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Laplace's Demon: seminar #1

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ABC-Gibbs, with an introduction to ABC

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Laplace's Demon: seminar #1

  1. 1. Approximate Bayesian computation Christian P. Robert U. Paris Dauphine PSL & Warwick U. & CREST ENSAE Laplace’s Demon, 13 May, 2020
  2. 2. Not to be confused with... The XKCD of ABC Christian P. Robert (Paris Dauphine PSL & Warwick U.) JSM 2019, Denver, Colorado
  3. 3. Co-authors Jean-Marie Cornuet, Arnaud Estoup (INRA Montpellier) David Frazier, Gael Martin (Monash U) Jean-Michel Marin (U Montpellier) Kerrie Mengersen (QUT) Espen Bernton, Pierre Jacob, Natesh Pillai (Harvard U) Judith Rousseau (U Oxford) Gr´egoire Clart´e, Robin Ryder, Julien Stoehr (U Paris Dauphine)
  4. 4. Outline 1 Motivation 2 Concept 3 Implementation 4 Big data issues 5 big parameter issues 6 ABC postdoc positions
  5. 5. another webinar
  6. 6. Posterior distribution Central concept of Bayesian inference: π( θ parameter | xobs observation ) proportional ∝ π(θ) prior × L(θ | xobs ) likelihood Drives derivation of optimal decisions assessment of uncertainty model selection prediction [McElreath, 2015]
  7. 7. Monte Carlo representation Exploration of Bayesian posterior π(θ|xobs) may (!) require to produce sample θ1, . . . , θT distributed from π(θ|xobs) (or asymptotically by Markov chain Monte Carlo aka MCMC) [McElreath, 2015]
  8. 8. Difficulties MCMC = workhorse of practical Bayesian analysis (BUGS, JAGS, Stan, &tc.), except when product π(θ) × L(θ | xobs ) well-defined but numerically unavailable or too costly to compute Only partial solutions are available: demarginalisation (latent variables) exchange algorithm (auxiliary variables) pseudo-marginal (unbiased estimator)
  9. 9. Difficulties MCMC = workhorse of practical Bayesian analysis (BUGS, JAGS, Stan, &tc.), except when product π(θ) × L(θ | xobs ) well-defined but numerically unavailable or too costly to compute Only partial solutions are available: demarginalisation (latent variables) exchange algorithm (auxiliary variables) pseudo-marginal (unbiased estimator)
  10. 10. Example 1: Dynamic mixture Mixture model {1 − wµ,τ(x)}fβ,λ(x) + wµ,τ(x)gε,σ(x) x > 0 where fβ,λ Weibull density, gε,σ generalised Pareto density, and wµ,τ Cauchy (arctan) cdf Intractable normalising constant C(µ, τ, β, λ, ε, σ) = ∞ 0 {(1 − wµ,τ(x))fβ,λ(x) + wµ,τ(x)gε,σ(x)} dx [Frigessi, Haug & Rue, 2002]
  11. 11. Example 2: truncated Normal Given set A ⊂ Rk (k large), truncated Normal model f(x | µ, Σ, A) ∝ exp{−(x − µ)T Σ−1 (x − µ)/2} IA(x) with intractable normalising constant C(µ, Σ, A) = A exp{−(x − µ)T Σ−1 (x − µ)/2}dx
  12. 12. Example 3: robust Normal statistics Normal sample x1, . . . , xn ∼ N(µ, σ2 ) summarised into (insufficient) ^µn = med(x1, . . . , xn) and ^σn = mad(x1, . . . , xn) = med |xi − ^µn| Under a conjugate prior π(µ, σ2), posterior close to intractable. but simulation of (^µn, ^σn) straightforward
  13. 13. Example 3: robust Normal statistics Normal sample x1, . . . , xn ∼ N(µ, σ2 ) summarised into (insufficient) ^µn = med(x1, . . . , xn) and ^σn = mad(x1, . . . , xn) = med |xi − ^µn| Under a conjugate prior π(µ, σ2), posterior close to intractable. but simulation of (^µn, ^σn) straightforward
  14. 14. Example 4: exponential random graph ERGM: binary random vector x indexed by all edges on set of nodes plus graph f(x | θ) = 1 C(θ) exp(θT S(x)) with S(x) vector of statistics and C(θ) intractable normalising constant [Grelaud & al., 2009; Everitt, 2012; Bouranis & al., 2017]
  15. 15. Realistic[er] applications Kingman’s coalescent in population genetics [Tavar´e et al., 1997; Beaumont et al., 2003] α-stable distributions [Peters et al, 2012] complex networks [Dutta et al., 2018] astrostatistics & cosmostatistics [Cameron & Pettitt, 2012; Ishida et al., 2015]
  16. 16. Concept 2 Concept algorithm summary statistics random forests calibration of tolerance 3 Implementation 4 Big data issues 5 big parameter issues 6 ABC postdoc positions
  17. 17. A?B?C? A stands for approximate [wrong likelihood] B stands for Bayesian [right prior] C stands for computation [producing a parameter sample]
