ABC Methods for Bayesian Model Choice                 ABC Methods for Bayesian Model Choice                               ...
ABC Methods for Bayesian Model ChoiceOutline                                        Approximate Bayesian computation      ...
ABC Methods for Bayesian Model Choice  Approximate Bayesian computationApproximate Bayesian computation Approximate Bayesi...
ABC Methods for Bayesian Model Choice  Approximate Bayesian computation     ABC basicsRegular Bayesian computation issues ...
ABC Methods for Bayesian Model Choice  Approximate Bayesian computation     ABC basicsUntractable likelihoods      Cases w...
ABC Methods for Bayesian Model Choice  Approximate Bayesian computation     ABC basicsIllustration      Example   Coalesce...
ABC Methods for Bayesian Model Choice  Approximate Bayesian computation     ABC basicsThe ABC method      Bayesian setting...
ABC Methods for Bayesian Model Choice  Approximate Bayesian computation     ABC basicsThe ABC method      Bayesian setting...
ABC Methods for Bayesian Model Choice  Approximate Bayesian computation     ABC basicsThe ABC method      Bayesian setting...
ABC Methods for Bayesian Model Choice  Approximate Bayesian computation     ABC basicsWhy does it work?!      The proof is...
ABC Methods for Bayesian Model Choice  Approximate Bayesian computation     ABC basicsA as approximative      When y is a ...
ABC Methods for Bayesian Model Choice  Approximate Bayesian computation     ABC basicsA as approximative      When y is a ...
ABC Methods for Bayesian Model Choice  Approximate Bayesian computation     ABC basicsABC algorithm      Algorithm 1 Likel...
ABC Methods for Bayesian Model Choice  Approximate Bayesian computation     ABC basicsOutput      The likelihood-free algo...
ABC Methods for Bayesian Model Choice  Approximate Bayesian computation     ABC basicsOutput      The likelihood-free algo...
ABC Methods for Bayesian Model Choice  Approximate Bayesian computation     ABC basicsMA example      Consider the MA(q) m...
ABC Methods for Bayesian Model Choice  Approximate Bayesian computation     ABC basicsMA example (2)      ABC algorithm th...
ABC Methods for Bayesian Model Choice  Approximate Bayesian computation     ABC basicsMA example (2)      ABC algorithm th...
ABC Methods for Bayesian Model Choice  Approximate Bayesian computation     ABC basicsComparison of distance impact      E...
ABC Methods for Bayesian Model Choice  Approximate Bayesian computation     ABC basicsComparison of distance impact       ...
ABC Methods for Bayesian Model Choice  Approximate Bayesian computation     ABC basicsComparison of distance impact       ...
ABC Methods for Bayesian Model Choice  Approximate Bayesian computation     ABC basicsABC advances      Simulating from th...
ABC Methods for Bayesian Model Choice  Approximate Bayesian computation     ABC basicsABC advances      Simulating from th...
ABC Methods for Bayesian Model Choice  Approximate Bayesian computation     ABC basicsABC advances      Simulating from th...
ABC Methods for Bayesian Model Choice  Approximate Bayesian computation     ABC basicsABC advances      Simulating from th...
ABC Methods for Bayesian Model Choice  Approximate Bayesian computation     Alphabet soupABC-MCMC      Markov chain (θ(t) ...
ABC Methods for Bayesian Model Choice  Approximate Bayesian computation     Alphabet soupABC-MCMC      Markov chain (θ(t) ...
ABC Methods for Bayesian Model Choice  Approximate Bayesian computation     Alphabet soupABC-MCMC (2)      Algorithm 2 Lik...
ABC Methods for Bayesian Model Choice  Approximate Bayesian computation     Alphabet soupWhy does it work?      Acceptance...
ABC Methods for Bayesian Model Choice  Approximate Bayesian computation     Alphabet soupWhich summary statistics?      Fu...
ABC Methods for Bayesian Model Choice  Approximate Bayesian computation     Alphabet soupWhich summary statistics?      Fu...
ABC Methods for Bayesian Model Choice  Approximate Bayesian computation     Alphabet soupWhich summary statistics?      Fu...
ABC Methods for Bayesian Model Choice  ABC for model choiceABC for model choice Approximate Bayesian computation ABC for m...
ABC Methods for Bayesian Model Choice  ABC for model choice     PrincipleBayesian model choice      Principle      Several...
ABC Methods for Bayesian Model Choice  ABC for model choice     PrincipleGeneric ABC for model choice      Algorithm 3 Lik...
ABC Methods for Bayesian Model Choice  ABC for model choice     PrincipleABC estimates      Posterior probability π(M = m|...
ABC Methods for Bayesian Model Choice  ABC for model choice     PrincipleABC estimates      Posterior probability π(M = m|...
ABC Methods for Bayesian Model Choice  ABC for model choice     PrincipleThe great ABC controversy      On-going controver...
ABC Methods for Bayesian Model Choice  ABC for model choice     PrincipleThe great ABC controversy     On-going controvery...
ABC Methods for Bayesian Model Choice  ABC for model choice     Gibbs random fieldsPotts model         Skip MRFs      Potts...
ABC Methods for Bayesian Model Choice  ABC for model choice     Gibbs random fieldsPotts model         Skip MRFs      Potts...
ABC Methods for Bayesian Model Choice  ABC for model choice     Gibbs random fieldsNeighbourhood relations      Setup      ...
ABC Methods for Bayesian Model Choice  ABC for model choice     Gibbs random fieldsModel index      Computational target:  ...
ABC Methods for Bayesian Model Choice  ABC for model choice     Gibbs random fieldsModel index      Computational target:  ...
ABC Methods for Bayesian Model Choice  ABC for model choice     Gibbs random fieldsSufficient statistics in Gibbs random field...
ABC Methods for Bayesian Model Choice  ABC for model choice     Gibbs random fieldsSufficient statistics in Gibbs random field...
ABC Methods for Bayesian Model Choice  ABC for model choice     Gibbs random fieldsToy example      iid Bernoulli model ver...
ABC Methods for Bayesian Model Choice  ABC for model choice     Gibbs random fieldsToy example (2)                         ...
ABC Methods for Bayesian Model Choice  ABC for model choice     Generic ABC model choiceMore about sufficiency      If η1 (x...
ABC Methods for Bayesian Model Choice  ABC for model choice     Generic ABC model choiceMore about sufficiency      If η1 (x...
ABC Methods for Bayesian Model Choice  ABC for model choice     Generic ABC model choiceLimiting behaviour of B12 (T → ∞) ...
ABC Methods for Bayesian Model Choice  ABC for model choice     Generic ABC model choiceLimiting behaviour of B12 (T → ∞) ...
ABC Methods for Bayesian Model Choice  ABC for model choice     Generic ABC model choiceLimiting behaviour of B12 ( → 0)  ...
ABC Methods for Bayesian Model Choice  ABC for model choice     Generic ABC model choiceLimiting behaviour of B12 ( → 0)  ...
ABC Methods for Bayesian Model Choice  ABC for model choice     Generic ABC model choiceLimiting behaviour of B12 (under s...
ABC Methods for Bayesian Model Choice  ABC for model choice     Generic ABC model choiceLimiting behaviour of B12 (under s...
ABC Methods for Bayesian Model Choice  ABC for model choice     Generic ABC model choicePoisson/geometric example      Sam...
ABC Methods for Bayesian Model Choice  ABC for model choice     Generic ABC model choicePoisson/geometric discrepancy     ...
