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- 1. MCMC and likelihood-free methods Part/day II: ABC MCMC and likelihood-free methods Part/day II: ABC Christian P. Robert Universit´ Paris-Dauphine, IuF, & CREST e Monash University, EBS, July 18, 2012
- 2. MCMC and likelihood-free methods Part/day II: ABCOutlinesimulation-based methods in EconometricsGenetics of ABCApproximate Bayesian computationABC for model choiceABC model choice consistency
- 3. MCMC and likelihood-free methods Part/day II: ABC simulation-based methods in EconometricsEcon’ections simulation-based methods in Econometrics Genetics of ABC Approximate Bayesian computation ABC for model choice ABC model choice consistency
- 4. MCMC and likelihood-free methods Part/day II: ABC simulation-based methods in EconometricsUsages of simulation in Econometrics 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
- 5. MCMC and likelihood-free methods Part/day II: ABC 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
- 6. MCMC and likelihood-free methods Part/day II: ABC 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
- 7. MCMC and likelihood-free methods Part/day II: ABC 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]
- 8. MCMC and likelihood-free methods Part/day II: ABC 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]
- 9. MCMC and likelihood-free methods Part/day II: ABC 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
- 10. MCMC and likelihood-free methods Part/day II: ABC 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
- 11. MCMC and likelihood-free methods Part/day II: ABC 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...
- 12. MCMC and likelihood-free methods Part/day II: ABC 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
- 13. MCMC and likelihood-free methods Part/day II: ABC 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)
- 14. MCMC and likelihood-free methods Part/day II: ABC 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
- 15. MCMC and likelihood-free methods Part/day II: ABC 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
- 16. MCMC and likelihood-free methods Part/day II: ABC 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]
- 17. MCMC and likelihood-free methods Part/day II: ABC simulation-based methods in EconometricsABC using indirect inference (1) 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...
- 18. MCMC and likelihood-free methods Part/day II: ABC simulation-based methods in EconometricsABC using indirect inference (2) ...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...
- 19. MCMC and likelihood-free methods Part/day II: ABC Genetics of ABCGenetics of ABC simulation-based methods in Econometrics Genetics of ABC Approximate Bayesian computation ABC for model choice ABC model choice consistency
- 20. MCMC and likelihood-free methods Part/day II: ABC 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
- 21. MCMC and likelihood-free methods Part/day II: ABC 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.
- 22. MCMC and likelihood-free methods Part/day II: ABC 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)
- 23. MCMC and likelihood-free methods Part/day II: ABC Genetics of ABC Le modèleWright-Fisher model de Wright-Fisher •! En l’absence de mutation et de A population of constant sélection, les fréquences size, in which individuals alléliques dérivent (augmentent reproduce at the same time. et diminuent) inévitablement jusqu’à Each gene in a generation is la fixation d’un allèle a copy of a gene of the •! La dérive conduit generation. previous donc à la perte de variation génétique à l’intérieur des populations mutation In the absence of and selection, allele frequencies derive inevitably until the ﬁxation of an allele.
- 24. MCMC and likelihood-free methods Part/day II: ABC Genetics of ABCCoalescent theory [Kingman, 1982; Tajima, Tavar´, &tc] e !"#$%&((")**+$,-"."/010234%".5"*$*%()23$15"6" Coalescence theory interested in the genealogy of a sample of genes7**+$,-",()5534%" common ancestor of the sample. " !! back in time to the " "!"7**+$,-"8",$)(5,1,"9"
- 25. MCMC and likelihood-free methods Part/day II: ABC 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
- 26. MCMC and likelihood-free methods Part/day II: ABC 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
- 27. MCMC and likelihood-free methods Part/day II: ABC 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).
- 28. MCMC and likelihood-free methods Part/day II: ABC 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.
- 29. MCMC and likelihood-free methods Part/day II: ABC 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?
- 30. MCMC and likelihood-free methods Part/day II: ABC Genetics of ABC !""#$%&()*+,(-*.&(/+0$"1)()&$/+2!,03!Scenarios 1/+*%*"4*+56(""4&7()&$/.+.1#+4*.+8-9:*.+ under competition Différents scénarios possibles, choix de scenario par ABC Verdu et al. 2009 96
- 31. 1/+*%*"4*+56(""4&7()&$/.+.1#+4*.+MCMC and likelihood-free methods Part/day II: ABC !""#$%&()*+,(-*.&(/+0$"1)()&$/+2!,03 Genetics of ABC 1/+*%*"4*+56(""4&7()&$/.+.1#+4*.+8-9:*.Différentsresults Simulation scénarios possibles, choix de scenari Différents scénarios possibles, choix de scenario par ABC c Scenario 1A is chosen. largement soutenu par rapport aux Le scenario 1a est autres ! plaide pour une origine commune des Le scenario 1a est largement soutenu par populations pygmées d’Afrique de l’Ouest rap Verdu e
- 32. MCMC and likelihood-free methods Part/day II: ABC Genetics of ABC !""#$%&()*+,(-*.&(/+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
- 33. MCMC and likelihood-free methods Part/day II: ABC Approximate Bayesian computationApproximate Bayesian computation simulation-based methods in Econometrics Genetics of ABC Approximate Bayesian computation ABC for model choice ABC model choice consistency
- 34. MCMC and likelihood-free methods Part/day II: ABC 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!