  18. 18. A?B?C? Rough version of the data [from dot to ball] Non-parametric approximation of the likelihood [near actual observation] Use of non-sufficient statistics [dimension reduction] Monte Carlo error [and no unbiasedness]
  19. 19. A seemingly na¨ıve representation When likelihood f(x|θ) not in closed form, likelihood-free rejection technique: ABC-AR algorithm For an observation xobs ∼ f(x|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 = xobs [Diggle & Gratton, 1984; Rubin, 1984; Tavar´e et al., 1997]
  20. 20. A seemingly na¨ıve representation When likelihood f(x|θ) not in closed form, likelihood-free rejection technique: ABC-AR algorithm For an observation xobs ∼ f(x|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 = xobs [Diggle & Gratton, 1984; Rubin, 1984; Tavar´e et al., 1997]
  21. 21. Why does it work? The mathematical proof is trivial: f(θi) ∝ z∈D π(θi)f(z|θi)Iy(z) ∝ π(θi)f(y|θi) = π(θi|y) [Accept–Reject 101] But very impractical when Pθ(Z = xobs ) ≈ 0
  22. 22. Why does it work? The mathematical proof is trivial: f(θi) ∝ z∈D π(θi)f(z|θi)Iy(z) ∝ π(θi)f(y|θi) = π(θi|y) [Accept–Reject 101] But very impractical when Pθ(Z = xobs ) ≈ 0
  23. 23. A as approximative When y is a continuous random variable, strict equality z = xobs is replaced with a tolerance zone ρ(xobs , z) ≤ ε where ρ is a distance Output distributed from π(θ) Pθ{ρ(xobs , z) < ε} def ∝ π(θ|ρ(xobs , z) < ε) [Pritchard et al., 1999]
  24. 24. A as approximative When y is a continuous random variable, strict equality z = xobs is replaced with a tolerance zone ρ(xobs , z) ≤ ε where ρ is a distance Output distributed from π(θ) Pθ{ρ(xobs , z) < ε} def ∝ π(θ|ρ(xobs , z) < ε) [Pritchard et al., 1999]
  25. 25. ABC algorithm Algorithm 1 Likelihood-free rejection sampler for i = 1 to N do repeat generate θ from prior π(·) generate z from sampling density f(·|θ ) until ρ{η(z), η(xobs)} ≤ ε set θi = θ end for where η(xobs) defines a (not necessarily sufficient) statistic Custom: η(xobs) called summary statistic
  26. 26. ABC algorithm Algorithm 2 Likelihood-free rejection sampler for i = 1 to N do repeat generate θ from prior π(·) generate z from sampling density f(·|θ ) until ρ{η(z), η(xobs)} ≤ ε set θi = θ end for where η(xobs) defines a (not necessarily sufficient) statistic Custom: η(xobs) called summary statistic
  27. 27. Example 3: robust Normal statistics mu=rnorm(N<-1e6) #prior sig=sqrt(rgamma(N,2,2)) medobs=median(obs) madobs=mad(obs) #summary for(t in diz<-1:N){ psud=rnorm(1e2)/sig[t]+mu[t] medpsu=median(psud)-medobs madpsu=mad(psud)-madobs diz[t]=medpsuˆ2+madpsuˆ2} #ABC subsample subz=which(diz<quantile(diz,.1))
  28. 28. Exact ABC posterior Algorithm samples from marginal in z of [exact] posterior πABC ε (θ, z|xobs ) = π(θ)f(z|θ)IAε,xobs (z) Aε,xobs ×Θ π(θ)f(z|θ)dzdθ , where Aε,xobs = {z ∈ D|ρ{η(z), η(xobs)} < ε}. Intuition that proper summary statistics coupled with small tolerance ε = εη should provide good approximation of the posterior distribution: πABC ε (θ|xobs ) = πABC ε (θ, z|xobs )dz ≈ π{θ|η(xobs )}
  29. 29. Exact ABC posterior Algorithm samples from marginal in z of [exact] posterior πABC ε (θ, z|xobs ) = π(θ)f(z|θ)IAε,xobs (z) Aε,xobs ×Θ π(θ)f(z|θ)dzdθ , where Aε,xobs = {z ∈ D|ρ{η(z), η(xobs)} < ε}. Intuition that proper summary statistics coupled with small tolerance ε = εη should provide good approximation of the posterior distribution: πABC ε (θ|xobs ) = πABC ε (θ, z|xobs )dz ≈ π{θ|η(xobs )}
  30. 30. Why summaries? reduction of dimension improvement of signal to noise ratio reduce tolerance ε considerably whole data may be unavailable (as in Example 3) medobs=median(obs) madobs=mad(obs) #summary
  31. 31. Example 6: MA inference Moving average model MA(2): xt = εt + θ1εt−1 + θ2εt−2 εt iid ∼ N(0, 1) Comparison of raw series:
  32. 32. Example 6: MA inference Moving average model MA(2): xt = εt + θ1εt−1 + θ2εt−2 εt iid ∼ N(0, 1) [Feller, 1970] Comparison of acf’s:
  33. 33. Example 6: MA inference Summary vs. raw:
  34. 34. Why not summaries? loss of sufficient information when πABC(θ|xobs) replaced with πABC(θ|η(xobs)) arbitrariness of summaries uncalibrated approximation whole data may be available (at same cost as summaries) (empirical) distributions may be compared (Wasserstein distances) [Bernton et al., 2019]