ABC Methods for Bayesian Model Choice  ABC for model choice     Generic ABC model choiceFormal recovery      Creating an e...
ABC Methods for Bayesian Model Choice  ABC for model choice     Generic ABC model choiceFormal recovery      Creating an e...
ABC Methods for Bayesian Model Choice  ABC for model choice     Generic ABC model choiceFormal recovery      Creating an e...
ABC Methods for Bayesian Model Choice  ABC for model choice     Generic ABC model choiceMeaning of the ABC-Bayes factor   ...
ABC Methods for Bayesian Model Choice  ABC for model choice     Generic ABC model choiceMeaning of the ABC-Bayes factor   ...
ABC Methods for Bayesian Model Choice  ABC for model choice     Generic ABC model choiceMA example                        ...
ABC Methods for Bayesian Model Choice  ABC for model choice     Generic ABC model choiceMA example                        ...
ABC Methods for Bayesian Model Choice  ABC for model choice     Generic ABC model choiceA population genetics evaluation  ...
ABC Methods for Bayesian Model Choice  ABC for model choice     Generic ABC model choiceA population genetics evaluation  ...
ABC Methods for Bayesian Model Choice  ABC for model choice     Generic ABC model choiceStability of importance sampling  ...
ABC Methods for Bayesian Model Choice  ABC for model choice     Generic ABC model choiceComparison with ABC      Use of 24...
ABC Methods for Bayesian Model Choice  ABC for model choice     Generic ABC model choiceComparison with ABC      Use of 15...
ABC Methods for Bayesian Model Choice  ABC for model choice     Generic ABC model choiceComparison with ABC      Use of 15...
ABC Methods for Bayesian Model Choice  ABC for model choice     Generic ABC model choiceThe only safe cases???      Beside...
ABC Methods for Bayesian Model Choice  ABC for model choice     Generic ABC model choiceThe only safe cases???      Beside...
ABC Methods for Bayesian Model Choice  Model choice consistencyABC model choice consistency Approximate Bayesian computati...
ABC Methods for Bayesian Model Choice  Model choice consistency     Formalised frameworkThe starting point      Central qu...
ABC Methods for Bayesian Model Choice  Model choice consistency     Formalised frameworkThe starting point      Central qu...
ABC Methods for Bayesian Model Choice  Model choice consistency     Formalised frameworkA benchmark if toy example      Co...
ABC Methods for Bayesian Model Choice  Model choice consistency     Formalised frameworkA benchmark if toy example      Co...
ABC Methods for Bayesian Model Choice  Model choice consistency     Formalised frameworkA benchmark if toy example      Co...
ABC Methods for Bayesian Model Choice  Model choice consistency     Formalised frameworkA benchmark if toy example      Co...
ABC Methods for Bayesian Model Choice  Model choice consistency     Formalised frameworkA benchmark if toy example      Co...
ABC Methods for Bayesian Model Choice  Model choice consistency     Formalised frameworkA benchmark if toy example      Co...
ABC Methods for Bayesian Model Choice  Model choice consistency     Formalised frameworkFramework      Starting from sampl...
ABC Methods for Bayesian Model Choice  Model choice consistency     Formalised frameworkFramework       c Comparison of   ...
ABC Methods for Bayesian Model Choice  Model choice consistency     Consistency resultsAssumptions      A collection of as...
ABC Methods for Bayesian Model Choice  Model choice consistency     Consistency resultsAssumptions      A collection of as...
ABC Methods for Bayesian Model Choice  Model choice consistency     Consistency resultsAssumptions      A collection of as...
ABC Methods for Bayesian Model Choice  Model choice consistency     Consistency resultsAssumptions      A collection of as...
ABC Methods for Bayesian Model Choice  Model choice consistency     Consistency resultsAssumptions      A collection of as...
ABC Methods for Bayesian Model Choice  Model choice consistency     Consistency resultsAssumptions      A collection of as...
ABC Methods for Bayesian Model Choice  Model choice consistency     Consistency resultsEffective dimension      Understandi...
ABC Methods for Bayesian Model Choice  Model choice consistency     Consistency resultsAsymptotic marginals      Asymptoti...
ABC Methods for Bayesian Model Choice  Model choice consistency     Consistency resultsWithin-model consistency      Under...
ABC Methods for Bayesian Model Choice  Model choice consistency     Consistency resultsWithin-model consistency      Under...
ABC Methods for Bayesian Model Choice  Model choice consistency     Consistency resultsBetween-model consistency      Cons...
ABC Methods for Bayesian Model Choice  Model choice consistency     Consistency resultsBetween-model consistency      Cons...
ABC Methods for Bayesian Model Choice  Model choice consistency     Consistency resultsBetween-model consistency      Cons...
ABC Methods for Bayesian Model Choice  Model choice consistency     Consistency resultsConsistency theorem      If        ...
ABC Methods for Bayesian Model Choice  Model choice consistency     Consistency resultsConsistency theorem      If Pn belo...
ABC Methods for Bayesian Model Choice  Model choice consistency     Summary statisticsConsequences on summary statistics  ...
ABC Methods for Bayesian Model Choice  Model choice consistency     Summary statisticsConsequences on summary statistics  ...
ABC Methods for Bayesian Model Choice  Model choice consistency     Summary statisticsToy example: Laplace versus Gauss [1...
ABC Methods for Bayesian Model Choice  Model choice consistency     Summary statisticsToy example: Laplace versus Gauss [1...
ABC Methods for Bayesian Model Choice  Model choice consistency     Summary statisticsToy example: Laplace versus Gauss [1...
ABC Methods for Bayesian Model Choice  Model choice consistency     Summary statisticsToy example: Laplace versus Gauss [1...
ABC Methods for Bayesian Model Choice  Model choice consistency     Summary statisticsToy example: Laplace versus Gauss [1...
ABC Methods for Bayesian Model Choice  Model choice consistency     Summary statisticsToy example: Laplace versus Gauss [0...
ABC Methods for Bayesian Model Choice  Model choice consistency     Summary statisticsToy example: Laplace versus Gauss [0...
ABC Methods for Bayesian Model Choice  Model choice consistency     Summary statisticsToy example: Laplace versus Gauss [0...
ABC Methods for Bayesian Model Choice  Model choice consistency     Summary statisticsChecking for adequate statistics    ...
ABC Methods for Bayesian Model Choice  Model choice consistency     Summary statisticsChecking for adequate statistics    ...
ABC Methods for Bayesian Model Choice  Model choice consistency     Summary statisticsA population genetic illustration   ...
ABC Methods for Bayesian Model Choice  Model choice consistency     Summary statisticsA population genetic illustration   ...
ABC Methods for Bayesian Model Choice  Model choice consistency     Summary statisticsA population genetic illustration   ...
ABC Methods for Bayesian Model Choice  Model choice consistency     Summary statisticsA population genetic illustration   ...
ABC Methods for Bayesian Model Choice  Model choice consistency     Summary statisticsA population genetic illustration   ...
ABC Methods for Bayesian Model Choice  Model choice consistency     Summary statisticsA population genetic illustration   ...
