Likelihood-free Methods Approximate Bayesian computation Alphabet soupWhich summary? Fundamental diﬃculty of the choice of the summary statistic when there is no non-trivial suﬃcient statistics [except when done by the experimenters in the ﬁeld]
• 188. Likelihood-free Methods Approximate Bayesian computation Alphabet soupWhich summary? Fundamental diﬃculty of the choice of the summary statistic when there is no non-trivial suﬃcient statistics [except when done by the experimenters in the ﬁeld] Starting from a large collection of summary statistics is available, Joyce and Marjoram (2008) consider the sequential inclusion into the ABC target, with a stopping rule based on a likelihood ratio test.
• 189. Likelihood-free Methods Approximate Bayesian computation Alphabet soupWhich summary? Fundamental diﬃculty of the choice of the summary statistic when there is no non-trivial suﬃcient statistics [except when done by the experimenters in the ﬁeld] Starting from a large collection of summary statistics is available, Joyce and Marjoram (2008) consider the sequential inclusion into the ABC target, with a stopping rule based on a likelihood ratio test. Does not taking into account the sequential nature of the tests Depends on parameterisation Order of inclusion matters.
• 190. Likelihood-free Methods Approximate Bayesian computation Alphabet soupA connected Monte Carlo study of the pseudo-data per simulated parameter Repeating simulations does not improve approximation Tolerance level does not seem to be highly inﬂuential Choice of distance / summary statistics / calibration factors are paramount to successful approximation ABC-SMC outperforms ABC-MCMC [Mckinley, Cook, Deardon, 2009]
• 191. Likelihood-free Methods Approximate Bayesian computation Automated summary statistic selectionSemi-automatic ABC Fearnhead and Prangle (2010) study ABC and the selection of the summary statistic in close proximity to Wilkinson’s proposal ABC then considered from a purely inferential viewpoint and calibrated for estimation purposes. Use of a randomised (or ‘noisy’) version of the summary statistics η (y) = η(y) + τ ˜ Derivation of a well-calibrated version of ABC, i.e. an algorithm that gives proper predictions for the distribution associated with this randomised summary statistic.
• 192. Likelihood-free Methods Approximate Bayesian computation Automated summary statistic selectionSemi-automatic ABC Fearnhead and Prangle (2010) study ABC and the selection of the summary statistic in close proximity to Wilkinson’s proposal ABC then considered from a purely inferential viewpoint and calibrated for estimation purposes. Use of a randomised (or ‘noisy’) version of the summary statistics η (y) = η(y) + τ ˜ Derivation of a well-calibrated version of ABC, i.e. an algorithm that gives proper predictions for the distribution associated with this randomised summary statistic. [calibration constraint: ABC approximation with same posterior mean as the true randomised posterior.]
• 193. Likelihood-free Methods Approximate Bayesian computation Automated summary statistic selectionSummary statistics Optimality of the posterior expectation E[θ|y] of the parameter of interest as summary statistics η(y)!
• 194. Likelihood-free Methods Approximate Bayesian computation Automated summary statistic selectionSummary statistics Optimality of the posterior expectation E[θ|y] of the parameter of interest as summary statistics η(y)! Use of the standard quadratic loss function (θ − θ0 )T A(θ − θ0 ) .
• 195. Likelihood-free Methods Approximate Bayesian computation Automated summary statistic selectionDetails on Fearnhead and Prangle (F&P) ABC Use of a summary statistic S(·), an importance proposal g (·), a kernel K (·) ≤ 1 and a bandwidth h > 0 such that (θ, ysim ) ∼ g (θ)f (ysim |θ) is accepted with probability (hence the bound) K [{S(ysim ) − sobs }/h] and the corresponding importance weight deﬁned by π(θ) g (θ) [Fearnhead & Prangle, 2012]
• 196. Likelihood-free Methods Approximate Bayesian computation Automated summary statistic selectionErrors, errors, and errors Three levels of approximation π(θ|yobs ) by π(θ|sobs ) loss of information [ignored] π(θ|sobs ) by π(s)K [{s − sobs }/h]π(θ|s) ds πABC (θ|sobs ) = π(s)K [{s − sobs }/h] ds noisy observations πABC (θ|sobs ) by importance Monte Carlo based on N simulations, represented by var(a(θ)|sobs )/Nacc [expected number of acceptances] [M. Twain/B. Disraeli]
• 197. Likelihood-free Methods Approximate Bayesian computation Automated summary statistic selectionAverage acceptance asymptotics For the average acceptance probability/approximate likelihood p(θ|sobs ) = f (ysim |θ) K [{S(ysim ) − sobs }/h] dysim , overall acceptance probability p(sobs ) = p(θ|sobs ) π(θ) dθ = π(sobs )hd + o(hd ) [F&P, Lemma 1]
• 198. Likelihood-free Methods Approximate Bayesian computation Automated summary statistic selectionOptimal importance proposal Best choice of importance proposal in terms of eﬀective sample size g (θ|sobs ) ∝ π(θ)p(θ|sobs )1/2 [Not particularly useful in practice]
• 199. Likelihood-free Methods Approximate Bayesian computation Automated summary statistic selectionOptimal importance proposal Best choice of importance proposal in terms of eﬀective sample size g (θ|sobs ) ∝ π(θ)p(θ|sobs )1/2 [Not particularly useful in practice] note that p(θ|sobs ) is an approximate likelihood reminiscent of parallel tempering could be approximately achieved by attrition of half of the data
• 200. Likelihood-free Methods Approximate Bayesian computation Automated summary statistic selectionCalibration of h “This result gives insight into how S(·) and h aﬀect the Monte Carlo error. To minimize Monte Carlo error, we need hd to be not too small. Thus ideally we want S(·) to be a low dimensional summary of the data that is suﬃciently informative about θ that π(θ|sobs ) is close, in some sense, to π(θ|yobs )” (F&P, p.5) turns h into an absolute value while it should be context-dependent and user-calibrated only addresses one term in the approximation error and acceptance probability (“curse of dimensionality”) h large prevents πABC (θ|sobs ) to be close to π(θ|sobs ) d small prevents π(θ|sobs ) to be close to π(θ|yobs ) (“curse of [dis]information”)
• 201. Likelihood-free Methods Approximate Bayesian computation Automated summary statistic selectionCalibrating ABC “If πABC is calibrated, then this means that probability statements that are derived from it are appropriate, and in particular that we can use πABC to quantify uncertainty in estimates” (F&P, p.5)
• 202. Likelihood-free Methods Approximate Bayesian computation Automated summary statistic selectionCalibrating ABC “If πABC is calibrated, then this means that probability statements that are derived from it are appropriate, and in particular that we can use πABC to quantify uncertainty in estimates” (F&P, p.5) Deﬁnition For 0 < q < 1 and subset A, event Eq (A) made of sobs such that PrABC (θ ∈ A|sobs ) = q. Then ABC is calibrated if Pr(θ ∈ A|Eq (A)) = q unclear meaning of conditioning on Eq (A)
• 203. Likelihood-free Methods Approximate Bayesian computation Automated summary statistic selectionCalibrated ABC Theorem (F&P) Noisy ABC, where sobs = S(yobs ) + h , ∼ K (·) is calibrated [Wilkinson, 2008] no condition on h!!
• 204. Likelihood-free Methods Approximate Bayesian computation Automated summary statistic selectionCalibrated ABC Consequence: when h = ∞ Theorem (F&P) The prior distribution is always calibrated is this a relevant property then?