  35. 35. Summary selection strategies Fundamental difficulty of selecting summary statistics when there is no non-trivial sufficient statistic [except when done by experimenters from the field]
  36. 36. Summary selection strategies Fundamental difficulty of selecting summary statistics when there is no non-trivial sufficient statistic [except when done by experimenters from the field] 1. Starting from large collection of summary statistics, Joyce and Marjoram (2008) consider the sequential inclusion into the ABC target, with a stopping rule based on a likelihood ratio test
  37. 37. Summary selection strategies Fundamental difficulty of selecting summary statistics when there is no non-trivial sufficient statistic [except when done by experimenters from the field] 2. Based on decision-theoretic principles, Fearnhead and Prangle (2012) end up with E[θ|xobs] as the optimal summary statistic
  38. 38. Summary selection strategies Fundamental difficulty of selecting summary statistics when there is no non-trivial sufficient statistic [except when done by experimenters from the field] 3. Use of indirect inference by Drovandi, Pettit, & Paddy (2011) with estimators of parameters of auxiliary model as summary statistics
  39. 39. Summary selection strategies Fundamental difficulty of selecting summary statistics when there is no non-trivial sufficient statistic [except when done by experimenters from the field] 4. Starting from large collection of summary statistics, Raynal & al. (2018, 2019) rely on random forests to build estimators and select summaries
  40. 40. Summary selection strategies Fundamental difficulty of selecting summary statistics when there is no non-trivial sufficient statistic [except when done by experimenters from the field] 5. Starting from large collection of summary statistics, Sedki & Pudlo (2012) use the Lasso to eliminate summaries
  41. 41. Semi-automated ABC Use of summary statistic η(·), importance proposal g(·), kernel K(·) ≤ 1 with bandwidth h ↓ 0 such that (θ, z) ∼ g(θ)f(z|θ) accepted with probability (hence the bound) K[{η(z) − η(xobs )}/h] and the corresponding importance weight defined by π(θ) g(θ) Theorem Optimality of posterior expectation E[θ|xobs] of parameter of interest as summary statistics η(xobs) [Fearnhead & Prangle, 2012; Sisson et al., 2019]
  42. 42. Semi-automated ABC Use of summary statistic η(·), importance proposal g(·), kernel K(·) ≤ 1 with bandwidth h ↓ 0 such that (θ, z) ∼ g(θ)f(z|θ) accepted with probability (hence the bound) K[{η(z) − η(xobs )}/h] and the corresponding importance weight defined by π(θ) g(θ) Theorem Optimality of posterior expectation E[θ|xobs] of parameter of interest as summary statistics η(xobs) [Fearnhead & Prangle, 2012; Sisson et al., 2019]
  43. 43. Random forests Technique that stemmed from Leo Breiman’s bagging (or bootstrap aggregating) machine learning algorithm for both classification [testing] and regression [estimation] [Breiman, 1996] Improved performances by averaging over classification schemes of randomly generated training sets, creating a “forest” of (CART) decision trees, inspired by Amit and Geman (1997) ensemble learning [Breiman, 2001]
  44. 44. Growing the forest Breiman’s solution for inducing random features in the trees of the forest: boostrap resampling of the dataset and random subset-ing [of size √ t] of the covariates driving the classification or regression at every node of each tree Covariate (summary) xτ that drives the node separation xτ cτ and the separation bound cτ chosen by minimising entropy or Gini index
  45. 45. ABC with random forests Idea: Starting with possibly large collection of summary statistics (η1, . . . , ηp) (from scientific theory input to available statistical softwares, methodologies, to machine-learning alternatives) ABC reference table involving model index, parameter values and summary statistics for the associated simulated pseudo-data run R randomforest to infer M or θ from (η1i, . . . , ηpi)