- 35. MCMC and likelihood-free methods Part/day II: ABC 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!
- 36. MCMC and likelihood-free methods Part/day II: ABC 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
- 37. MCMC and likelihood-free methods Part/day II: ABC 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
- 38. MCMC and likelihood-free methods Part/day II: ABC Approximate Bayesian computation ABC basicsIllustrations Example (Genesis) 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]
- 39. MCMC and likelihood-free methods Part/day II: ABC Approximate Bayesian computation ABC basicsThe ABC method Bayesian setting: target is π(θ)f (x|θ)
- 40. MCMC and likelihood-free methods Part/day II: ABC 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:
- 41. MCMC and likelihood-free methods Part/day II: ABC 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
- 42. MCMC and likelihood-free methods Part/day II: ABC 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]
- 43. MCMC and likelihood-free methods Part/day II: ABC 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.
- 44. MCMC and likelihood-free methods Part/day II: ABC 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
- 45. MCMC and likelihood-free methods Part/day II: ABC 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]
- 46. MCMC and likelihood-free methods Part/day II: ABC 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
- 47. MCMC and likelihood-free methods Part/day II: ABC 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)) < }.
- 48. MCMC and likelihood-free methods Part/day II: ABC 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) .
- 49. MCMC and likelihood-free methods Part/day II: ABC Approximate Bayesian computation ABC basicsConvergence of ABC (ﬁrst attempt) What happens when → 0?
- 50. MCMC and likelihood-free methods Part/day II: ABC 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 )
- 51. MCMC and likelihood-free methods Part/day II: ABC 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
- 52. MCMC and likelihood-free methods Part/day II: ABC 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 ]
- 53. MCMC and likelihood-free methods Part/day II: ABC Approximate Bayesian computation ABC basicsConvergence of ABC (second attempt) What happens when → 0?
- 54. MCMC and likelihood-free methods Part/day II: ABC 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).
- 55. MCMC and likelihood-free methods Part/day II: ABC 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
- 56. MCMC and likelihood-free methods Part/day II: ABC 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 ) ,
- 57. MCMC and likelihood-free methods Part/day II: ABC 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
- 58. MCMC and likelihood-free methods Part/day II: ABC 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)
- 59. MCMC and likelihood-free methods Part/day II: ABC 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) .
- 60. MCMC and likelihood-free methods Part/day II: ABC 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
- 61. MCMC and likelihood-free methods Part/day II: ABC 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)
- 62. MCMC and likelihood-free methods Part/day II: ABC 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
- 63. MCMC and likelihood-free methods Part/day II: ABC 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
- 64. MCMC and likelihood-free methods Part/day II: ABC 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
- 65. MCMC and likelihood-free methods Part/day II: ABC 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
- 66. MCMC and likelihood-free methods Part/day II: ABC 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
- 67. MCMC and likelihood-free methods Part/day II: ABC 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]
- 68. MCMC and likelihood-free methods Part/day II: ABC Approximate Bayesian computation ABC basicsABC advances Simulating from the prior is often poor in eﬃciency
- 69. MCMC and likelihood-free methods Part/day II: ABC 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]
- 70. MCMC and likelihood-free methods Part/day II: ABC 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]
- 71. MCMC and likelihood-free methods Part/day II: ABC 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]
- 72. MCMC and likelihood-free methods Part/day II: ABC 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]
- 73. MCMC and likelihood-free methods Part/day II: ABC 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)]
- 74. MCMC and likelihood-free methods Part/day II: ABC 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...
- 75. MCMC and likelihood-free methods Part/day II: ABC 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]
- 76. MCMC and likelihood-free methods Part/day II: ABC 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]
- 77. MCMC and likelihood-free methods Part/day II: ABC 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))}
- 78. MCMC and likelihood-free methods Part/day II: ABC 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]
- 79. MCMC and likelihood-free methods Part/day II: ABC 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]
- 80. MCMC and likelihood-free methods Part/day II: ABC 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
- 81. MCMC and likelihood-free methods Part/day II: ABC 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
- 82. MCMC and likelihood-free methods Part/day II: ABC Approximate Bayesian computation Alphabet soupABC-NCH (2) Why neural network?
