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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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