  46. 46. ABC with random forests Idea: Starting with possibly large collection of summary statistics (η1, . . . , ηp) (from scientific theory input to available statistical softwares, methodologies, to machine-learning alternatives) ABC reference table involving model index, parameter values and summary statistics for the associated simulated pseudo-data run R randomforest to infer M or θ from (η1i, . . . , ηpi) Average of the trees is resulting summary statistics, highly non-linear predictor of the model index
  47. 47. ABC with random forests Idea: Starting with possibly large collection of summary statistics (η1, . . . , ηp) (from scientific theory input to available statistical softwares, methodologies, to machine-learning alternatives) ABC reference table involving model index, parameter values and summary statistics for the associated simulated pseudo-data run R randomforest to infer M or θ from (η1i, . . . , ηpi) Potential selection of most active summaries, calibrated against pure noise
  48. 48. Classification of summaries by random forests Given large collection of summary statistics, rather than selecting a subset and excluding the others, estimate each parameter by random forests handles thousands of predictors ignores useless components fast estimation with good local properties automatised with few calibration steps substitute to Fearnhead and Prangle (2012) preliminary estimation of ^θ(yobs) includes a natural (classification) distance measure that avoids choice of either distance or tolerance [Marin et al., 2016, 2018]
  49. 49. Calibration of tolerance Calibration of threshold ε from scratch [how small is small?] from k-nearest neighbour perspective [quantile of prior predictive] subz=which(diz<quantile(diz,.1)) from asymptotics [convergence speed] related with choice of distance [automated selection by random forests] Fearnhead & Prangle, 2012; Biau et al., 2013; Liu & Fearnhead 2018]
  50. 50. Implementation 3 Implementation ABC-MCMC ABC-PMC post-processed ABC 4 Big data issues 5 big parameter issues 6 ABC postdoc positions
  51. 51. Sofware Several ABC R packages for performing parameter estimation and model selection [Nunes & Prangle, 2017]
  52. 52. Sofware Several ABC R packages for performing parameter estimation and model selection [Nunes & Prangle, 2017] abctools R package tuning ABC analyses
  53. 53. Sofware Several ABC R packages for performing parameter estimation and model selection [Nunes & Prangle, 2017] abcrf R package ABC via random forests
  54. 54. Sofware Several ABC R packages for performing parameter estimation and model selection [Nunes & Prangle, 2017] EasyABC R package several algorithms for performing efficient ABC sampling schemes, including four sequential sampling schemes and 3 MCMC schemes
  55. 55. Sofware Several ABC R packages for performing parameter estimation and model selection [Nunes & Prangle, 2017] DIYABC non R software for population genetics
  56. 56. ABC-IS Basic ABC algorithm limitations blind [no learning] inefficient [curse of dimension] inapplicable to improper priors Importance sampling version importance density g(θ) bounded kernel function Kh with bandwidth h acceptance probability of Kh{ρ[η(xobs ), η(x{θ})]} π(θ) g(θ) max θ Aθ [Fearnhead & Prangle, 2012]
  57. 57. ABC-IS Basic ABC algorithm limitations blind [no learning] inefficient [curse of dimension] inapplicable to improper priors Importance sampling version importance density g(θ) bounded kernel function Kh with bandwidth h acceptance probability of Kh{ρ[η(xobs ), η(x{θ})]} π(θ) g(θ) max θ Aθ [Fearnhead & Prangle, 2012]
  58. 58. ABC-MCMC Markov chain (θ(t)) created via transition function θ(t+1) =    θ ∼ Kω(θ |θ(t)) if x ∼ f(x|θ ) is such that x ≈ y and u ∼ U(0, 1) ≤ π(θ )Kω(θ(t)|θ ) π(θ(t))Kω(θ |θ(t)) , θ(t) otherwise, has the posterior π(θ|y) as stationary distribution [Marjoram et al, 2003]
  59. 59. ABC-MCMC Algorithm 3 Likelihood-free MCMC sampler get (θ(0), z(0)) by Algorithm 1 for t = 1 to N do generate θ from Kω ·|θ(t−1) , z from f(·|θ ), u from U[0,1], if u ≤ π(θ )Kω(θ(t−1)|θ ) π(θ(t−1)Kω(θ |θ(t−1)) IAε,xobs (z ) then set (θ(t), z(t)) = (θ , z ) else (θ(t), z(t))) = (θ(t−1), z(t−1)), end if end for
  60. 60. ABC-PMC Generate a sample at iteration t by ^πt(θ(t) ) ∝ N j=1 ω (t−1) j Kt(θ(t) |θ (t−1) j ) modulo acceptance of the associated xt, with tolerance εt ↓, and use importance weight associated with accepted simulation θ (t) i ω (t) i ∝ π(θ (t) i ) ^πt(θ (t) i ) © Still likelihood free [Sisson et al., 2007; Beaumont et al., 2009]