• 205. Likelihood-free Methods Approximate Bayesian computation Automated summary statistic selectionMore about calibrated ABC “Calibration is not universally accepted by Bayesians. It is even more questionable here as we care how statements we make relate to the real world, not to a mathematically deﬁned posterior.” R. Wilkinson Same reluctance about the prior being calibrated Property depending on prior, likelihood, and summary Calibration is a frequentist property (almost a p-value!) More sensible to account for the simulator’s imperfections than using noisy-ABC against a meaningless based measure [Wilkinson, 2012]
• 206. Likelihood-free Methods Approximate Bayesian computation Automated summary statistic selectionConverging ABC Theorem (F&P) For noisy ABC, the expected noisy-ABC log-likelihood, E {log[p(θ|sobs )]} = log[p(θ|S(yobs ) + )]π(yobs |θ0 )K ( )dyobs d , has its maximum at θ = θ0 . True for any choice of summary statistic? even ancilary statistics?! [Imposes at least identiﬁability...] Relevant in asymptotia and not for the data
• 207. Likelihood-free Methods Approximate Bayesian computation Automated summary statistic selectionConverging ABC Corollary For noisy ABC, the ABC posterior converges onto a point mass on the true parameter value as m → ∞. For standard ABC, not always the case (unless h goes to zero). Strength of regularity conditions (c1) and (c2) in Bernardo & Smith, 1994? [out-of-reach constraints on likelihood and posterior] Again, there must be conditions imposed upon summary statistics...
• 208. Likelihood-free Methods Approximate Bayesian computation Automated summary statistic selectionLoss motivated statistic Under quadratic loss function, Theorem (F&P) ˆ (i) The minimal posterior error E[L(θ, θ)|yobs ] occurs when ˆ = E(θ|yobs ) (!) θ (ii) When h → 0, EABC (θ|sobs ) converges to E(θ|yobs ) ˆ (iii) If S(yobs ) = E[θ|yobs ] then for θ = EABC [θ|sobs ] ˆ E[L(θ, θ)|yobs ] = trace(AΣ) + h2 xT AxK (x)dx + o(h2 ). measure-theoretic diﬃculties? dependence of sobs on h makes me uncomfortable inherent to noisy ABC Relevant for choice of K ?
• 209. Likelihood-free Methods Approximate Bayesian computation Automated summary statistic selectionOptimal summary statistic “We take a diﬀerent approach, and weaken the requirement for πABC to be a good approximation to π(θ|yobs ). We argue for πABC to be a good approximation solely in terms of the accuracy of certain estimates of the parameters.” (F&P, p.5) From this result, F&P derive their choice of summary statistic, S(y) = E(θ|y) [almost suﬃcient] suggest h = O(N −1/(2+d) ) and h = O(N −1/(4+d) ) as optimal bandwidths for noisy and standard ABC.
• 210. Likelihood-free Methods Approximate Bayesian computation Automated summary statistic selectionOptimal summary statistic “We take a diﬀerent approach, and weaken the requirement for πABC to be a good approximation to π(θ|yobs ). We argue for πABC to be a good approximation solely in terms of the accuracy of certain estimates of the parameters.” (F&P, p.5) From this result, F&P derive their choice of summary statistic, S(y) = E(θ|y) [wow! EABC [θ|S(yobs )] = E[θ|yobs ]] suggest h = O(N −1/(2+d) ) and h = O(N −1/(4+d) ) as optimal bandwidths for noisy and standard ABC.
• 211. Likelihood-free Methods Approximate Bayesian computation Automated summary statistic selectionCaveat Since E(θ|yobs ) is most usually unavailable, F&P suggest (i) use a pilot run of ABC to determine a region of non-negligible posterior mass; (ii) simulate sets of parameter values and data; (iii) use the simulated sets of parameter values and data to estimate the summary statistic; and (iv) run ABC with this choice of summary statistic.
• 212. Likelihood-free Methods Approximate Bayesian computation Automated summary statistic selectionCaveat Since E(θ|yobs ) is most usually unavailable, F&P suggest (i) use a pilot run of ABC to determine a region of non-negligible posterior mass; (ii) simulate sets of parameter values and data; (iii) use the simulated sets of parameter values and data to estimate the summary statistic; and (iv) run ABC with this choice of summary statistic. where is the assessment of the ﬁrst stage error?
• 213. Likelihood-free Methods Approximate Bayesian computation Automated summary statistic selectionApproximating the summary statistic As Beaumont et al. (2002) and Blum and Fran¸ois (2010), F&P c use a linear regression to approximate E(θ|yobs ): (i) θi = β0 + β (i) f (yobs ) + i with f made of standard transforms [Further justiﬁcations?]
• 214. Likelihood-free Methods Approximate Bayesian computation Automated summary statistic selection[my]questions about semi-automatic ABC dependence on h and S(·) in the early stage reduction of Bayesian inference to point estimation approximation error in step (i) not accounted for not parameterisation invariant practice shows that proper approximation to genuine posterior distributions stems from using a (much) larger number of summary statistics than the dimension of the parameter the validity of the approximation to the optimal summary statistic depends on the quality of the pilot run important inferential issues like model choice are not covered by this approach. [Robert, 2012]
• 215. Likelihood-free Methods Approximate Bayesian computation Automated summary statistic selectionMore about semi-automatic ABC [End of section derived from comments on Read Paper, to appear] “The apparently arbitrary nature of the choice of summary statistics has always been perceived as the Achilles heel of ABC.” M. Beaumont
• 216. Likelihood-free Methods Approximate Bayesian computation Automated summary statistic selectionMore about semi-automatic ABC [End of section derived from comments on Read Paper, to appear] “The apparently arbitrary nature of the choice of summary statistics has always been perceived as the Achilles heel of ABC.” M. Beaumont “Curse of dimensionality” linked with the increase of the dimension of the summary statistic Connection with principal component analysis [Itan et al., 2010] Connection with partial least squares [Wegman et al., 2009] Beaumont et al. (2002) postprocessed output is used as input by F&P to run a second ABC
• 217. Likelihood-free Methods Approximate Bayesian computation Automated summary statistic selectionWood’s alternative Instead of a non-parametric kernel approximation to the likelihood 1 K {η(yr ) − η(yobs )} R r Wood (2010) suggests a normal approximation η(y(θ)) ∼ Nd (µθ , Σθ ) whose parameters can be approximated based on the R simulations (for each value of θ).
• 218. Likelihood-free Methods Approximate Bayesian computation Automated summary statistic selectionWood’s alternative Instead of a non-parametric kernel approximation to the likelihood 1 K {η(yr ) − η(yobs )} R r Wood (2010) suggests a normal approximation η(y(θ)) ∼ Nd (µθ , Σθ ) whose parameters can be approximated based on the R simulations (for each value of θ). Parametric versus non-parametric rate [Uh?!] Automatic weighting of components of η(·) through Σθ Dependence on normality assumption (pseudo-likelihood?) [Cornebise, Girolami & Kosmidis, 2012]
• 219. Likelihood-free Methods Approximate Bayesian computation Automated summary statistic selectionReinterpretation and extensions Reinterpretation of ABC output as joint simulation from π (x, y |θ) = f (x|θ)¯Y |X (y |x) ¯ π where πY |X (y |x) = K (y − x) ¯
• 220. Likelihood-free Methods Approximate Bayesian computation Automated summary statistic selectionReinterpretation and extensions Reinterpretation of ABC output as joint simulation from π (x, y |θ) = f (x|θ)¯Y |X (y |x) ¯ π where πY |X (y |x) = K (y − x) ¯ Reinterpretation of noisy ABC if y |y obs ∼ πY |X (·|y obs ), then marginally ¯ ¯ y ∼ πY |θ (·|θ0 ) ¯ ¯ c Explain for the consistency of Bayesian inference based on y and π ¯ ¯ [Lee, Andrieu & Doucet, 2012]
• 221. Likelihood-free Methods Approximate Bayesian computation Automated summary statistic selectionReinterpretation and extensions Also ﬁts within the pseudo-marginal of Andrieu and Roberts (2009) Algorithm 5 Rejuvenating GIMH-ABC At time t, given θt = θ: Sample θ ∼ g (·|θ). ⊗N Sample z1:N ∼ πY |Θ (·|θ ). ⊗(N−1) Sample x1:N−1 ∼ πY |Θ (·|θ). With probability N π(θ )g (θ|θ ) i=1 IBh (zi ) (y ) min 1, × N−1 π(θ)g (θ |θ) 1+ i=1 IBh (xi ) (y ) set θt+1 = θ . Otherwise set θt+1 = θ.