- 83. MCMC and likelihood-free methods Part/day II: ABC 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
- 84. MCMC and likelihood-free methods Part/day II: ABC 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,
- 85. MCMC and likelihood-free methods Part/day II: ABC 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]
- 86. MCMC and likelihood-free methods Part/day II: ABC 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
- 87. MCMC and likelihood-free methods Part/day II: ABC 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
- 88. MCMC and likelihood-free methods Part/day II: ABC 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
- 89. MCMC and likelihood-free methods Part/day II: ABC 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) )
- 90. MCMC and likelihood-free methods Part/day II: ABC 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)
- 91. MCMC and likelihood-free methods Part/day II: ABC 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]
- 92. MCMC and likelihood-free methods Part/day II: ABC 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]} }
- 93. MCMC and likelihood-free methods Part/day II: ABC Approximate Bayesian computation Alphabet soupA toy example x =2
- 94. MCMC and likelihood-free methods Part/day II: ABC 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
- 95. MCMC and likelihood-free methods Part/day II: ABC Approximate Bayesian computation Alphabet soupA toy example x = 10
- 96. MCMC and likelihood-free methods Part/day II: ABC 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
- 97. MCMC and likelihood-free methods Part/day II: ABC Approximate Bayesian computation Alphabet soupA toy example x = 50
- 98. MCMC and likelihood-free methods Part/day II: ABC 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
- 99. MCMC and likelihood-free methods Part/day II: ABC 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 y when z ∼ f (z|θ)
- 100. MCMC and likelihood-free methods Part/day II: ABC 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 y when z ∼ f (z|θ) Warning! Replacement of ξ( |y, θ) with a non-parametric kernel approximation.
- 101. MCMC and likelihood-free methods Part/day II: ABC 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)
- 102. MCMC and likelihood-free methods Part/day II: ABC 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)
- 103. MCMC and likelihood-free methods Part/day II: ABC Approximate Bayesian computation Alphabet soupABCµ multiple errors [ c Ratmann et al., PNAS, 2009]
- 104. MCMC and likelihood-free methods Part/day II: ABC Approximate Bayesian computation Alphabet soupABCµ for model choice [ c Ratmann et al., PNAS, 2009]
- 105. MCMC and likelihood-free methods Part/day II: ABC 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).
- 106. MCMC and likelihood-free methods Part/day II: ABC 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]
- 107. MCMC and likelihood-free methods Part/day II: ABC 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 )
- 108. MCMC and likelihood-free methods Part/day II: ABC 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]
- 109. MCMC and likelihood-free methods Part/day II: ABC 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]
- 110. MCMC and likelihood-free methods Part/day II: ABC 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...
- 111. MCMC and likelihood-free methods Part/day II: ABC 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.
- 112. MCMC and likelihood-free methods Part/day II: ABC 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.
- 113. MCMC and likelihood-free methods Part/day II: ABC 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
- 114. MCMC and likelihood-free methods Part/day II: ABC Approximate Bayesian computation Alphabet soupABC-PRC bias Lack of unbiasedness of the method
- 115. MCMC and likelihood-free methods Part/day II: ABC 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 .
- 116. MCMC and likelihood-free methods Part/day II: ABC 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)) .
- 117. MCMC and likelihood-free methods Part/day II: ABC 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
- 118. MCMC and likelihood-free methods Part/day II: ABC 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]
- 119. MCMC and likelihood-free methods Part/day II: ABC 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
- 120. MCMC and likelihood-free methods Part/day II: ABC 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]
- 121. MCMC and likelihood-free methods Part/day II: ABC 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
- 122. MCMC and likelihood-free methods Part/day II: ABC 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) , ·)
- 123. MCMC and likelihood-free methods Part/day II: ABC 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]
- 124. MCMC and likelihood-free methods Part/day II: ABC 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]
- 125. MCMC and likelihood-free methods Part/day II: ABC 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
- 126. MCMC and likelihood-free methods Part/day II: ABC 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
- 127. MCMC and likelihood-free methods Part/day II: ABC 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]
- 128. MCMC and likelihood-free methods Part/day II: ABC 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
- 129. MCMC and likelihood-free methods Part/day II: ABC 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]
- 130. MCMC and likelihood-free methods Part/day II: ABC 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).
- 131. MCMC and likelihood-free methods Part/day II: ABC 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]

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