  61. 61. ABC-SMC Use of a kernel Kt associated with target πεt and derivation of the backward kernel Lt−1(z, z ) = πεt (z )Kt(z , z) πεt (z) Update of the weights ω (t) i ∝ ω (t−1) i M m=1 IAεt (x (t) im) M m=1 IAεt−1 (x (t−1) im ) when x (t) im ∼ Kt(x (t−1) i , ·) [Del Moral, Doucet & Jasra, 2009]
  62. 62. ABC-NP Better usage of [prior] simulations by adjustement: instead of throwing away θ such that ρ(η(z), η(xobs)) > ε, replace θ’s with locally regressed transforms θ∗ = θ − {η(z) − η(xobs )}T ^β [Csill´ery et al., TEE, 2010] where ^β is obtained by [NP] weighted least square regression on (η(z) − η(xobs)) with weights Kδ ρ(η(z), η(xobs )) [Beaumont et al., 2002, Genetics]
  63. 63. ABC-NN Incorporating non-linearities and heterocedasticities: θ∗ = ^m(η(xobs )) + [θ − ^m(η(z))] ^σ(η(xobs)) ^σ(η(z)) where ^m(η) estimated by non-linear regression (e.g., neural network) ^σ(η) estimated by non-linear regression on residuals log{θi − ^m(ηi)}2 = log σ2 (ηi) + ξi [Blum & Franc¸ois, 2009]
  64. 64. ABC-NN Incorporating non-linearities and heterocedasticities: θ∗ = ^m(η(xobs )) + [θ − ^m(η(z))] ^σ(η(xobs)) ^σ(η(z)) where ^m(η) estimated by non-linear regression (e.g., neural network) ^σ(η) estimated by non-linear regression on residuals log{θi − ^m(ηi)}2 = log σ2 (ηi) + ξi [Blum & Franc¸ois, 2009]
  65. 65. Big data issues 4 Big data issues 5 big parameter issues 6 ABC postdoc positions
  66. 66. Computational bottleneck Time per iteration increases with sample size n of the data: cost of sampling O(n1+?) associated with a reasonable acceptance probability makes ABC infeasible for large datasets surrogate models to get samples (e.g., using copulas) direct sampling of summary statistics (e.g., synthetic likelihood) [Wood, 2010] borrow from proposals for scalable MCMC (e.g., divide & conquer)
  67. 67. Approximate ABC [AABC] Idea approximations on both parameter and model spaces by resorting to bootstrap techniques. [Buzbas & Rosenberg, 2015] Procedure scheme 1. Sample (θi, xi), i = 1, . . . , m, from prior predictive 2. Simulate θ∗ ∼ π(·) and assign weight wi to dataset x(i) simulated under k-closest θi to θ∗ 3. Generate dataset x∗ as bootstrap weighted sample from (x(1), . . . , x(k)) Drawbacks If m too small, prior predictive sample may miss informative parameters large error and misleading representation of true posterior
  68. 68. Approximate ABC [AABC] Procedure scheme 1. Sample (θi, xi), i = 1, . . . , m, from prior predictive 2. Simulate θ∗ ∼ π(·) and assign weight wi to dataset x(i) simulated under k-closest θi to θ∗ 3. Generate dataset x∗ as bootstrap weighted sample from (x(1), . . . , x(k)) Drawbacks If m too small, prior predictive sample may miss informative parameters large error and misleading representation of true posterior only suited for models with very few parameters
  69. 69. Divide-and-conquer perspectives 1. divide the large dataset into smaller batches 2. sample from the batch posterior 3. combine the result to get a sample from the targeted posterior Alternative via ABC-EP [Barthelm´e & Chopin, 2014]
  70. 70. Divide-and-conquer perspectives 1. divide the large dataset into smaller batches 2. sample from the batch posterior 3. combine the result to get a sample from the targeted posterior Alternative via ABC-EP [Barthelm´e & Chopin, 2014]
  71. 71. Geometric combination: WASP Subset posteriors given partition xobs [1] , . . . , xobs [B] of observed data xobs, let define π(θ | xobs [b] ) ∝ π(θ) j∈[b] f(xobs j | θ)B . [Srivastava et al., 2015] Subset posteriors are combined via Wasserstein barycenter [Cuturi, 2014]
  72. 72. Geometric combination: WASP Subset posteriors given partition xobs [1] , . . . , xobs [B] of observed data xobs, let define π(θ | xobs [b] ) ∝ π(θ) j∈[b] f(xobs j | θ)B . [Srivastava et al., 2015] Subset posteriors are combined via Wasserstein barycenter [Cuturi, 2014] Drawback require sampling from f(· | θ)B by ABC means. Should be feasible for latent variable (z) representations when f(x | z, θ) available in closed form [Doucet & Robert, 2001]