ABC model choice
ABC model choice
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ABC model choice
ABC model choice
ABC model choice
ABC model choice
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ABC model choice
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ABC model choice

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talk at the Confronting Intractability workshop in Bristol, April 18, 2012

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Transcript of "ABC model choice"

  1. 1. ABC Methods for Bayesian Model Choice ABC Methods for Bayesian Model Choice Christian P. Robert Universit´ Paris-Dauphine, IuF, & CREST e http://www.ceremade.dauphine.fr/~xian Confronting intractability in Bristol Joint work with J.-M. Cornuet, A. Grelaud, J.-M. Marin, N. Pillai, & J. Rousseau
  2. 2. ABC Methods for Bayesian Model ChoiceOutline Approximate Bayesian computation ABC for model choice Model choice consistency
  3. 3. ABC Methods for Bayesian Model Choice Approximate Bayesian computationApproximate Bayesian computation Approximate Bayesian computation ABC basics Alphabet soup ABC for model choice Model choice consistency
  4. 4. ABC Methods for Bayesian Model Choice Approximate Bayesian computation ABC basicsRegular Bayesian computation issues When faced with a non-standard posterior distribution π(θ|y) ∝ π(θ)L(θ|y) the standard solution is to use simulation (Monte Carlo) to produce a sample θ1 , . . . , θ T from π(θ|y) (or approximately by Markov chain Monte Carlo methods) [Robert & Casella, 2004]
  5. 5. ABC Methods for Bayesian Model Choice 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!
  6. 6. ABC Methods for Bayesian Model Choice Approximate Bayesian computation ABC basicsIllustration Example Coalescence 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]
  7. 7. ABC Methods for Bayesian Model Choice Approximate Bayesian computation ABC basicsThe ABC method Bayesian setting: target is π(θ)f (x|θ)
  8. 8. ABC Methods for Bayesian Model Choice 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:
  9. 9. ABC Methods for Bayesian Model Choice 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
  10. 10. ABC Methods for Bayesian Model Choice 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]
  11. 11. ABC Methods for Bayesian Model Choice Approximate Bayesian computation ABC basicsA as approximative When y is a continuous random variable, equality z = y is replaced with a tolerance condition, (y, z) ≤ where is a distance
  12. 12. ABC Methods for Bayesian Model Choice Approximate Bayesian computation ABC basicsA as approximative 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) < )
  13. 13. ABC Methods for Bayesian Model Choice Approximate Bayesian computation ABC basicsABC algorithm Algorithm 1 Likelihood-free rejection sampler 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 (maybe in-sufficient) statistic
  14. 14. ABC Methods for Bayesian Model Choice 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)) < }.
  15. 15. ABC Methods for Bayesian Model Choice 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) .
  16. 16. ABC Methods for Bayesian Model Choice Approximate Bayesian computation ABC basicsMA example Consider 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)
  17. 17. ABC Methods for Bayesian Model Choice Approximate Bayesian computation ABC basicsMA example (2) ABC algorithm thus made of 1. picking a new value (ϑ1 , ϑ2 ) in the triangle 2. generating an iid sequence ( t )−q<t≤T 3. producing a simulated series (xt )1≤t≤T
  18. 18. ABC Methods for Bayesian Model Choice Approximate Bayesian computation ABC basicsMA example (2) ABC algorithm thus made of 1. picking a new value (ϑ1 , ϑ2 ) in the triangle 2. generating an iid sequence ( t )−q<t≤T 3. producing a simulated series (xt )1≤t≤T Distance: basic distance between the series T ρ((xt )1≤t≤T , (xt )1≤t≤T ) = (xt − xt )2 t=1 or between summary statistics like the first q autocorrelations T τj = xt xt−j t=j+1
  19. 19. ABC Methods for Bayesian Model Choice 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
  20. 20. ABC Methods for Bayesian Model Choice 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
  21. 21. ABC Methods for Bayesian Model Choice 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
  22. 22. ABC Methods for Bayesian Model Choice Approximate Bayesian computation ABC basicsABC advances Simulating from the prior is often poor in efficiency
  23. 23. ABC Methods for Bayesian Model Choice 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]
  24. 24. ABC Methods for Bayesian Model Choice 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]
  25. 25. ABC Methods for Bayesian Model Choice 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]
  26. 26. ABC Methods for Bayesian Model Choice 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,
  27. 27. ABC Methods for Bayesian Model Choice 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]
  28. 28. ABC Methods for Bayesian Model Choice 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
  29. 29. ABC Methods for Bayesian Model Choice Approximate Bayesian computation Alphabet soupWhy does it work? Acceptance probability that does not involve the calculation of the likelihood and π (θ , z |y) Kω (θ(t−1) |θ )f (z(t−1) |θ(t−1) ) × π (θ(t−1) , z(t−1) |y) Kω (θ |θ(t−1) )f (z |θ ) π(θ ) f (z |θ ) IA ,y (z ) = (t−1) ) f (z(t−1) |θ (t−1) )I (t−1) ) π(θ A ,y (z Kω (θ(t−1) |θ ) f (z(t−1) |θ(t−1) ) × Kω (θ |θ(t−1) ) f (z |θ ) π(θ )Kω (θ(t−1) |θ ) = IA (z ) . π(θ(t−1) Kω (θ |θ(t−1) ) ,y
  30. 30. ABC Methods for Bayesian Model Choice Approximate Bayesian computation Alphabet soupWhich summary statistics? Fundamental difficulty of the choice of the summary statistic when there is no non-trivial sufficient statistic [except when done by the experimenters in the field]
  31. 31. ABC Methods for Bayesian Model Choice Approximate Bayesian computation Alphabet soupWhich summary statistics? Fundamental difficulty of the choice of the summary statistic when there is no non-trivial sufficient statistic [except when done by the experimenters in the field] Starting from a large collection of summary statistics is available, Joyce and Marjoram (2008) consider the sequential inclusion into the ABC target, with a stopping rule based on a likelihood ratio test.
  32. 32. ABC Methods for Bayesian Model Choice Approximate Bayesian computation Alphabet soupWhich summary statistics? Fundamental difficulty of the choice of the summary statistic when there is no non-trivial sufficient statistic [except when done by the experimenters in the field] Based on decision-theoretic principles, Fearnhead and Prangle (2012) end up with E[θ|y] as the optimal summary statistic
  33. 33. ABC Methods for Bayesian Model Choice ABC for model choiceABC for model choice Approximate Bayesian computation ABC for model choice Principle Gibbs random fields Generic ABC model choice Model choice consistency
  34. 34. ABC Methods for Bayesian Model Choice ABC for model choice PrincipleBayesian model choice Principle Several models M1 , M2 , . . . are considered simultaneously for dataset y and model index M central to inference. Use of a prior π(M = m), plus a prior distribution on the parameter conditional on the value m of the model index, πm (θ m ) Goal is to derive the posterior distribution of M, π(M = m|data) a challenging computational target when models are complex.
  35. 35. ABC Methods for Bayesian Model Choice ABC for model choice PrincipleGeneric ABC for model choice Algorithm 3 Likelihood-free model choice sampler (ABC-MC) for t = 1 to T do repeat Generate m from the prior π(M = m) Generate θ m from the prior πm (θ m ) Generate z from the model fm (z|θ m ) until ρ{η(z), η(y)} < Set m(t) = m and θ (t) = θ m end for [Grelaud & al., 2009; Toni & al., 2009]
  36. 36. ABC Methods for Bayesian Model Choice ABC for model choice PrincipleABC estimates Posterior probability π(M = m|y) approximated by the frequency of acceptances from model m T 1 Im(t) =m . T t=1
  37. 37. ABC Methods for Bayesian Model Choice ABC for model choice PrincipleABC estimates Posterior probability π(M = m|y) approximated by the frequency of acceptances from model m T 1 Im(t) =m . T t=1 Extension to a weighted polychotomous logistic regression estimate of π(M = m|y), with non-parametric kernel weights [Cornuet et al., DIYABC, 2009]
  38. 38. ABC Methods for Bayesian Model Choice ABC for model choice PrincipleThe great ABC controversy On-going controvery in phylogeographic genetics about the validity of using ABC for testing Against: Templeton, 2008, 2009, 2010a, 2010b, 2010c, &tc argues that nested hypotheses cannot have higher probabilities than nesting hypotheses (!)