• 222. Likelihood-free Methods Approximate Bayesian computation Automated summary statistic selectionReinterpretation and extensions Also ﬁts within the pseudo-marginal of Andrieu and Roberts (2009) Algorithm 6 One-hit MCMC-ABC At time t, with θt = θ: Sample θ ∼ g (·|θ). With probability 1 − min 1, π(θ )g(θ |θ)) , set θt+1 = θ and go to π(θ)g (θ|θ time t + 1. Sample zi ∼ πY |Θ (·|θ ) and xi ∼ πY |Θ (·|θ) for i = 1, . . . until y ∈ Bh (zi ) and/or y ∈ Bh (xi ). If y ∈ Bh (zi ) set θt+1 = θ and go to time t + 1. If y ∈ Bh (xi ) set θt+1 = θ and go to time t + 1. [Lee, Andrieu & Doucet, 2012]
• 223. Likelihood-free Methods Approximate Bayesian computation Automated summary statistic selectionReinterpretation and extensions Also ﬁts within the pseudo-marginal of Andrieu and Roberts (2009) Validation c The invariant distribution of θ is πθ|Y (·|y ) ¯ for both algorithms. [Lee, Andrieu & Doucet, 2012]
• 224. Likelihood-free Methods Approximate Bayesian computation Automated summary statistic selectionABC for Markov chains Rewriting the posterior as π(θ)1−n (θ)π(θ|x1 ) π(θ|xt−1 , xt ) where π(θ|xt−1 , xt ) ∝ f (xt |xt−1 , θ)π(θ)
• 225. Likelihood-free Methods Approximate Bayesian computation Automated summary statistic selectionABC for Markov chains Rewriting the posterior as π(θ)1−n (θ)π(θ|x1 ) π(θ|xt−1 , xt ) where π(θ|xt−1 , xt ) ∝ f (xt |xt−1 , θ)π(θ) Allows for a stepwise ABC, replacing each π(θ|xt−1 , xt ) by an ABC approximation Similarity with F&P’s multiple sources of data (and also with Dean et al., 2011 ) [White et al., 2010, 2012]
• 226. Likelihood-free Methods Approximate Bayesian computation Automated summary statistic selectionBack to suﬃciency Diﬀerence between regular suﬃciency, equivalent to π(θ|y) = π(θ|η(y)) for all θ’s and all priors π, and
• 227. Likelihood-free Methods Approximate Bayesian computation Automated summary statistic selectionBack to suﬃciency Diﬀerence between regular suﬃciency, equivalent to π(θ|y) = π(θ|η(y)) for all θ’s and all priors π, and marginal suﬃciency, stated as π(µ(θ)|y) = π(µ(θ)|η(y)) for all θ’s, the given prior π and a subvector µ(θ) [Basu, 1977]
• 228. Likelihood-free Methods Approximate Bayesian computation Automated summary statistic selectionBack to suﬃciency Diﬀerence between regular suﬃciency, equivalent to π(θ|y) = π(θ|η(y)) for all θ’s and all priors π, and marginal suﬃciency, stated as π(µ(θ)|y) = π(µ(θ)|η(y)) for all θ’s, the given prior π and a subvector µ(θ) [Basu, 1977] Relates to F & P’s main result, but could event be reduced to conditional suﬃciency π(µ(θ)|yobs ) = π(µ(θ)|η(yobs )) (if feasible at all...) [Dawson, 2012]
• 229. Likelihood-free Methods Approximate Bayesian computation Automated summary statistic selectionABC and BCa’s Arguments in favour of a comparison of ABC with bootstrap in the event π(θ) is not deﬁned from prior information but as conventional non-informative prior. [Efron, 2012]
• 230. Likelihood-free Methods Approximate Bayesian computation Automated summary statistic selectionPredictive performances Instead of posterior means, other aspects of posterior to explore. E.g., look at minimising loss of information p(θ, y) p(θ, η(y)) p(θ, y) log dθdy − p(θ, η(y)) log dθdη(y) p(θ)p(y) p(θ)p(η(y)) for selection of summary statistics. [Filippi, Barnes, & Stumpf, 2012]
• 231. Likelihood-free Methods Approximate Bayesian computation Automated summary statistic selectionAuxiliary variables Auxiliary variable method avoids computations of untractable constant in likelihood ˜ f (y|θ) = Zθ f (y|θ)
• 232. Likelihood-free Methods Approximate Bayesian computation Automated summary statistic selectionAuxiliary variables Auxiliary variable method avoids computations of untractable constant in likelihood ˜ f (y|θ) = Zθ f (y|θ) Introduce pseudo-data z with artiﬁcial target g (z|θ, y) Generate θ ∼ K (θ, θ ) and z ∼ f (z|θ ) Accept with probability π(θ )f (y|θ )g (z |θ , y) K (θ , θ)f (z|θ) ∧1 π(θ)f (y|θ)g (z|θ, y) K (θ, θ )f (z |θ ) [Møller, Pettitt, Berthelsen, & Reeves, 2006]
• 233. Likelihood-free Methods Approximate Bayesian computation Automated summary statistic selectionAuxiliary variables Auxiliary variable method avoids computations of untractable constant in likelihood ˜ f (y|θ) = Zθ f (y|θ) Introduce pseudo-data z with artiﬁcial target g (z|θ, y) Generate θ ∼ K (θ, θ ) and z ∼ f (z|θ ) Accept with probability ˜ ˜ π(θ )f (y|θ )g (z |θ , y) K (θ , θ)f (z|θ) ∧1 ˜ ˜ π(θ)f (y|θ)g (z|θ, y) K (θ, θ )f (z |θ ) [Møller, Pettitt, Berthelsen, & Reeves, 2006]
• 234. Likelihood-free Methods Approximate Bayesian computation Automated summary statistic selectionAuxiliary variables Auxiliary variable method avoids computations of untractable constant in likelihood ˜ f (y|θ) = Zθ f (y|θ) Introduce pseudo-data z with artiﬁcial target g (z|θ, y) Generate θ ∼ K (θ, θ ) and z ∼ f (z|θ ) For Gibbs random ﬁelds , existence of a genuine suﬃcient statistic η(y). [Møller, Pettitt, Berthelsen, & Reeves, 2006]
• 235. Likelihood-free Methods Approximate Bayesian computation Automated summary statistic selectionAuxiliary variables and ABC Special case of ABC when g (z|θ, y) = K (η(bz) − η(y)) ˜ ˜ ˜ ˜ f (y|θ )f (z|θ)/f (y|θ)f (z |θ ) replaced by one [or not?!]
• 236. Likelihood-free Methods Approximate Bayesian computation Automated summary statistic selectionAuxiliary variables and ABC Special case of ABC when g (z|θ, y) = K (η(bz) − η(y)) ˜ ˜ ˜ ˜ f (y|θ )f (z|θ)/f (y|θ)f (z |θ ) replaced by one [or not?!] Consequences likelihood-free (ABC) versus constant-free (AVM) in ABC, K (·) should be allowed to depend on θ for Gibbs random ﬁelds, the auxiliary approach should be prefered to ABC [Møller, 2012]
• 237. Likelihood-free Methods Approximate Bayesian computation Automated summary statistic selectionABC and BIC Idea of applying BIC during the local regression : Run regular ABC Select summary statistics during local regression Recycle the prior simulation sample (reference table) with those summary statistics Rerun the corresponding local regression (low cost) [Pudlo & Sedki, 2012]
• 238. Likelihood-free Methods Approximate Bayesian computation Automated summary statistic selectionA Brave New World?!
• 239. Likelihood-free Methods ABC for model choiceABC for model choice Introduction The Metropolis-Hastings Algorithm simulation-based methods in Econometrics Genetics of ABC Approximate Bayesian computation ABC for model choice Model choice Gibbs random ﬁelds
• 240. Likelihood-free Methods ABC for model choice Model choiceBayesian model choice Several models M1 , M2 , . . . are considered simultaneously for a dataset y and the model index M is part of the inference. Use of a prior distribution. π(M = m), plus a prior distribution on the parameter conditional on the value m of the model index, πm (θ m ) Goal is to derive the posterior distribution of M, challenging computational target when models are complex.
