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### Xian's abc

1. 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. 2. Likelihood-free MethodsOutlineIntroductionThe Metropolis-Hastings Algorithmsimulation-based methods in EconometricsGenetics of ABCApproximate Bayesian computationABC for model choiceABC model choice consistency
3. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 79. Likelihood-free Methods Approximate Bayesian computation ABC basicsUntractable likelihoods
80. 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. 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. 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. 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. 84. Likelihood-free Methods Approximate Bayesian computation ABC basicsThe ABC method Bayesian setting: target is π(θ)f (x|θ)
85. 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. 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. 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. 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. 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. 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. 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. 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. 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. 94. Likelihood-free Methods Approximate Bayesian computation ABC basicsConvergence of ABC (ﬁrst attempt) What happens when → 0?
95. 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. 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. 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. 98. Likelihood-free Methods Approximate Bayesian computation ABC basicsConvergence of ABC (second attempt) What happens when → 0?
99. 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. 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. 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. 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. 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. 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. 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. 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. 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 )−qt≤T 3. producing a simulated series (xt )1≤t≤T
108. 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 )−qt≤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. 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. 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. 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. 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. 113. Likelihood-free Methods Approximate Bayesian computation ABC basicsABC advances Simulating from the prior is often poor in eﬃciency
114. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 127. Likelihood-free Methods Approximate Bayesian computation Alphabet soupABC-NCH (2) Why neural network?
128. 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. 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. 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. 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. 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. 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. 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) IAX (( 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. 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. 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. 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. 138. Likelihood-free Methods Approximate Bayesian computation Alphabet soupA toy example x =2
139. 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. 140. Likelihood-free Methods Approximate Bayesian computation Alphabet soupA toy example x = 10
141. 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. 142. Likelihood-free Methods Approximate Bayesian computation Alphabet soupA toy example x = 50
143. 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. 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. 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. 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. 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. 148. Likelihood-free Methods Approximate Bayesian computation Alphabet soupABCµ multiple errors [ c Ratmann et al., PNAS, 2009]
149. 149. Likelihood-free Methods Approximate Bayesian computation Alphabet soupABCµ for model choice [ c Ratmann et al., PNAS, 2009]
150. 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. 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. 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. 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. 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. 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. 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.