  39. 39. ABC Methods for Bayesian Model Choice ABC for model choice PrincipleThe great ABC controversy On-going controvery in phylogeographic genetics about the validity of using ABC for testing Replies: Fagundes et al., 2008, Against: Templeton, 2008, Beaumont et al., 2010, Berger et 2009, 2010a, 2010b, 2010c, &tc al., 2010, Csill`ry et al., 2010 e argues that nested hypotheses point out that the criticisms are cannot have higher probabilities addressed at [Bayesian] than nesting hypotheses (!) model-based inference and have nothing to do with ABC...
  40. 40. ABC Methods for Bayesian Model Choice ABC for model choice Gibbs random fieldsPotts model Skip MRFs Potts model Distribution with an energy function of the form θS(y) = θ δyl =yi l∼i where l∼i denotes a neighbourhood structure
  41. 41. ABC Methods for Bayesian Model Choice ABC for model choice Gibbs random fieldsPotts model Skip MRFs Potts model Distribution with an energy function of the form θS(y) = θ δyl =yi l∼i where l∼i denotes a neighbourhood structure In most realistic settings, summation Zθ = exp{θS(x)} x∈X involves too many terms to be manageable and numerical approximations cannot always be trusted
  42. 42. ABC Methods for Bayesian Model Choice ABC for model choice Gibbs random fieldsNeighbourhood relations Setup Choice to be made between M neighbourhood relations m i∼i (0 ≤ m ≤ M − 1) with Sm (x) = I{xi =xi } m i∼i driven by the posterior probabilities of the models.
  43. 43. ABC Methods for Bayesian Model Choice ABC for model choice Gibbs random fieldsModel index Computational target: P(M = m|x) ∝ fm (x|θm )πm (θm ) dθm π(M = m) Θm
  44. 44. ABC Methods for Bayesian Model Choice ABC for model choice Gibbs random fieldsModel index Computational target: P(M = m|x) ∝ fm (x|θm )πm (θm ) dθm π(M = m) Θm If S(x) sufficient statistic for the joint parameters (M, θ0 , . . . , θM −1 ), P(M = m|x) = P(M = m|S(x)) .
  45. 45. ABC Methods for Bayesian Model Choice ABC for model choice Gibbs random fieldsSufficient statistics in Gibbs random fields Each model m has its own sufficient statistic Sm (·) and S(·) = (S0 (·), . . . , SM −1 (·)) is also (model-)sufficient.
  46. 46. ABC Methods for Bayesian Model Choice ABC for model choice Gibbs random fieldsSufficient statistics in Gibbs random fields Each model m has its own sufficient statistic Sm (·) and S(·) = (S0 (·), . . . , SM −1 (·)) is also (model-)sufficient. Explanation: For Gibbs random fields, 1 2 x|M = m ∼ fm (x|θm ) = fm (x|S(x))fm (S(x)|θm ) 1 = f 2 (S(x)|θm ) n(S(x)) m where n(S(x)) = {˜ ∈ X : S(˜ ) = S(x)} x x c S(x) is sufficient for the joint parameters
  47. 47. ABC Methods for Bayesian Model Choice ABC for model choice Gibbs random fieldsToy example iid Bernoulli model versus two-state first-order Markov chain, i.e. n f0 (x|θ0 ) = exp θ0 I{xi =1} {1 + exp(θ0 )}n , i=1 versus n 1 f1 (x|θ1 ) = exp θ1 I{xi =xi−1 } {1 + exp(θ1 )}n−1 , 2 i=2 with priors θ0 ∼ U(−5, 5) and θ1 ∼ U(0, 6) (inspired by “phase transition” boundaries).
  48. 48. ABC Methods for Bayesian Model Choice ABC for model choice Gibbs random fieldsToy example (2) 10 5 5 BF01 BF01 0 ^ ^ 0 −5 −5 −40 −20 0 10 −10 −40 −20 0 10 BF01 BF01 (left) Comparison of the true BF m0 /m1 (x0 ) with BF m0 /m1 (x0 ) (in logs) over 2, 000 simulations and 4.106 proposals from the prior. (right) Same when using tolerance corresponding to the 1% quantile on the distances.
  49. 49. ABC Methods for Bayesian Model Choice ABC for model choice Generic ABC model choiceMore about sufficiency If η1 (x) sufficient statistic for model m = 1 and parameter θ1 and η2 (x) sufficient statistic for model m = 2 and parameter θ2 , (η1 (x), η2 (x)) is not always sufficient for (m, θm )
  50. 50. ABC Methods for Bayesian Model Choice ABC for model choice Generic ABC model choiceMore about sufficiency If η1 (x) sufficient statistic for model m = 1 and parameter θ1 and η2 (x) sufficient statistic for model m = 2 and parameter θ2 , (η1 (x), η2 (x)) is not always sufficient for (m, θm ) c Potential loss of information at the testing level
  51. 51. ABC Methods for Bayesian Model Choice ABC for model choice Generic ABC model choiceLimiting behaviour of B12 (T → ∞) ABC approximation T t=1 Imt =1 Iρ{η(zt ),η(y)}≤ B12 (y) = T , t=1 Imt =2 Iρ{η(zt ),η(y)}≤ where the (mt , z t )’s are simulated from the (joint) prior
  52. 52. ABC Methods for Bayesian Model Choice ABC for model choice Generic ABC model choiceLimiting behaviour of B12 (T → ∞) ABC approximation T t=1 Imt =1 Iρ{η(zt ),η(y)}≤ B12 (y) = T , t=1 Imt =2 Iρ{η(zt ),η(y)}≤ where the (mt , z t )’s are simulated from the (joint) prior As T go to infinity, limit Iρ{η(z),η(y)}≤ π1 (θ 1 )f1 (z|θ 1 ) dz dθ 1 B12 (y) = Iρ{η(z),η(y)}≤ π2 (θ 2 )f2 (z|θ 2 ) dz dθ 2 η Iρ{η,η(y)}≤ π1 (θ 1 )f1 (η|θ 1 ) dη dθ 1 = η , Iρ{η,η(y)}≤ π2 (θ 2 )f2 (η|θ 2 ) dη dθ 2 η η where f1 (η|θ 1 ) and f2 (η|θ 2 ) distributions of η(z)
  53. 53. ABC Methods for Bayesian Model Choice ABC for model choice Generic ABC model choiceLimiting behaviour of B12 ( → 0) When goes to zero, η η π1 (θ 1 )f1 (η(y)|θ 1 ) dθ 1 B12 (y) = η π2 (θ 2 )f2 (η(y)|θ 2 ) dθ 2
  54. 54. ABC Methods for Bayesian Model Choice ABC for model choice Generic ABC model choiceLimiting behaviour of B12 ( → 0) When goes to zero, η η π1 (θ 1 )f1 (η(y)|θ 1 ) dθ 1 B12 (y) = η π2 (θ 2 )f2 (η(y)|θ 2 ) dθ 2 c Bayes factor based on the sole observation of η(y)
  55. 55. ABC Methods for Bayesian Model Choice ABC for model choice Generic ABC model choiceLimiting behaviour of B12 (under sufficiency) If η(y) sufficient statistic in both models, fi (y|θ i ) = gi (y)fiη (η(y)|θ i ) Thus η Θ1 π(θ 1 )g1 (y)f1 (η(y)|θ 1 ) dθ 1 B12 (y) = η Θ2 π(θ 2 )g2 (y)f2 (η(y)|θ 2 ) dθ 2 η g1 (y) π1 (θ 1 )f1 (η(y)|θ 1 ) dθ 1 g1 (y) η = η = B (y) . g2 (y) π2 (θ 2 )f2 (η(y)|θ 2 ) dθ 2 g2 (y) 12 [Didelot, Everitt, Johansen & Lawson, 2011]
  56. 56. ABC Methods for Bayesian Model Choice ABC for model choice Generic ABC model choiceLimiting behaviour of B12 (under sufficiency) If η(y) sufficient statistic in both models, fi (y|θ i ) = gi (y)fiη (η(y)|θ i ) Thus η Θ1 π(θ 1 )g1 (y)f1 (η(y)|θ 1 ) dθ 1 B12 (y) = η Θ2 π(θ 2 )g2 (y)f2 (η(y)|θ 2 ) dθ 2 η g1 (y) π1 (θ 1 )f1 (η(y)|θ 1 ) dθ 1 g1 (y) η = η = B (y) . g2 (y) π2 (θ 2 )f2 (η(y)|θ 2 ) dθ 2 g2 (y) 12 [Didelot, Everitt, Johansen & Lawson, 2011] c No discrepancy only when cross-model sufficiency
  57. 57. ABC Methods for Bayesian Model Choice ABC for model choice Generic ABC model choicePoisson/geometric example Sample x = (x1 , . . . , xn ) from either a Poisson P(λ) or from a geometric G(p) Sum n S= xi = η(x) i=1 sufficient statistic for either model but not simultaneously Discrepancy ratio g1 (x) S!n−S / i xi ! = g2 (x) 1 n+S−1 S
  58. 58. ABC Methods for Bayesian Model Choice ABC for model choice Generic ABC model choicePoisson/geometric discrepancy η Range of B12 (x) versus B12 (x): The values produced have nothing in common.