• 241. Likelihood-free Methods ABC for model choice Model choiceGeneric ABC for model choice Algorithm 7 Likelihood-free model choice sampler (ABC-MC) for t = 1 to T do repeat Generate m from the prior π(M = m) Generate θ m from the prior πm (θ m ) Generate z from the model fm (z|θ m ) until ρ{η(z), η(y)} < Set m(t) = m and θ (t) = θ m end for
• 242. Likelihood-free Methods ABC for model choice Model choiceABC estimates Posterior probability π(M = m|y) approximated by the frequency of acceptances from model m T 1 Im(t) =m . T t=1 Issues with implementation: should tolerances be the same for all models? should summary statistics vary across models (incl. their dimension)? should the distance measure ρ vary as well?
• 243. Likelihood-free Methods ABC for model choice Model choiceABC estimates Posterior probability π(M = m|y) approximated by the frequency of acceptances from model m T 1 Im(t) =m . T t=1 Extension to a weighted polychotomous logistic regression estimate of π(M = m|y), with non-parametric kernel weights [Cornuet et al., DIYABC, 2009]
• 244. Likelihood-free Methods ABC for model choice Model choiceThe Great ABC controversy On-going controvery in phylogeographic genetics about the validity of using ABC for testing Against: Templeton, 2008, 2009, 2010a, 2010b, 2010c argues that nested hypotheses cannot have higher probabilities than nesting hypotheses (!)
• 245. Likelihood-free Methods ABC for model choice Model choiceThe Great ABC controversy On-going controvery in phylogeographic genetics about the validity of using ABC for testing Replies: Fagundes et al., 2008, Against: Templeton, 2008, Beaumont et al., 2010, Berger et 2009, 2010a, 2010b, 2010c al., 2010, Csill`ry et al., 2010 e argues that nested hypotheses point out that the criticisms are cannot have higher probabilities addressed at [Bayesian] than nesting hypotheses (!) model-based inference and have nothing to do with ABC...
• 246. Likelihood-free Methods ABC for model choice Gibbs random ﬁeldsGibbs random ﬁelds Gibbs distribution The rv y = (y1 , . . . , yn ) is a Gibbs random ﬁeld associated with the graph G if 1 f (y) = exp − Vc (yc ) , Z c∈C where Z is the normalising constant, C is the set of cliques of G and Vc is any function also called potential suﬃcient statistic U(y) = c∈C Vc (yc ) is the energy function
• 247. Likelihood-free Methods ABC for model choice Gibbs random ﬁeldsGibbs random ﬁelds Gibbs distribution The rv y = (y1 , . . . , yn ) is a Gibbs random ﬁeld associated with the graph G if 1 f (y) = exp − Vc (yc ) , Z c∈C where Z is the normalising constant, C is the set of cliques of G and Vc is any function also called potential suﬃcient statistic U(y) = c∈C Vc (yc ) is the energy function c Z is usually unavailable in closed form
• 248. Likelihood-free Methods ABC for model choice Gibbs random ﬁeldsPotts model Potts model Vc (y) is of the form Vc (y) = θS(y) = θ δyl =yi l∼i where l∼i denotes a neighbourhood structure
• 249. Likelihood-free Methods ABC for model choice Gibbs random ﬁeldsPotts model Potts model Vc (y) is of the form Vc (y) = θS(y) = θ δyl =yi l∼i where l∼i denotes a neighbourhood structure In most realistic settings, summation Zθ = exp{θ T S(x)} x∈X involves too many terms to be manageable and numerical approximations cannot always be trusted [Cucala, Marin, CPR & Titterington, 2009]
• 250. Likelihood-free Methods ABC for model choice Gibbs random ﬁeldsBayesian Model Choice Comparing a model with potential S0 taking values in Rp0 versus a model with potential S1 taking values in Rp1 can be done through the Bayes factor corresponding to the priors π0 and π1 on each parameter space exp{θ T S0 (x)}/Zθ 0 ,0 π0 (dθ 0 ) 0 Bm0 /m1 (x) = exp{θ T S1 (x)}/Zθ 1 ,1 π1 (dθ 1 ) 1
• 251. Likelihood-free Methods ABC for model choice Gibbs random ﬁeldsBayesian Model Choice Comparing a model with potential S0 taking values in Rp0 versus a model with potential S1 taking values in Rp1 can be done through the Bayes factor corresponding to the priors π0 and π1 on each parameter space exp{θ T S0 (x)}/Zθ 0 ,0 π0 (dθ 0 ) 0 Bm0 /m1 (x) = exp{θ T S1 (x)}/Zθ 1 ,1 π1 (dθ 1 ) 1 Use of Jeﬀreys’ scale to select most appropriate model
• 252. Likelihood-free Methods ABC for model choice Gibbs random ﬁeldsNeighbourhood relations Choice to be made between M neighbourhood relations m i ∼i (0 ≤ m ≤ M − 1) with Sm (x) = I{xi =xi } m i ∼i driven by the posterior probabilities of the models.
• 253. Likelihood-free Methods ABC for model choice Gibbs random ﬁeldsModel index Formalisation via a model index M that appears as a new parameter with prior distribution π(M = m) and π(θ|M = m) = πm (θm )
• 254. Likelihood-free Methods ABC for model choice Gibbs random ﬁeldsModel index Formalisation via a model index M that appears as a new parameter with prior distribution π(M = m) and π(θ|M = m) = πm (θm ) Computational target: P(M = m|x) ∝ fm (x|θm )πm (θm ) dθm π(M = m) , Θm
• 255. Likelihood-free Methods ABC for model choice Gibbs random ﬁeldsSuﬃcient statistics By deﬁnition, if S(x) suﬃcient statistic for the joint parameters (M, θ0 , . . . , θM−1 ), P(M = m|x) = P(M = m|S(x)) .
• 256. Likelihood-free Methods ABC for model choice Gibbs random ﬁeldsSuﬃcient statistics By deﬁnition, if S(x) suﬃcient statistic for the joint parameters (M, θ0 , . . . , θM−1 ), P(M = m|x) = P(M = m|S(x)) . For each model m, own suﬃcient statistic Sm (·) and S(·) = (S0 (·), . . . , SM−1 (·)) also suﬃcient.
• 257. Likelihood-free Methods ABC for model choice Gibbs random ﬁeldsSuﬃcient statistics in Gibbs random ﬁelds For Gibbs random ﬁelds, 1 2 x|M = m ∼ fm (x|θm ) = fm (x|S(x))fm (S(x)|θm ) 1 = f 2 (S(x)|θm ) n(S(x)) m where n(S(x)) = {˜ ∈ X : S(˜) = S(x)} x x c S(x) is therefore also suﬃcient for the joint parameters [Speciﬁc to Gibbs random ﬁelds!]
• 258. Likelihood-free Methods ABC for model choice Gibbs random ﬁeldsABC model choice Algorithm ABC-MC Generate m∗ from the prior π(M = m). ∗ Generate θm∗ from the prior πm∗ (·). Generate x ∗ from the model fm∗ (·|θm∗ ). ∗ Compute the distance ρ(S(x0 ), S(x∗ )). Accept (θm∗ , m∗ ) if ρ(S(x0 ), S(x∗ )) < . ∗ Note When = 0 the algorithm is exact
• 259. Likelihood-free Methods ABC for model choice Gibbs random ﬁeldsABC approximation to the Bayes factor Frequency ratio: ˆ P(M = m0 |x0 ) π(M = m1 ) BF m0 /m1 (x0 ) = × ˆ P(M = m1 |x0 ) π(M = m0 ) {mi∗ = m0 } π(M = m1 ) = × , {mi∗ = m1 } π(M = m0 )
• 260. Likelihood-free Methods ABC for model choice Gibbs random ﬁeldsABC approximation to the Bayes factor Frequency ratio: ˆ P(M = m0 |x0 ) π(M = m1 ) BF m0 /m1 (x0 ) = × ˆ P(M = m1 |x0 ) π(M = m0 ) {mi∗ = m0 } π(M = m1 ) = × , {mi∗ = m1 } π(M = m0 ) replaced with 1 + {mi∗ = m0 } π(M = m1 ) BF m0 /m1 (x0 ) = × 1 + {mi∗ = m1 } π(M = m0 ) to avoid indeterminacy (also Bayes estimate).