  73. 73. Geometric combination: WASP Subset posteriors given partition xobs [1] , . . . , xobs [B] of observed data xobs, let define π(θ | xobs [b] ) ∝ π(θ) j∈[b] f(xobs j | θ)B . [Srivastava et al., 2015] Subset posteriors are combined via Wasserstein barycenter [Cuturi, 2014] Alternative backfeed subset posteriors as priors to other subsets, partitioning summaries
  74. 74. Consensus ABC Na¨ıve scheme For each data batch b = 1, . . . , B 1. Sample (θ [b] 1 , . . . , θ [b] n ) from diffused prior ˜π(·) ∝ π(·)1/B 2. Run ABC to sample from batch posterior ^π(· | d(S(xobs [b] ), S(x[b])) < ε) 3. Compute sample posterior variance Σ−1 b Combine batch posterior approximations θj = B b=1 Σbθ [b] j B b=1 Σb Remark Diffuse prior ˜π(·) non informative calls for ABC-MCMC steps
  75. 75. Consensus ABC Na¨ıve scheme For each data batch b = 1, . . . , B 1. Sample (θ [b] 1 , . . . , θ [b] n ) from diffused prior ˜π(·) ∝ π(·)1/B 2. Run ABC to sample from batch posterior ^π(· | d(S(xobs [b] ), S(x[b])) < ε) 3. Compute sample posterior variance Σ−1 b Combine batch posterior approximations θj = B b=1 Σbθ [b] j B b=1 Σb Remark Diffuse prior ˜π(·) non informative calls for ABC-MCMC steps
  76. 76. Big parameter issues Curse of dimension: as dim(Θ) = kθ increases exploration of parameter space gets harder summary statistic η forced to increase, since at least of dimension kη ≥ dim(Θ) Some solutions adopt more local algorithms like ABC-MCMC or ABC-SMC aim at posterior marginals and approximate joint posterior by copula [Li et al., 2016] run ABC-Gibbs [Clart´e et al., 2016]
  77. 77. Example 11: Hierarchical MA(2) xi iid ∼ MA2(µi, σi) µi = (βi,1 − βi,2, 2(βi,1 + βi,2) − 1) with (βi,1, βi,2, βi,3) iid ∼ Dir(α1, α2, α3) σi iid ∼ IG(σ1, σ2) α = (α1, α2, α3), with prior E(1)⊗3 σ = (σ1, σ2) with prior C(1)+⊗2 α µ1 µ2 µn. . . x1 x2 xn. . . σ σ1 σ2 σn. . .
  78. 78. Example 11: Hierarchical MA(2) © Based on 106 prior and 103 posteriors simulations, 3n summary statistics, and series of length 100, ABC-Rej posterior hardly distinguishable from prior!
  79. 79. ABC within Gibbs: Hierarchical models α µ1 µ2 µn. . . x1 x2 xn. . . Hierarchical Bayes models often allow for simplified conditional distributions thanks to partial independence properties, e.g., xj | µj ∼ π(xj | µj), µj | α i.i.d. ∼ π(µj | α), α ∼ π(α).
  80. 80. ABC-Gibbs When parameter decomposed into θ = (θ1, . . . , θn) Algorithm 4 ABC-Gibbs sampler starting point θ(0) = (θ (0) 1 , . . . , θ (0) n ), observation xobs for i = 1, . . . , N do for j = 1, . . . , n do θ (i) j ∼ πεj (· | x , sj, θ (i) 1 , . . . , θ (i) j−1, θ (i−1) j+1 , . . . , θ (i−1) n ) end for end for Divide & conquer: one tolerance εj for each parameter θj one statistic sj for each parameter θj [Clart´e et al., 2019]
  81. 81. ABC-Gibbs When parameter decomposed into θ = (θ1, . . . , θn) Algorithm 5 ABC-Gibbs sampler starting point θ(0) = (θ (0) 1 , . . . , θ (0) n ), observation xobs for i = 1, . . . , N do for j = 1, . . . , n do θ (i) j ∼ πεj (· | x , sj, θ (i) 1 , . . . , θ (i) j−1, θ (i−1) j+1 , . . . , θ (i−1) n ) end for end for Divide & conquer: one tolerance εj for each parameter θj one statistic sj for each parameter θj [Clart´e et al., 2019]
  82. 82. Compatibility When using ABC-Gibbs conditionals with different acceptance events, e.g., different statistics π(α)π(sα(µ) | α) and π(µ)f(sµ(x ) | α, µ). conditionals are incompatible ABC-Gibbs does not necessarily converge (even for tolerance equal to zero) potential limiting distribution not a genuine posterior (double use of data) unknown [except for a specific version] possibly far from genuine posterior(s) [Clart´e et al., 2016]
  83. 83. Compatibility When using ABC-Gibbs conditionals with different acceptance events, e.g., different statistics π(α)π(sα(µ) | α) and π(µ)f(sµ(x ) | α, µ). conditionals are incompatible ABC-Gibbs does not necessarily converge (even for tolerance equal to zero) potential limiting distribution not a genuine posterior (double use of data) unknown [except for a specific version] possibly far from genuine posterior(s) [Clart´e et al., 2016]
  84. 84. Convergence In hierarchical case n = 2, Theorem If there exists 0 < κ < 1/2 such that sup θ1,˜θ1 πε2 (· | x , s2, θ1) − πε2 (· | x , s2, ˜θ1) TV = κ ABC-Gibbs Markov chain geometrically converges in total variation to stationary distribution νε, with geometric rate 1 − 2κ.