  59. 59. ABC Methods for Bayesian Model Choice ABC for model choice Generic ABC model choiceFormal recovery Creating an encompassing exponential family T T f (x|θ1 , θ2 , α1 , α2 ) ∝ exp{θ1 η1 (x) + θ2 η2 (x) + α1 t1 (x) + α2 t2 (x)} leads to a sufficient statistic (η1 (x), η2 (x), t1 (x), t2 (x)) [Didelot, Everitt, Johansen & Lawson, 2011]
  60. 60. ABC Methods for Bayesian Model Choice ABC for model choice Generic ABC model choiceFormal recovery Creating an encompassing exponential family T T f (x|θ1 , θ2 , α1 , α2 ) ∝ exp{θ1 η1 (x) + θ2 η2 (x) + α1 t1 (x) + α2 t2 (x)} leads to a sufficient statistic (η1 (x), η2 (x), t1 (x), t2 (x)) [Didelot, Everitt, Johansen & Lawson, 2011] In the Poisson/geometric case, if i xi ! is added to S, no discrepancy
  61. 61. ABC Methods for Bayesian Model Choice ABC for model choice Generic ABC model choiceFormal recovery Creating an encompassing exponential family T T f (x|θ1 , θ2 , α1 , α2 ) ∝ exp{θ1 η1 (x) + θ2 η2 (x) + α1 t1 (x) + α2 t2 (x)} leads to a sufficient statistic (η1 (x), η2 (x), t1 (x), t2 (x)) [Didelot, Everitt, Johansen & Lawson, 2011] Only applies in genuine sufficiency settings... c Inability to evaluate information loss due to summary statistics
  62. 62. ABC Methods for Bayesian Model Choice ABC for model choice Generic ABC model choiceMeaning of the ABC-Bayes factor The ABC approximation to the Bayes Factor is based solely on the summary statistics....
  63. 63. ABC Methods for Bayesian Model Choice ABC for model choice Generic ABC model choiceMeaning of the ABC-Bayes factor The ABC approximation to the Bayes Factor is based solely on the summary statistics.... In the Poisson/geometric case, if E[yi ] = θ0 > 0, η (θ0 + 1)2 −θ0 lim B12 (y) = e n→∞ θ0
  64. 64. ABC Methods for Bayesian Model Choice ABC for model choice Generic ABC model choiceMA example 0.7 0.6 0.6 0.6 0.6 0.5 0.5 0.5 0.5 0.4 0.4 0.4 0.4 0.3 0.3 0.3 0.3 0.2 0.2 0.2 0.2 0.1 0.1 0.1 0.1 0.0 0.0 0.0 0.0 1 2 1 2 1 2 1 2 Evolution [against ] of ABC Bayes factor, in terms of frequencies of visits to models MA(1) (left) and MA(2) (right) when equal to 10, 1, .1, .01% quantiles on insufficient autocovariance distances. Sample of 50 points from a MA(2) with θ1 = 0.6, θ2 = 0.2. True Bayes factor equal to 17.71.
  65. 65. ABC Methods for Bayesian Model Choice ABC for model choice Generic ABC model choiceMA example 0.8 0.6 0.6 0.6 0.6 0.4 0.4 0.4 0.4 0.2 0.2 0.2 0.2 0.0 0.0 0.0 0.0 1 2 1 2 1 2 1 2 Evolution [against ] of ABC Bayes factor, in terms of frequencies of visits to models MA(1) (left) and MA(2) (right) when equal to 10, 1, .1, .01% quantiles on insufficient autocovariance distances. Sample of 50 points from a MA(1) model with θ1 = 0.6. True Bayes factor B21 equal to .004.