• 261. Likelihood-free Methods ABC for model choice Gibbs random ﬁeldsToy example iid Bernoulli model versus two-state ﬁrst-order Markov chain, i.e. n f0 (x|θ0 ) = exp θ0 I{xi =1} {1 + exp(θ0 )}n , i=1 versus n 1 f1 (x|θ1 ) = exp θ1 I{xi =xi−1 } {1 + exp(θ1 )}n−1 , 2 i=2 with priors θ0 ∼ U(−5, 5) and θ1 ∼ U(0, 6) (inspired by “phase transition” boundaries).
• 262. Likelihood-free Methods ABC for model choice Gibbs random ﬁeldsToy example (2) 10 5 5 BF01 BF01 0 ^ ^ 0 −5 −5 −40 −20 0 10 −10 −40 −20 0 10 BF01 BF01 (left) Comparison of the true BF m0 /m1 (x0 ) with BF m0 /m1 (x0 ) (in logs) over 2, 000 simulations and 4.106 proposals from the prior. (right) Same when using tolerance corresponding to the 1% quantile on the distances.
• 263. Likelihood-free Methods ABC for model choice Generic ABC model choiceBack to suﬃciency ‘Suﬃcient statistics for individual models are unlikely to be very informative for the model probability.’ [Scott Sisson, Jan. 31, 2011, X.’Og]
• 264. Likelihood-free Methods ABC for model choice Generic ABC model choiceBack to suﬃciency ‘Suﬃcient statistics for individual models are unlikely to be very informative for the model probability.’ [Scott Sisson, Jan. 31, 2011, X.’Og] If η1 (x) suﬃcient statistic for model m = 1 and parameter θ1 and η2 (x) suﬃcient statistic for model m = 2 and parameter θ2 , (η1 (x), η2 (x)) is not always suﬃcient for (m, θm )
• 265. Likelihood-free Methods ABC for model choice Generic ABC model choiceBack to suﬃciency ‘Suﬃcient statistics for individual models are unlikely to be very informative for the model probability.’ [Scott Sisson, Jan. 31, 2011, X.’Og] If η1 (x) suﬃcient statistic for model m = 1 and parameter θ1 and η2 (x) suﬃcient statistic for model m = 2 and parameter θ2 , (η1 (x), η2 (x)) is not always suﬃcient for (m, θm ) c Potential loss of information at the testing level
• 266. Likelihood-free Methods ABC for model choice Generic ABC model choiceLimiting behaviour of B12 (T → ∞) ABC approximation T t=1 Imt =1 Iρ{η(zt ),η(y)}≤ B12 (y) = T , t=1 Imt =2 Iρ{η(zt ),η(y)}≤ where the (mt , z t )’s are simulated from the (joint) prior
• 267. Likelihood-free Methods ABC for model choice Generic ABC model choiceLimiting behaviour of B12 (T → ∞) ABC approximation T t=1 Imt =1 Iρ{η(zt ),η(y)}≤ B12 (y) = T , t=1 Imt =2 Iρ{η(zt ),η(y)}≤ where the (mt , z t )’s are simulated from the (joint) prior As T go to inﬁnity, limit Iρ{η(z),η(y)}≤ π1 (θ 1 )f1 (z|θ 1 ) dz dθ 1 B12 (y) = Iρ{η(z),η(y)}≤ π2 (θ 2 )f2 (z|θ 2 ) dz dθ 2 Iρ{η,η(y)}≤ π1 (θ 1 )f1η (η|θ 1 ) dη dθ 1 = , Iρ{η,η(y)}≤ π2 (θ 2 )f2η (η|θ 2 ) dη dθ 2 where f1η (η|θ 1 ) and f2η (η|θ 2 ) distributions of η(z)
• 268. Likelihood-free Methods ABC for model choice Generic ABC model choiceLimiting behaviour of B12 ( → 0) When goes to zero, η π1 (θ 1 )f1η (η(y)|θ 1 ) dθ 1 B12 (y) = , π2 (θ 2 )f2η (η(y)|θ 2 ) dθ 2
• 269. Likelihood-free Methods ABC for model choice Generic ABC model choiceLimiting behaviour of B12 ( → 0) When goes to zero, η π1 (θ 1 )f1η (η(y)|θ 1 ) dθ 1 B12 (y) = , π2 (θ 2 )f2η (η(y)|θ 2 ) dθ 2 c Bayes factor based on the sole observation of η(y)
• 270. Likelihood-free Methods ABC for model choice Generic ABC model choiceLimiting behaviour of B12 (under suﬃciency) If η(y) suﬃcient statistic for both models, fi (y|θ i ) = gi (y)fi η (η(y)|θ i ) Thus Θ1 π(θ 1 )g1 (y)f1η (η(y)|θ 1 ) dθ 1 B12 (y) = Θ2 π(θ 2 )g2 (y)f2η (η(y)|θ 2 ) dθ 2 g1 (y) π1 (θ 1 )f1η (η(y)|θ 1 ) dθ 1 g1 (y) η = η = B (y) . g2 (y) π2 (θ 2 )f2 (η(y)|θ 2 ) dθ 2 g2 (y) 12 [Didelot, Everitt, Johansen & Lawson, 2011]
• 271. Likelihood-free Methods ABC for model choice Generic ABC model choiceLimiting behaviour of B12 (under suﬃciency) If η(y) suﬃcient statistic for both models, fi (y|θ i ) = gi (y)fi η (η(y)|θ i ) Thus Θ1 π(θ 1 )g1 (y)f1η (η(y)|θ 1 ) dθ 1 B12 (y) = Θ2 π(θ 2 )g2 (y)f2η (η(y)|θ 2 ) dθ 2 g1 (y) π1 (θ 1 )f1η (η(y)|θ 1 ) dθ 1 g1 (y) η = η = B (y) . g2 (y) π2 (θ 2 )f2 (η(y)|θ 2 ) dθ 2 g2 (y) 12 [Didelot, Everitt, Johansen & Lawson, 2011] c No discrepancy only when cross-model suﬃciency
• 272. Likelihood-free Methods ABC for model choice Generic ABC model choicePoisson/geometric example Sample x = (x1 , . . . , xn ) from either a Poisson P(λ) or from a geometric G(p) Then n S= yi = η(x) i=1 suﬃcient statistic for either model but not simultaneously Discrepancy ratio g1 (x) S!n−S / i yi ! = g2 (x) 1 n+S−1 S
• 273. Likelihood-free Methods ABC for model choice Generic ABC model choicePoisson/geometric discrepancy η Range of B12 (x) versus B12 (x) B12 (x): The values produced have nothing in common.