  85. 85. Example 11: Hierarchical MA(2) Separation from the prior for identical number of simulations
  86. 86. Explicit limiting distribution For model xj | µj ∼ π(xj | µj) , µj | α i.i.d. ∼ π(µj | α) , α ∼ π(α) alternative ABC based on: ˜π(α, µ | x ) ∝ π(α)q(µ) generate a new µ π(˜µ | α)1η(sα(µ),sα(˜µ))<εα d˜µ × f(˜x | µ)π(x | µ)1η(sµ(x ),sµ(˜x))<εµ d˜x, with q arbitrary distribution on µ
  87. 87. Explicit limiting distribution For model xj | µj ∼ π(xj | µj) , µj | α i.i.d. ∼ π(µj | α) , α ∼ π(α) induces full conditionals ˜π(α | µ) ∝ π(α) π(˜µ | α)1η(sα(µ),sα(˜µ))<εα d˜x and ˜π(µ | α, x ) ∝ q(µ) π(˜µ | α)1η(sα(µ),sα(˜µ))<εα d˜µ × f(˜x | µ)π(x | µ)1η(sµ(x ),sµ(˜x))<εµ d˜x now compatible with new artificial joint
  88. 88. Explicit limiting distribution For model xj | µj ∼ π(xj | µj) , µj | α i.i.d. ∼ π(µj | α) , α ∼ π(α) that is, prior simulations of α ∼ π(α) and of ˜µ ∼ π(˜µ | α) until η(sα(µ), sα(˜µ)) < εα simulation of µ from instrumental q(µ) and of auxiliary variables ˜µ and ˜x until both constraints satisfied
  89. 89. Explicit limiting distribution For model xj | µj ∼ π(xj | µj) , µj | α i.i.d. ∼ π(µj | α) , α ∼ π(α) Resulting Gibbs sampler stationary for posterior proportional to π(α, µ) q(sα(µ)) projection f(sµ(x ) | µ) projection that is, for likelihood associated with sµ(x ) and prior distribution proportional to π(α, µ)q(sα(µ)) [exact!]
  90. 90. Gibbs with ABC approximations Souza Rodrigues et al. (arxiv:1906.04347) alternative ABC-ed Gibbs: Further references to earlier occurrences of Gibbs versions of ABC ABC version of Gibbs sampling with approximations to the conditionals with no further corrections Related to regression post-processing `a la Beaumont et al. (2002) used to designing approximate full conditional, possibly involving neural networks Requires preliminary ABC step Drawing from approximate full conditionals done exactly, possibly via a bootstrapped version.
  91. 91. Hierarchical models: toy example Model α ∼ U([0 ; 20]), (µ1, . . . , µn) | α ∼ N(α, 1)⊗n, (xi,1, . . . , xi,K) | µi ∼ N (µi, 0.1)⊗K . Numerical experiment n = 20, K = 10, Pseudo observation generated for α = 1.7, Algorithms runs for a constant budget: Ntot = N × Nε = 21000. We look at the estimates for µ1 whose value for the pseudo observations is 3.04.
  92. 92. Hierarchical models: toy example Illustration Assumptions of convergence theorem hold Tolerance as quantile of distances at each call, i..e, selection of simulation with smallest distance out of Nα = Nµ = 30 Summary statistic as empirical mean (sufficient) Setting when SMC-ABC fails as well
  93. 93. Hierarchical models: toy example Figure comparison of the sampled densities of µ1 (left) and α (right) [dot-dashed line as true posterior] 0 1 2 3 4 0 2 4 6 0.0 0.5 1.0 1.5 2.0 −4 −2 0 2 4 Method ABC Gibbs Simple ABC
  94. 94. Hierarchical models: toy example Figure posterior densities of µ1 and α for 100 replicas of 3 ABC algorithms [true marginal posterior as dashed line] ABC−GibbsSimpleABCSMC−ABC 0 2 4 6 0 1 2 3 4 0 1 2 3 4 0 10 20 30 40 mu1 density ABC−GibbsSimpleABCSMC−ABC 0 1 2 3 4 5 0.0 0.5 1.0 1.5 2.0 2.5 0.0 0.5 1.0 1.5 2.0 0 5 10 15 20 hyperparameter density
  95. 95. Hierarchical models: moving average example Pseudo observations xobs 1 generated for µ1 = (−0.06, −0.22). 0 1 2 3 −1.0 −0.5 0.0 0.5 1.0 value density type ABCGibbs ABCsimple prior 1st parameter, 1st coordinate −1.0 −0.5 0.0 0.5 1.0 −1.0 −0.5 0.0 0.5 1.0 b1 b2 0.2 0.4 0.6 0.8 level 1st parameter simple −1.0 −0.5 0.0 0.5 1.0 −1.0 −0.5 0.0 0.5 1.0 b1 b2 2.5 5.0 7.5 10.0 level 1st parameter gibbs Separation from the prior for identical number of simulations.