  66. 66. ABC Methods for Bayesian Model Choice ABC for model choice Generic ABC model choiceA population genetics evaluation Gene free Population genetics example with 3 populations 2 scenari 15 individuals 5 loci single mutation parameter
  67. 67. ABC Methods for Bayesian Model Choice ABC for model choice Generic ABC model choiceA population genetics evaluation Gene free Population genetics example with 3 populations 2 scenari 15 individuals 5 loci single mutation parameter 24 summary statistics 2 million ABC proposal importance [tree] sampling alternative
  68. 68. ABC Methods for Bayesian Model Choice ABC for model choice Generic ABC model choiceStability of importance sampling 1.0 1.0 1.0 1.0 1.0 q q 0.8 0.8 0.8 0.8 0.8 0.6 0.6 0.6 0.6 0.6 q 0.4 0.4 0.4 0.4 0.4 0.2 0.2 0.2 0.2 0.2 0.0 0.0 0.0 0.0 0.0
  69. 69. ABC Methods for Bayesian Model Choice ABC for model choice Generic ABC model choiceComparison with ABC Use of 24 summary statistics and DIY-ABC logistic correction 1.0 qq qq qq qq qqq qq q q qq q q qq q q q q q q q q q q qq q q qq q q q qq qq qq q q q q qq q q q q q q q q q q q qq q q q qq q q 0.8 q q q q q q q q q q q q qq q q q q q q q qq q q q q q qqq q q qq q ABC direct and logistic q q qq q q q q q q q 0.6 q q q qq q q qq q q q q q q q q q q q q q q q q q q q q q q q q q qq qq q q qq 0.4 qq q q q q q q q q q q q q q q q q q 0.2 q qq q q q q q q q q q q 0.0 qq 0.0 0.2 0.4 0.6 0.8 1.0 importance sampling
  70. 70. ABC Methods for Bayesian Model Choice ABC for model choice Generic ABC model choiceComparison with ABC Use of 15 summary statistics 6 4 q q q q q q q qq 2 q q q q qq ABC direct q q q qq q qq q q q q qq qq q qq q q q q q qq q q q q q qq q q qq q qq q q qq q q q qqq q q q q q q q q 0 qq qq q q q q qq qq q q qq q q q q q q q q −2 −4 −4 −2 0 2 4 6 importance sampling
  71. 71. ABC Methods for Bayesian Model Choice ABC for model choice Generic ABC model choiceComparison with ABC Use of 15 summary statistics and DIY-ABC logistic correction qq 6 q q q q q q q q q q q 4 q qq q qq qq q q q q q q q q q q qq q q q q q q q ABC direct and logistic q q q q q q q qq q q q q q q q qq q 2 q q q q q qq q qq q q q qq qq q qq qq q q q qq q q qq qq qq q q qqqq qq q q q qqqqq qq q q q qq q q q q qq q qq q q qqq q q qqqqq q q qq q q qq q q q qq 0 qq qqq q q qq q qqq q q q q q q q q q q q q q q q q q q q q qq q q −2 q q q q q −4 q q q −4 −2 0 2 4 6 importance sampling
  72. 72. ABC Methods for Bayesian Model Choice ABC for model choice Generic ABC model choiceThe only safe cases??? Besides specific models like Gibbs random fields, using distances over the data itself escapes the discrepancy... [Toni & Stumpf, 2010; Sousa & al., 2009]
  73. 73. ABC Methods for Bayesian Model Choice ABC for model choice Generic ABC model choiceThe only safe cases??? Besides specific models like Gibbs random fields, using distances over the data itself escapes the discrepancy... [Toni & Stumpf, 2010; Sousa & al., 2009] ...and so does the use of more informal model fitting measures [Ratmann & al., 2009]
  74. 74. ABC Methods for Bayesian Model Choice Model choice consistencyABC model choice consistency Approximate Bayesian computation ABC for model choice Model choice consistency Formalised framework Consistency results Summary statistics Conclusions
  75. 75. ABC Methods for Bayesian Model Choice Model choice consistency Formalised frameworkThe starting point Central question to the validation of ABC for model choice: When is a Bayes factor based on an insufficient statistic T (y) consistent?
  76. 76. ABC Methods for Bayesian Model Choice Model choice consistency Formalised frameworkThe starting point Central question to the validation of ABC for model choice: When is a Bayes factor based on an insufficient statistic T (y) consistent? T Note: c drawn on T (y) through B12 (y) necessarily differs from c drawn on y through B12 (y)
  77. 77. ABC Methods for Bayesian Model Choice Model choice consistency Formalised frameworkA benchmark if toy example Comparison suggested by referee of PNAS paper [thanks]: [X, Cornuet, Marin, & Pillai, Aug. 2011] Model M1 : y ∼ N (θ1 , 1) opposed √ to model M2 : y ∼ L(θ√ 1/ 2), Laplace distribution with mean θ2 2, and scale parameter 1/ 2 (variance one).
  78. 78. ABC Methods for Bayesian Model Choice Model choice consistency Formalised frameworkA benchmark if toy example Comparison suggested by referee of PNAS paper [thanks]: [X, Cornuet, Marin, & Pillai, Aug. 2011] Model M1 : y ∼ N (θ1 , 1) opposed √ to model M2 : y ∼ L(θ√ 1/ 2), Laplace distribution with mean θ2 2, and scale parameter 1/ 2 (variance one). Four possible statistics 1. sample mean y (sufficient for M1 if not M2 ); 2. sample median med(y) (insufficient); 3. sample variance var(y) (ancillary); 4. median absolute deviation mad(y) = med(y − med(y));
  79. 79. ABC Methods for Bayesian Model Choice Model choice consistency Formalised frameworkA benchmark if toy example Comparison suggested by referee of PNAS paper [thanks]: [X, Cornuet, Marin, & Pillai, Aug. 2011] Model M1 : y ∼ N (θ1 , 1) opposed √ to model M2 : y ∼ L(θ√ 1/ 2), Laplace distribution with mean θ2 2, and scale parameter 1/ 2 (variance one). 6 5 4 Density 3 2 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 posterior probability
  80. 80. ABC Methods for Bayesian Model Choice Model choice consistency Formalised frameworkA benchmark if toy example Comparison suggested by referee of PNAS paper [thanks]: [X, Cornuet, Marin, & Pillai, Aug. 2011] Model M1 : y ∼ N (θ1 , 1) opposed √ to model M2 : y ∼ L(θ√ 1/ 2), Laplace distribution with mean θ2 2, and scale parameter 1/ 2 (variance one). n=100 0.7 0.6 0.5 0.4 q 0.3 q q q 0.2 0.1 q q q q q 0.0 q q Gauss Laplace
  81. 81. ABC Methods for Bayesian Model Choice Model choice consistency Formalised frameworkA benchmark if toy example Comparison suggested by referee of PNAS paper [thanks]: [X, Cornuet, Marin, & Pillai, Aug. 2011] Model M1 : y ∼ N (θ1 , 1) opposed √ to model M2 : y ∼ L(θ√ 1/ 2), Laplace distribution with mean θ2 2, and scale parameter 1/ 2 (variance one). 6 3 5 4 Density Density 2 3 2 1 1 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.0 0.2 0.4 0.6 0.8 1.0 posterior probability probability
  82. 82. ABC Methods for Bayesian Model Choice Model choice consistency Formalised frameworkA benchmark if toy example Comparison suggested by referee of PNAS paper [thanks]: [X, Cornuet, Marin, & Pillai, Aug. 2011] Model M1 : y ∼ N (θ1 , 1) opposed √ to model M2 : y ∼ L(θ√ 1/ 2), Laplace distribution with mean θ2 2, and scale parameter 1/ 2 (variance one). n=100 n=100 1.0 0.7 q q 0.6 q 0.8 q q 0.5 q q q 0.6 q 0.4 q q q q 0.3 q q 0.4 q q 0.2 0.2 q 0.1 q q q q q q q 0.0 0.0 q q Gauss Laplace Gauss Laplace
  83. 83. ABC Methods for Bayesian Model Choice Model choice consistency Formalised frameworkFramework Starting from sample y = (y1 , . . . , yn ) the observed sample, not necessarily iid with true distribution y ∼ Pn Summary statistics T (y) = T n = (T1 (y), T2 (y), · · · , Td (y)) ∈ Rd with true distribution T n ∼ Gn .