• 274. Likelihood-free Methods ABC for model choice Generic ABC model choiceFormal recovery Creating an encompassing exponential family T T f (x|θ1 , θ2 , α1 , α2 ) ∝ exp{θ1 η1 (x) + θ1 η1 (x) + α1 t1 (x) + α2 t2 (x)} leads to a suﬃcient statistic (η1 (x), η2 (x), t1 (x), t2 (x)) [Didelot, Everitt, Johansen & Lawson, 2011]
• 275. Likelihood-free Methods ABC for model choice Generic ABC model choiceFormal recovery Creating an encompassing exponential family T T f (x|θ1 , θ2 , α1 , α2 ) ∝ exp{θ1 η1 (x) + θ1 η1 (x) + α1 t1 (x) + α2 t2 (x)} leads to a suﬃcient statistic (η1 (x), η2 (x), t1 (x), t2 (x)) [Didelot, Everitt, Johansen & Lawson, 2011] In the Poisson/geometric case, if i xi ! is added to S, no discrepancy
• 276. Likelihood-free Methods ABC for model choice Generic ABC model choiceFormal recovery Creating an encompassing exponential family T T f (x|θ1 , θ2 , α1 , α2 ) ∝ exp{θ1 η1 (x) + θ1 η1 (x) + α1 t1 (x) + α2 t2 (x)} leads to a suﬃcient statistic (η1 (x), η2 (x), t1 (x), t2 (x)) [Didelot, Everitt, Johansen & Lawson, 2011] Only applies in genuine suﬃciency settings... c Inability to evaluate loss brought by summary statistics
• 277. Likelihood-free Methods ABC for model choice Generic ABC model choiceMeaning of the ABC-Bayes factor ‘This is also why focus on model discrimination typically (...) proceeds by (...) accepting that the Bayes Factor that one obtains is only derived from the summary statistics and may in no way correspond to that of the full model.’ [Scott Sisson, Jan. 31, 2011, X.’Og]
• 278. Likelihood-free Methods ABC for model choice Generic ABC model choiceMeaning of the ABC-Bayes factor ‘This is also why focus on model discrimination typically (...) proceeds by (...) accepting that the Bayes Factor that one obtains is only derived from the summary statistics and may in no way correspond to that of the full model.’ [Scott Sisson, Jan. 31, 2011, X.’Og] In the Poisson/geometric case, if E[yi ] = θ0 > 0, η (θ0 + 1)2 −θ0 lim B12 (y) = e n→∞ θ0
• 279. Likelihood-free Methods ABC for model choice Generic ABC model choiceMA(q) divergence 0.7 0.6 0.6 0.6 0.6 0.5 0.5 0.5 0.5 0.4 0.4 0.4 0.4 0.3 0.3 0.3 0.3 0.2 0.2 0.2 0.2 0.1 0.1 0.1 0.1 0.0 0.0 0.0 0.0 1 2 1 2 1 2 1 2 Evolution [against ] of ABC Bayes factor, in terms of frequencies of visits to models MA(1) (left) and MA(2) (right) when equal to 10, 1, .1, .01% quantiles on insuﬃcient autocovariance distances. Sample of 50 points from a MA(2) with θ1 = 0.6, θ2 = 0.2. True Bayes factor equal to 17.71.
• 280. Likelihood-free Methods ABC for model choice Generic ABC model choiceMA(q) divergence 0.8 0.6 0.6 0.6 0.6 0.4 0.4 0.4 0.4 0.2 0.2 0.2 0.2 0.0 0.0 0.0 0.0 1 2 1 2 1 2 1 2 Evolution [against ] of ABC Bayes factor, in terms of frequencies of visits to models MA(1) (left) and MA(2) (right) when equal to 10, 1, .1, .01% quantiles on insuﬃcient autocovariance distances. Sample of 50 points from a MA(1) model with θ1 = 0.6. True Bayes factor B21 equal to .004.
• 281. Likelihood-free Methods ABC for model choice Generic ABC model choiceFurther comments ‘There should be the possibility that for the same model, but diﬀerent (non-minimal) [summary] statistics (so ∗ diﬀerent η’s: η1 and η1 ) the ratio of evidences may no longer be equal to one.’ [Michael Stumpf, Jan. 28, 2011, ’Og] Using diﬀerent summary statistics [on diﬀerent models] may indicate the loss of information brought by each set but agreement does not lead to trustworthy approximations.
• 282. Likelihood-free Methods ABC for model choice Generic ABC model choiceA population genetics evaluation Population genetics example with 3 populations 2 scenari 15 individuals 5 loci single mutation parameter
• 283. Likelihood-free Methods ABC for model choice Generic ABC model choiceA population genetics evaluation Population genetics example with 3 populations 2 scenari 15 individuals 5 loci single mutation parameter 24 summary statistics 2 million ABC proposal importance [tree] sampling alternative
• 284. Likelihood-free Methods ABC for model choice Generic ABC model choiceStability of importance sampling 1.0 1.0 1.0 1.0 1.0 q q 0.8 0.8 0.8 0.8 0.8 0.6 0.6 0.6 0.6 0.6 q 0.4 0.4 0.4 0.4 0.4 0.2 0.2 0.2 0.2 0.2 0.0 0.0 0.0 0.0 0.0
• 285. Likelihood-free Methods ABC for model choice Generic ABC model choiceComparison with ABC Use of 24 summary statistics and DIY-ABC logistic correction 1.0 q q q q q q q q q q q q q q q q qq q q 0.8 q q q q q qq q q q q q q qq q ABC estimates of P(M=1|y) q q q q q q qq qq q qq q q q q q 0.6 q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q qq 0.4 qq q q q q q q q q q 0.2 q q 0.0 0.0 0.2 0.4 0.6 0.8 1.0 Importance Sampling estimates of P(M=1|y)
• 286. Likelihood-free Methods ABC for model choice Generic ABC model choiceComparison with ABC Use of 15 summary statistics and DIY-ABC logistic correction 1.0 q q q q qq q qq q qq q qqq qq q qq q q q q qq q q q q q q qq q q q q q 0.8 q q q q qq q q q q q q q q q ABC estimates of P(M=1|y) q q q q q q q q q q q q q q 0.6 q q q qq q q q q q q qq 0.4 q q q q q q q q q q q 0.2 q q q q q q q 0.0 0.0 0.2 0.4 0.6 0.8 1.0 Importance Sampling estimates of P(M=1|y)
• 287. Likelihood-free Methods ABC for model choice Generic ABC model choiceComparison with ABC Use of 24 summary statistics and DIY-ABC logistic correction 1.0 q q q q q q qq q q q qq q q q q q q q q q q q 0.8 qq q q q q q q q q q q q qq qq ABC estimates of P(M=1|y) q q q q q 0.6 q q q q q q q q q q q qq q q q q q q q q q q q q q q 0.4 q q q q q q q q q q q q q q q q q q q q 0.2 q q q q q q 0.0 0.0 0.2 0.4 0.6 0.8 1.0 Importance Sampling estimates of P(M=1|y)
• 288. Likelihood-free Methods ABC for model choice Generic ABC model choiceThe only safe cases??? Besides speciﬁc models like Gibbs random ﬁelds, using distances over the data itself escapes the discrepancy... [Toni & Stumpf, 2010; Sousa & al., 2009]
• 289. Likelihood-free Methods ABC for model choice Generic ABC model choiceThe only safe cases??? Besides speciﬁc models like Gibbs random ﬁelds, using distances over the data itself escapes the discrepancy... [Toni & Stumpf, 2010; Sousa & al., 2009] ...and so does the use of more informal model ﬁtting measures [Ratmann & al., 2009]
• 290. Likelihood-free Methods ABC model choice consistencyABC model choice consistency Introduction The Metropolis-Hastings Algorithm simulation-based methods in Econometrics Genetics of ABC Approximate Bayesian computation ABC for model choice ABC model choice consistency
• 291. Likelihood-free Methods ABC model choice consistency Formalised frameworkThe starting point Central question to the validation of ABC for model choice: When is a Bayes factor based on an insuﬃcient statistic T(y) consistent?
• 292. Likelihood-free Methods ABC model choice consistency Formalised frameworkThe starting point Central question to the validation of ABC for model choice: When is a Bayes factor based on an insuﬃcient statistic T(y) consistent? T Note: c drawn on T(y) through B12 (y) necessarily diﬀers from c drawn on y through B12 (y)
• 293. Likelihood-free Methods ABC model choice consistency Formalised frameworkA benchmark if toy example Comparison suggested by referee of PNAS paper [thanks]: [X, Cornuet, Marin, & Pillai, Aug. 2011] Model M1 : y ∼ N (θ1 , 1) opposed to model M2 : √ y ∼ L(θ2 , 1/ √ Laplace distribution with mean θ2 and scale 2), parameter 1/ 2 (variance one).