  96. 96. Hierarchical models: moving average example Real dataset measures of 8GHz daily flux intensity emitted by 7 stellar objects from the NRL GBI website: http://ese.nrl.navy.mil/. [?] 0 1 2 3 −1.0 −0.5 0.0 0.5 1.0 value density type ABCGibbs ABCsimple prior 1st parameter, 1st coordinate −1.0 −0.5 0.0 0.5 1.0 −1.0 −0.5 0.0 0.5 1.0 b1 b2 0.2 0.4 0.6 level 1st parameter simple −1.0 −0.5 0.0 0.5 1.0 −1.0 −0.5 0.0 0.5 1.0 b1 b2 2 4 6 8 level 1st parameter gibbs Separation from the prior for identical number of simulations.
  97. 97. Hierarchical models: g&k example Model g-and-k distribution (due to Tukey) defined through inverse cdf: easy to simulate but with no closed-form pdf: A + B 1 + 0.8 1 − exp(−gΦ−1(r) 1 + exp(−gΦ−1(r) 1 + Φ−1 (r)2 k Φ−1 (r) Note: MCMC feasible in this setting (Pierre Jacob) α A1 A2 ... An x1 x2 xn ... B g k
  98. 98. Hierarchical models: g&k example Assumption B, g and k known, inference on α and Ai solely. 1 2 3 4 Hyperparameter −7.5 −7.0 −6.5 −6.0 −5.5 −5.0−7.5 −7.0 −6.5 −6.0 −5.5 −5.0−7.5 −7.0 −6.5 −6.0 −5.5 −5.0−7.5 −7.0 −6.5 −6.0 −5.5 −5.0−7.5 −7.0 −6.5 −6.0 −5.5 −5.0 0 2 4 6 8 value density Method ABC Gibbs ABC−SMC vanilla ABC
  99. 99. General case: inverse problem Example inspired by deterministic inverse problems: difficult to tackle with traditional methods pseudo-likelihood extremely expensive to compute requiring the use of surrogate models Let y be solution of heat equation on a circle defined for (τ, z) ∈]0, T] × [0, 1[ by ∂τy(z, τ) = ∂z (θ(z)∂zy(z, τ)) , with θ(z) = n j=1 θj1[(j−1)/n,j/n](z) and boundary conditions y(z, 0) = y0(z) and y(0, τ) = y(1, τ) [?]
  100. 100. General case: inverse problem Assume y0 known and parameter θ = (θ1, . . . , θn), with discretized equation via first order finite elements of size 1/n for z and ∆ for τ. Stepwise approximation of solution ^y(z, t) = n j=1 yj,tϕj(z) where, for j < n, ϕj(z) = (1 − |nz − j|)1|nz−j|<1 and ϕn(z) = (1 − nz)10<z<1/n + (nz − n + 1)11−1/n<z<1 and with yj,t defined by yj,t+1 − yj,t 3∆ + yj+1,t+1 − yj+1,t 6∆ + yj−1,t+1 − yj−1,t 6∆ = yj,t+1(θj+1 + θj) − yj−1,t+1θj − yj+1,t+1θj+1.
  101. 101. General case: inverse problem In ABC-Gibbs, each parameter θm updated with summary statistics observations at locations m − 2, m − 1, m, m + 1 while ABC-SMC relies on the whole data as statistic. In experiment, n = 20 and ∆ = 0.1, with a prior θj ∼ U[0, 1], independently. Convergence theorem applies to this setting. Comparison with ABC, using same simulation budget, keeping the total number of simulations constant at Nε · N = 8 · 106. Choice of ABC reference table size critical: for a fixed computational budget reach balance between quality of the approximations of the conditionals (improved by increasing Nε), and Monte-Carlo error and convergence of the algorithm, (improved by increasing N).
  102. 102. General case: inverse problem Figure mean and variance of the ABC and ABC-Gibbs estimators of θ1 as Nε increases [horizontal line shows true value] 0.5 0.6 0.7 0.8 0 10 20 30 40 0.000 0.002 0.004 0 10 20 30 40 Method ABC Gibbs Simple ABC
  103. 103. General case: inverse problem Figure histograms of ABC-Gibbs and ABC outputs compared with uniform prior ABC Gibbs Simple ABC 0.25 0.50 0.75 1.00 0.25 0.50 0.75 1.00 0 1 2 3
  104. 104. ABC postdoc positions 2 post-doc positions with the ABSint research grant: Focus on approximate Bayesian techniques like ABC, variational Bayes, PAC-Bayes, Bayesian non-parametrics, scalable MCMC, and related topics. A potential direction of research would be the derivation of new Bayesian tools for model checking in such complex environments. Terms: up to 24 months, no teaching duty attached, primarily located in Universit´e Paris-Dauphine, with supported periods in Oxford (J. Rousseau) and visits to Montpellier (J.-M. Marin). No hard deadline. If interested, send application to me: bayesianstatistics@gmail.com

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