  84. 84. ABC Methods for Bayesian Model Choice Model choice consistency Formalised frameworkFramework c Comparison of – under M1 , y ∼ F1,n (·|θ1 ) where θ1 ∈ Θ1 ⊂ Rp1 – under M2 , y ∼ F2,n (·|θ2 ) where θ2 ∈ Θ2 ⊂ Rp2 turned into – under M1 , T (y) ∼ G1,n (·|θ1 ), and θ1 |T (y) ∼ π1 (·|T n ) – under M2 , T (y) ∼ G2,n (·|θ2 ), and θ2 |T (y) ∼ π2 (·|T n )
  85. 85. ABC Methods for Bayesian Model Choice Model choice consistency Consistency resultsAssumptions A collection of asymptotic “standard” assumptions: [A1] is a standard central limit theorem under the true model [A2] controls the large deviations of the estimator T n from the estimand µ(θ) [A3] is the standard prior mass condition found in Bayesian asymptotics (di effective dimension of the parameter) [A4] restricts the behaviour of the model density against the true density
  86. 86. ABC Methods for Bayesian Model Choice Model choice consistency Consistency resultsAssumptions A collection of asymptotic “standard” assumptions: [A1] There exist a sequence {vn } converging to +∞, a distribution Q, a symmetric, d × d positive definite matrix V0 and a vector µ0 ∈ Rd such that −1/2 n→∞ vn V0 (T n − µ0 ) Q, under Gn
  87. 87. ABC Methods for Bayesian Model Choice Model choice consistency Consistency resultsAssumptions A collection of asymptotic “standard” assumptions: [A2] For i = 1, 2, there exist sets Fn,i ⊂ Θi and constants i , τi , αi >0 such that for all τ > 0, Gi,n |T n − µ(θi )| > τ |µi (θi ) − µ0 | ∧ i |θi sup −αi θi ∈Fn,i (|µi (θi ) − µ0 | ∧ i ) −α vn i with c −τ πi (Fn,i ) = o(vn i ).
  88. 88. ABC Methods for Bayesian Model Choice Model choice consistency Consistency resultsAssumptions A collection of asymptotic “standard” assumptions: [A3] For u > 0 −1 Sn,i (u) = θi ∈ Fn,i ; |µ(θi ) − µ0 | ≤ u vn if inf{|µi (θi ) − µ0 |; θi ∈ Θi } = 0, there exist constants di < τi ∧ αi − 1 such that −d πi (Sn,i (u)) ∼ udi vn i , ∀u vn
  89. 89. ABC Methods for Bayesian Model Choice Model choice consistency Consistency resultsAssumptions A collection of asymptotic “standard” assumptions: [A4] If inf{|µi (θi ) − µ0 |; θi ∈ Θi } = 0, for any > 0, there exist U, δ > 0 and (En ) such that, if θi ∈ Sn,i (U ) c En ⊂ {t; gi (t|θi ) < δgn (t)} and Gn (En ) < .
  90. 90. ABC Methods for Bayesian Model Choice Model choice consistency Consistency resultsAssumptions A collection of asymptotic “standard” assumptions: Again [A1] is a standard central limit theorem under the true model [A2] controls the large deviations of the estimator T n from the estimand µ(θ) [A3] is the standard prior mass condition found in Bayesian asymptotics (di effective dimension of the parameter) [A4] restricts the behaviour of the model density against the true density
  91. 91. ABC Methods for Bayesian Model Choice Model choice consistency Consistency resultsEffective dimension Understanding di in [A3]: defined only when µ0 ∈ {µi (θi ), θi ∈ Θi }, πi (θi : |µi (θi ) − µ0 | < n−1/2 ) = O(n−di /2 ) is the effective dimension of the model Θi around µ0
  92. 92. ABC Methods for Bayesian Model Choice Model choice consistency Consistency resultsAsymptotic marginals Asymptotically, under [A1]–[A4] mi (t) = gi (t|θi ) πi (θi ) dθi Θi is such that (i) if inf{|µi (θi ) − µ0 |; θi ∈ Θi } = 0, Cl vn i ≤ mi (T n ) ≤ Cu vn i d−d d−d and (ii) if inf{|µi (θi ) − µ0 |; θi ∈ Θi } > 0 mi (T n ) = oPn [vn i + vn i ]. d−τ d−α
  93. 93. ABC Methods for Bayesian Model Choice Model choice consistency Consistency resultsWithin-model consistency Under same assumptions, if inf{|µi (θi ) − µ0 |; θi ∈ Θi } = 0, the posterior distribution of µi (θi ) given T n is consistent at rate 1/vn provided αi ∧ τi > di .
  94. 94. ABC Methods for Bayesian Model Choice Model choice consistency Consistency resultsWithin-model consistency Under same assumptions, if inf{|µi (θi ) − µ0 |; θi ∈ Θi } = 0, the posterior distribution of µi (θi ) given T n is consistent at rate 1/vn provided αi ∧ τi > di . Note: di can truly be seen as an effective dimension of the model under the posterior πi (.|T n ), since if µ0 ∈ {µi (θi ); θi ∈ Θi }, mi (T n ) ∼ vn i d−d
  95. 95. ABC Methods for Bayesian Model Choice Model choice consistency Consistency resultsBetween-model consistency Consequence of above is that asymptotic behaviour of the Bayes factor is driven by the asymptotic mean value of T n under both models. And only by this mean value!
  96. 96. ABC Methods for Bayesian Model Choice Model choice consistency Consistency resultsBetween-model consistency Consequence of above is that asymptotic behaviour of the Bayes factor is driven by the asymptotic mean value of T n under both models. And only by this mean value! Indeed, if inf{|µ0 − µ2 (θ2 )|; θ2 ∈ Θ2 } = inf{|µ0 − µ1 (θ1 )|; θ1 ∈ Θ1 } = 0 then Cl vn 1 −d2 ) ≤ m1 (T n ) m2 (T n ) ≤ Cu vn 1 −d2 ) , −(d −(d where Cl , Cu = OPn (1), irrespective of the true model. c Only depends on the difference d1 − d2 : no consistency
  97. 97. ABC Methods for Bayesian Model Choice Model choice consistency Consistency resultsBetween-model consistency Consequence of above is that asymptotic behaviour of the Bayes factor is driven by the asymptotic mean value of T n under both models. And only by this mean value! Else, if inf{|µ0 − µ2 (θ2 )|; θ2 ∈ Θ2 } > inf{|µ0 − µ1 (θ1 )|; θ1 ∈ Θ1 } = 0 then m1 (T n ) ≥ Cu min vn 1 −α2 ) , vn 1 −τ2 ) −(d −(d m2 (T n )
  98. 98. ABC Methods for Bayesian Model Choice Model choice consistency Consistency resultsConsistency theorem If inf{|µ0 − µ2 (θ2 )|; θ2 ∈ Θ2 } = inf{|µ0 − µ1 (θ1 )|; θ1 ∈ Θ1 } = 0, Bayes factor B12 = O(vn 1 −d2 ) ) T −(d irrespective of the true model. It is inconsistent since it always picks the model with the smallest dimension
  99. 99. ABC Methods for Bayesian Model Choice Model choice consistency Consistency resultsConsistency theorem If Pn belongs to one of the two models and if µ0 cannot be attained by the other one : 0 = min (inf{|µ0 − µi (θi )|; θi ∈ Θi }, i = 1, 2) < max (inf{|µ0 − µi (θi )|; θi ∈ Θi }, i = 1, 2) , T then the Bayes factor B12 is consistent
  100. 100. ABC Methods for Bayesian Model Choice Model choice consistency Summary statisticsConsequences on summary statistics Bayes factor driven by the means µi (θi ) and the relative position of µ0 wrt both sets {µi (θi ); θi ∈ Θi }, i = 1, 2. For ABC, this implies the most likely statistics T n are ancillary statistics with different mean values under both models