• 294. Likelihood-free Methods ABC model choice consistency Formalised frameworkA benchmark if toy example Comparison suggested by referee of PNAS paper [thanks]: [X, Cornuet, Marin, & Pillai, Aug. 2011] Model M1 : y ∼ N (θ1 , 1) opposed to model M2 : √ y ∼ L(θ2 , 1/ √ Laplace distribution with mean θ2 and scale 2), parameter 1/ 2 (variance one). Four possible statistics 1. sample mean y (suﬃcient for M1 if not M2 ); 2. sample median med(y) (insuﬃcient); 3. sample variance var(y) (ancillary); 4. median absolute deviation mad(y) = med(y − med(y));
• 295. Likelihood-free Methods ABC model choice consistency Formalised frameworkA benchmark if toy example Comparison suggested by referee of PNAS paper [thanks]: [X, Cornuet, Marin, & Pillai, Aug. 2011] Model M1 : y ∼ N (θ1 , 1) opposed to model M2 : √ y ∼ L(θ2 , 1/ √ Laplace distribution with mean θ2 and scale 2), parameter 1/ 2 (variance one). 6 5 4 Density 3 2 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 posterior probability
• 296. Likelihood-free Methods ABC model choice consistency Formalised frameworkA benchmark if toy example Comparison suggested by referee of PNAS paper [thanks]: [X, Cornuet, Marin, & Pillai, Aug. 2011] Model M1 : y ∼ N (θ1 , 1) opposed to model M2 : √ y ∼ L(θ2 , 1/ √ Laplace distribution with mean θ2 and scale 2), parameter 1/ 2 (variance one). n=100 0.7 0.6 0.5 0.4 q 0.3 q q q 0.2 0.1 q q q q q 0.0 q q Gauss Laplace
• 297. Likelihood-free Methods ABC model choice consistency Formalised frameworkA benchmark if toy example Comparison suggested by referee of PNAS paper [thanks]: [X, Cornuet, Marin, & Pillai, Aug. 2011] Model M1 : y ∼ N (θ1 , 1) opposed to model M2 : √ y ∼ L(θ2 , 1/ √ Laplace distribution with mean θ2 and scale 2), parameter 1/ 2 (variance one). 6 3 5 4 Density Density 2 3 2 1 1 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.0 0.2 0.4 0.6 0.8 1.0 posterior probability probability
• 298. Likelihood-free Methods ABC model choice consistency Formalised frameworkA benchmark if toy example Comparison suggested by referee of PNAS paper [thanks]: [X, Cornuet, Marin, & Pillai, Aug. 2011] Model M1 : y ∼ N (θ1 , 1) opposed to model M2 : √ y ∼ L(θ2 , 1/ √ Laplace distribution with mean θ2 and scale 2), parameter 1/ 2 (variance one). n=100 n=100 1.0 0.7 q q 0.6 q 0.8 q q 0.5 q q q 0.6 q 0.4 q q q q 0.3 q q 0.4 q q 0.2 0.2 q 0.1 q q q q q q q 0.0 0.0 q q Gauss Laplace Gauss Laplace
• 299. Likelihood-free Methods ABC model choice consistency Formalised frameworkFramework Starting from sample y = (y1 , . . . , yn ) the observed sample, not necessarily iid with true distribution y ∼ Pn Summary statistics T(y) = Tn = (T1 (y), T2 (y), · · · , Td (y)) ∈ Rd with true distribution Tn ∼ Gn .
• 300. Likelihood-free Methods ABC model choice consistency Formalised frameworkFramework c Comparison of – under M1 , y ∼ F1,n (·|θ1 ) where θ1 ∈ Θ1 ⊂ Rp1 – under M2 , y ∼ F2,n (·|θ2 ) where θ2 ∈ Θ2 ⊂ Rp2 turned into – under M1 , T(y) ∼ G1,n (·|θ1 ), and θ1 |T(y) ∼ π1 (·|Tn ) – under M2 , T(y) ∼ G2,n (·|θ2 ), and θ2 |T(y) ∼ π2 (·|Tn )
• 301. Likelihood-free Methods ABC model choice consistency Consistency resultsAssumptions A collection of asymptotic “standard” assumptions: [A1] There exist a sequence {vn } converging to +∞, an a.c. distribution Q with continuous bounded density q(·), a symmetric, d × d positive deﬁnite matrix V0 and a vector µ0 ∈ Rd such that −1/2 n→∞ vn V0 (Tn − µ0 ) Q, under Gn and for all M > 0 −d −1/2 sup |V0 |1/2 vn gn (t) − q vn V0 {t − µ0 } = o(1) vn |t−µ0 |<M
• 302. Likelihood-free Methods ABC model choice consistency Consistency resultsAssumptions A collection of asymptotic “standard” assumptions: [A2] For i = 1, 2, there exist d × d symmetric positive deﬁnite matrices Vi (θi ) and µi (θi ) ∈ Rd such that n→∞ vn Vi (θi )−1/2 (Tn − µi (θi )) Q, under Gi,n (·|θi ) .
• 303. Likelihood-free Methods ABC model choice consistency Consistency resultsAssumptions A collection of asymptotic “standard” assumptions: [A3] For i = 1, 2, there exist sets Fn,i ⊂ Θi and constants i , τi , αi >0 such that for all τ > 0, sup Gi,n |Tn − µ(θi )| > τ |µi (θi ) − µ0 | ∧ i |θi θi ∈Fn,i −α −αi vn i sup (|µi (θi ) − µ0 | ∧ i ) θi ∈Fn,i with c −τ πi (Fn,i ) = o(vn i ).
• 304. Likelihood-free Methods ABC model choice consistency Consistency resultsAssumptions A collection of asymptotic “standard” assumptions: [A4] For u > 0 −1 Sn,i (u) = θi ∈ Fn,i ; |µ(θi ) − µ0 | ≤ u vn if inf{|µi (θi ) − µ0 |; θi ∈ Θi } = 0, there exist constants di < τi ∧ αi − 1 such that −d πi (Sn,i (u)) ∼ u di vn i , ∀u vn
• 305. Likelihood-free Methods ABC model choice consistency Consistency resultsAssumptions A collection of asymptotic “standard” assumptions: [A5] If inf{|µi (θi ) − µ0 |; θi ∈ Θi } = 0, there exists U > 0 such that for any M > 0, −d sup sup |Vi (θi )|1/2 vn gi (t|θi ) vn |t−µ0 |<M θi ∈Sn,i (U) −q vn Vi (θi )−1/2 (t − µ(θi ) = o(1) and πi Sn,i (U) ∩ ||Vi (θi )−1 || + ||Vi (θi )|| > M lim lim sup =0. M→∞ n πi (Sn,i (U))
• 306. Likelihood-free Methods ABC model choice consistency Consistency resultsAssumptions A collection of asymptotic “standard” assumptions: [A1]–[A2] are standard central limit theorems ([A1] redundant when one model is “true”) [A3] controls the large deviations of the estimator Tn from the estimand µ(θ) [A4] is the standard prior mass condition found in Bayesian asymptotics (di eﬀective dimension of the parameter) [A5] controls more tightly convergence esp. when µi is not one-to-one
• 307. Likelihood-free Methods ABC model choice consistency Consistency resultsEﬀective dimension [A4] Understanding d1 , d2 : deﬁned only when µ0 ∈ {µi (θi ), θi ∈ Θi }, πi (θi : |µi (θi ) − µ0 | < n−1/2 ) = O(n−di /2 ) is the eﬀective dimension of the model Θi around µ0
• 308. Likelihood-free Methods ABC model choice consistency Consistency resultsAsymptotic marginals Asymptotically, under [A1]–[A5] mi (t) = gi (t|θi ) πi (θi ) dθi Θi is such that (i) if inf{|µi (θi ) − µ0 |; θi ∈ Θi } = 0, Cl vn i ≤ mi (Tn ) ≤ Cu vn i d−d d−d and (ii) if inf{|µi (θi ) − µ0 |; θi ∈ Θi } > 0 mi (Tn ) = oPn [vn i + vn i ]. d−τ d−α
• 309. Likelihood-free Methods ABC model choice consistency Consistency resultsWithin-model consistency Under same assumptions, if inf{|µi (θi ) − µ0 |; θi ∈ Θi } = 0, the posterior distribution of µi (θi ) given Tn is consistent at rate 1/vn provided αi ∧ τi > di .