  101. 101. ABC Methods for Bayesian Model Choice Model choice consistency Summary statisticsConsequences on summary statistics Bayes factor driven by the means µi (θi ) and the relative position of µ0 wrt both sets {µi (θi ); θi ∈ Θi }, i = 1, 2. For ABC, this implies the most likely statistics T n are ancillary statistics with different mean values under both models Else, if T n asymptotically depends on some of the parameters of the models, it is quite likely that there exists θi ∈ Θi such that µi (θi ) = µ0 even though model M1 is misspecified
  102. 102. ABC Methods for Bayesian Model Choice Model choice consistency Summary statisticsToy example: Laplace versus Gauss [1] If n T n = n−1 Xi4 , µ1 (θ) = 3 + θ4 + 6θ2 , µ2 (θ) = 6 + · · · i=1 and the true distribution is Laplace with mean θ0 = 1, under the √ Gaussian model the value θ∗ = 2 3 − 3 leads to µ0 = µ(θ∗ ) [here d1 = d2 = d = 1]
  103. 103. ABC Methods for Bayesian Model Choice Model choice consistency Summary statisticsToy example: Laplace versus Gauss [1] If n T n = n−1 Xi4 , µ1 (θ) = 3 + θ4 + 6θ2 , µ2 (θ) = 6 + · · · i=1 and the true distribution is Laplace with mean θ0 = 1, under the √ Gaussian model the value θ∗ = 2 3 − 3 leads to µ0 = µ(θ∗ ) [here d1 = d2 = d = 1] c A Bayes factor associated with such a statistic is inconsistent
  104. 104. ABC Methods for Bayesian Model Choice Model choice consistency Summary statisticsToy example: Laplace versus Gauss [1] If n n −1 T =n Xi4 , µ1 (θ) = 3 + θ4 + 6θ2 , µ2 (θ) = 6 + · · · i=1 8 6 4 2 0 0.0 0.2 0.4 0.6 0.8 1.0 probability Fourth moment
  105. 105. ABC Methods for Bayesian Model Choice Model choice consistency Summary statisticsToy example: Laplace versus Gauss [1] If n T n = n−1 Xi4 , µ1 (θ) = 3 + θ4 + 6θ2 , µ2 (θ) = 6 + · · · i=1 n=1000 0.45 q 0.40 0.35 q q q 0.30 q M1 M2
  106. 106. ABC Methods for Bayesian Model Choice Model choice consistency Summary statisticsToy example: Laplace versus Gauss [1] If n T n = n−1 Xi4 , µ1 (θ) = 3 + θ4 + 6θ2 , µ2 (θ) = 6 + · · · i=1 Details: Comparison of the distributions of the posterior probabilities that the data is from a normal model (as opposed to a Laplace model) with unknown mean when the data is made of n = 1000 observations either from a normal (M1) or Laplace (M2) distribution with mean one and when the summary statistic in the ABC algorithm is restricted to the empirical fourth moment. The ABC algorithm uses proposals from the prior N (0, 4) and selects the tolerance as the 1% distance quantile.
  107. 107. ABC Methods for Bayesian Model Choice Model choice consistency Summary statisticsToy example: Laplace versus Gauss [0] When ¯(4) ¯(6) T (y) = yn , yn and the true distribution is Laplace with mean θ0 = 0, then √ ∗ ∗ µ0 = 6, µ1 (θ1 ) = 6 with θ1 = 2 3 − 3 [d1 = 1 and d2 = 1/2] thus B12 ∼ n−1/4 → 0 : consistent
  108. 108. ABC Methods for Bayesian Model Choice Model choice consistency Summary statisticsToy example: Laplace versus Gauss [0] When ¯(4) ¯(6) T (y) = yn , yn and the true distribution is Laplace with mean θ0 = 0, then √ ∗ ∗ µ0 = 6, µ1 (θ1 ) = 6 with θ1 = 2 3 − 3 [d1 = 1 and d2 = 1/2] thus B12 ∼ n−1/4 → 0 : consistent Under the Gaussian model µ0 = 3, µ2 (θ2 ) ≥ 6 > 3 ∀θ2 B12 → +∞ : consistent
  109. 109. ABC Methods for Bayesian Model Choice Model choice consistency Summary statisticsToy example: Laplace versus Gauss [0] When ¯(4) ¯(6) T (y) = yn , yn 5 4 3 Density 2 1 0 0.0 0.2 0.4 0.6 0.8 1.0 posterior probabilities Fourth and sixth moments
  110. 110. ABC Methods for Bayesian Model Choice Model choice consistency Summary statisticsChecking for adequate statistics After running ABC, i.e. creating reference tables of (θi , xi ) from ˆ both joints, straightforward derivation of ABC estimates θ1 and θ2 . ˆ Evaluation of E1 [T (X)] and E2 [T (X)] allows for detection of ˆ θ1 ˆ θ2 different means under both models.
  111. 111. ABC Methods for Bayesian Model Choice Model choice consistency Summary statisticsChecking for adequate statistics q q q 40 30 20 10 q q q 0 Gauss Laplace Gauss Laplace Normalised χ2 without and with mad
  112. 112. ABC Methods for Bayesian Model Choice Model choice consistency Summary statisticsA population genetic illustration Two populations (1 and 2) having diverged at a fixed known time in the past and third population (3) which diverged from one of those two populations (models 1 and 2, respectively). Observation of 50 diploid individuals/population genotyped at 5, 50 or 100 independent microsatellite loci. Stepwise mutation model: the number of repeats of the mutated gene increases or decreases by one. Mutation rate µ common to all loci set to 0.005 (single parameter) with uniform prior distribution µ ∼ U[0.0001, 0.01]
  113. 113. ABC Methods for Bayesian Model Choice Model choice consistency Summary statisticsA population genetic illustration Two populations (1 and 2) having diverged at a fixed known time in the past and third population (3) which diverged from one of those two populations (models 1 and 2, respectively). Observation of 50 diploid individuals/population genotyped at 5, 50 or 100 independent microsatellite loci. Model 2
  114. 114. ABC Methods for Bayesian Model Choice Model choice consistency Summary statisticsA population genetic illustration Summary statistics associated to the (δµ )2 distance
  115. 115. ABC Methods for Bayesian Model Choice Model choice consistency Summary statisticsA population genetic illustration Summary statistics associated to the (δµ )2 distance xl,i,j repeated number of allele in locus l = 1, . . . , L for individual i = 1, . . . , 100 within the population j = 1, 2, 3. Then L 100 100 2 1 1 1 (δµ )21 ,j2 j = xl,i1 ,j1 − xl,i2 ,j2 . L 100 100 l=1 i1 =1 i2 =1
  116. 116. ABC Methods for Bayesian Model Choice Model choice consistency Summary statisticsA population genetic illustration For two copies of locus l with allele sizes xl,i,j1 and xl,i ,j2 , most recent common ancestor at coalescence time τj1 ,j2 , gene genealogy distance of 2τj1 ,j2 , hence number of mutation Poisson with parameter 2µτj1 ,j2 . Therefore, 2 E xl,i,j1 − xl,i ,j2 |τj1 ,j2 = 2µτj1 ,j2 and Model 1 Model 2 E (δµ )2 1,2 2µ1 t 2µ2 t E (δµ )2 1,3 2µ1 t 2µ2 t E (δµ )2 2,3 2µ1 t 2µ2 t
  117. 117. ABC Methods for Bayesian Model Choice Model choice consistency Summary statisticsA population genetic illustration Thus, Bayes factor based only on distance (δµ )2 not convergent: if 1,2 µ1 = µ2 , same expectations. Bayes factor based only on distance (δµ )2 or (δµ )2 not 1,3 2,3 convergent: if µ1 = 2µ2 or 2µ1 = µ2 same expectation. if two of the three distances are used, Bayes factor converges: there is no (µ1 , µ2 ) for which all expectations are equal.
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