• 310. Likelihood-free Methods ABC model choice consistency Consistency resultsWithin-model consistency Under same assumptions, if inf{|µi (θi ) − µ0 |; θi ∈ Θi } = 0, the posterior distribution of µi (θi ) given Tn is consistent at rate 1/vn provided αi ∧ τi > di . Note: di can truly be seen as an eﬀective dimension of the model under the posterior πi (.|Tn ), since if µ0 ∈ {µi (θi ); θi ∈ Θi }, mi (Tn ) ∼ vn i d−d
• 311. Likelihood-free Methods ABC model choice consistency Consistency resultsBetween-model consistency Consequence of above is that asymptotic behaviour of the Bayes factor is driven by the asymptotic mean value of Tn under both models. And only by this mean value!
• 312. Likelihood-free Methods ABC model choice consistency Consistency resultsBetween-model consistency Consequence of above is that asymptotic behaviour of the Bayes factor is driven by the asymptotic mean value of Tn under both models. And only by this mean value! Indeed, if inf{|µ0 − µ2 (θ2 )|; θ2 ∈ Θ2 } = inf{|µ0 − µ1 (θ1 )|; θ1 ∈ Θ1 } = 0 then −(d1 −d2 ) −(d1 −d2 ) Cl vn ≤ m1 (Tn ) m2 (Tn ) ≤ Cu vn , where Cl , Cu = OPn (1), irrespective of the true model. c Only depends on the diﬀerence d1 − d2
• 313. Likelihood-free Methods ABC model choice consistency Consistency resultsBetween-model consistency Consequence of above is that asymptotic behaviour of the Bayes factor is driven by the asymptotic mean value of Tn under both models. And only by this mean value! Else, if inf{|µ0 − µ2 (θ2 )|; θ2 ∈ Θ2 } > inf{|µ0 − µ1 (θ1 )|; θ1 ∈ Θ1 } = 0 then m1 (Tn ) −(d1 −α2 ) −(d1 −τ2 ) n ≥ Cu min vn , vn , m2 (T )
• 314. Likelihood-free Methods ABC model choice consistency Consistency resultsConsistency theorem If inf{|µ0 − µ2 (θ2 )|; θ2 ∈ Θ2 } = inf{|µ0 − µ1 (θ1 )|; θ1 ∈ Θ1 } = 0, Bayes factor T −(d1 −d2 ) B12 = O(vn ) irrespective of the true model. It is consistent iﬀ Pn is within the model with the smallest dimension
• 315. Likelihood-free Methods ABC model choice consistency Consistency resultsConsistency theorem If Pn belongs to one of the two models and if µ0 cannot be attained by the other one : 0 = min (inf{|µ0 − µi (θi )|; θi ∈ Θi }, i = 1, 2) < max (inf{|µ0 − µi (θi )|; θi ∈ Θi }, i = 1, 2) , T then the Bayes factor B12 is consistent
• 316. Likelihood-free Methods ABC model choice consistency Summary statisticsConsequences on summary statistics Bayes factor driven by the means µi (θi ) and the relative position of µ0 wrt both sets {µi (θi ); θi ∈ Θi }, i = 1, 2. For ABC, this implies the most likely statistics Tn are ancillary statistics with diﬀerent mean values under both models Else, if Tn asymptotically depends on some of the parameters of the models, it is quite likely that there exists θi ∈ Θi such that µi (θi ) = µ0 even though model M1 is misspeciﬁed
• 317. Likelihood-free Methods ABC model choice consistency Summary statisticsToy example: Laplace versus Gauss [1] If n Tn = n−1 Xi4 , µ1 (θ) = 3 + θ4 + 6θ2 , µ2 (θ) = 6 + · · · i=1 and the true distribution is Laplace with mean θ0 = 1, under the √ Gaussian model the value θ∗ = 2 3 − 3 leads to µ0 = µ(θ∗ ) [here d1 = d2 = d = 1]
• 318. Likelihood-free Methods ABC model choice consistency Summary statisticsToy example: Laplace versus Gauss [1] If n Tn = n−1 Xi4 , µ1 (θ) = 3 + θ4 + 6θ2 , µ2 (θ) = 6 + · · · i=1 and the true distribution is Laplace with mean θ0 = 1, under the √ Gaussian model the value θ∗ = 2 3 − 3 leads to µ0 = µ(θ∗ ) [here d1 = d2 = d = 1] c a Bayes factor associated with such a statistic is inconsistent
• 319. Likelihood-free Methods ABC model choice consistency Summary statisticsToy example: Laplace versus Gauss [1] If n n −1 T =n Xi4 , µ1 (θ) = 3 + θ4 + 6θ2 , µ2 (θ) = 6 + · · · i=1 8 6 4 2 0 0.0 0.2 0.4 0.6 0.8 1.0 probability Fourth moment
• 320. Likelihood-free Methods ABC model choice consistency Summary statisticsToy example: Laplace versus Gauss [1] If n n −1 T =n Xi4 , µ1 (θ) = 3 + θ4 + 6θ2 , µ2 (θ) = 6 + · · · i=1 n=1000 0.45 q 0.40 0.35 q q q 0.30 q M1 M2
• 321. Likelihood-free Methods ABC model choice consistency Summary statisticsToy example: Laplace versus Gauss [1] If n n −1 T =n Xi4 , µ1 (θ) = 3 + θ4 + 6θ2 , µ2 (θ) = 6 + · · · i=1 Caption: Comparison of the distributions of the posterior probabilities that the data is from a normal model (as opposed to a Laplace model) with unknown mean when the data is made of n = 1000 observations either from a normal (M1) or Laplace (M2) distribution with mean one and when the summary statistic in the ABC algorithm is restricted to the empirical fourth moment. The ABC algorithm uses proposals from the prior N (0, 4) and selects the tolerance as the 1% distance quantile.
• 322. Likelihood-free Methods ABC model choice consistency Summary statisticsToy example: Laplace versus Gauss [0] When (4) (6) T(y) = yn , yn ¯ ¯ and the true distribution is Laplace with mean θ0 = 0, then √ ∗ ∗ µ0 = 6, µ1 (θ1 ) = 6 with θ1 = 2 3 − 3 [d1 = 1 and d2 = 1/2] thus B12 ∼ n−1/4 → 0 : consistent
• 323. Likelihood-free Methods ABC model choice consistency Summary statisticsToy example: Laplace versus Gauss [0] When (4) (6) T(y) = yn , yn ¯ ¯ and the true distribution is Laplace with mean θ0 = 0, then √ ∗ ∗ µ0 = 6, µ1 (θ1 ) = 6 with θ1 = 2 3 − 3 [d1 = 1 and d2 = 1/2] thus B12 ∼ n−1/4 → 0 : consistent Under the Gaussian model µ0 = 3 µ2 (θ2 ) ≥ 6 > 3 ∀θ2 B12 → +∞ : consistent
• 324. Likelihood-free Methods ABC model choice consistency Summary statisticsToy example: Laplace versus Gauss [0] When (4) (6) T(y) = yn , yn ¯ ¯ 5 4 3 Density 2 1 0 0.0 0.2 0.4 0.6 0.8 1.0 posterior probabilities Fourth and sixth moments
• 325. Likelihood-free Methods ABC model choice consistency Summary statisticsEmbedded models When M1 submodel of M2 , and if the true distribution belongs to the smaller model M1 , Bayes factor is of order −(d1 −d2 ) vn
• 326. Likelihood-free Methods ABC model choice consistency Summary statisticsEmbedded models When M1 submodel of M2 , and if the true distribution belongs to the smaller model M1 , Bayes factor is of order −(d1 −d2 ) vn If summary statistic only informative on a parameter that is the same under both models, i.e if d1 = d2 , then c the Bayes factor is not consistent
• 327. Likelihood-free Methods ABC model choice consistency Summary statisticsEmbedded models When M1 submodel of M2 , and if the true distribution belongs to the smaller model M1 , Bayes factor is of order −(d1 −d2 ) vn Else, d1 < d2 and Bayes factor is consistent under M1 . If true distribution not in M1 , then c Bayes factor is consistent only if µ1 = µ2 = µ0