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Recent results on ABC algorithms

Statistician
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University of Warwick

Recent results on ABC algorithms

- 1. Approximative Bayesian Computation (ABC) Methods Approximative Bayesian Computation (ABC) Methods Christian P. Robert Universit´ Paris Dauphine and CREST-INSEE e http://www.ceremade.dauphine.fr/~xian Joint works with M. Beaumont, J.-M. Cornuet, A. Grelaud, J.-M. Marin, F. Rodolphe, & J.-F. Tally Colloquium, Universit´ de Montpellier 2, 12 f´vrier 2009 e e
- 2. Approximative Bayesian Computation (ABC) Methods Outline Introduction 1 Population Monte Carlo 2 ABC 3 ABC-PMC 4 ABC for model choice in GRFs 5
- 3. Approximative Bayesian Computation (ABC) Methods Introduction General purpose Given a density π known up to a normalizing constant, and an integrable function h, compute h(x)˜ (x)µ(dx) π Π(h) = h(x)π(x)µ(dx) = π (x)µ(dx) ˜ when h(x)˜ (x)µ(dx) is intractable. π
- 4. Approximative Bayesian Computation (ABC) Methods Introduction Monte Carlo basics Monte Carlo basics Generate an iid sample x1 , . . . , xN from π and estimate Π(h) by N ˆN ΠM C (h) = N −1 h(xi ). i=1 as ˆN LLN: ΠM C (h) −→ Π(h) If Π(h2 ) = h2 (x)π(x)µ(dx) < ∞, √ L ˆN N ΠM C (h) − Π(h) N 0, Π [h − Π(h)]2 . CLT: Caveat Often impossible or ineﬃcient to simulate directly from Π
- 5. Approximative Bayesian Computation (ABC) Methods Introduction Importance Sampling Importance Sampling For Q proposal distribution such that Q(dx) = q(x)µ(dx), alternative representation Π(h) = h(x){π/q}(x)q(x)µ(dx). Principle Generate an iid sample x1 , . . . , xN ∼ Q and estimate Π(h) by N ˆ IS ΠQ,N (h) = N −1 h(xi ){π/q}(xi ). i=1
- 6. Approximative Bayesian Computation (ABC) Methods Introduction Importance Sampling Then as ˆ LLN: ΠIS (h) −→ Π(h) and if Q((hπ/q)2 ) < ∞, Q,N √ L ˆ Q,N N (ΠIS (h) − Π(h)) N 0, Q{(hπ/q − Π(h))2 } . CLT : Caveat ˆ Q,N If normalizing constant unknown, impossible to use ΠIS Generic problem in Bayesian Statistics: π(θ|x) ∝ f (x|θ)π(θ).
- 7. Approximative Bayesian Computation (ABC) Methods Introduction Importance Sampling Self-Normalised Importance Sampling Self normalized version −1 N N ˆ Q,N ΠSN IS (h) {π/q}(xi ) = h(xi ){π/q}(xi ). i=1 i=1 as ˆ ΠSN IS (h) −→ Π(h) LLN : Q,N and if Π((1 + h2 )(π/q)) < ∞, √ L 0, π {(π/q)(h − Π(h)}2 ) . ˆ Q,N N (ΠSN IS (h) − Π(h)) CLT : N c The quality of the SNIS approximation depends on the choice of Q
- 8. Approximative Bayesian Computation (ABC) Methods Introduction Importance Sampling Iterated importance sampling Introduction of an algorithmic temporal dimension : (t) (t−1) xi ∼ qt (x|xi ) i = 1, . . . , n, t = 1, . . . and n 1 (t) (t) ˆ It = ̺i h(xi ) n i=1 is still unbiased for (t) πt (xi ) (t) ̺i = , i = 1, . . . , n (t) (t−1) qt (xi |xi )
- 9. Approximative Bayesian Computation (ABC) Methods Population Monte Carlo PMCA: Population Monte Carlo Algorithm At time t = 0 iid Generate (xi,0 )1≤i≤N ∼ Q0 Set ωi,0 = {π/q0 }(xi,0 ) iid Generate (Ji,0 )1≤i≤N ∼ M(1, (¯ i,0 )1≤i≤N ) ω Set xi,0 = xJi ,0 ˜ At time t (t = 1, . . . , T ), ind Generate xi,t ∼ Qi,t (˜i,t−1 , ·) x Set ωi,t = {π(xi,t )/qi,t (˜i,t−1 , xi,t )} x iid Generate (Ji,t )1≤i≤N ∼ M(1, (¯ i,t )1≤i≤N ) ω Set xi,t = xJi,t ,t . ˜ [Capp´, Douc, Guillin, Marin, & CPR, 2009] e
- 10. Approximative Bayesian Computation (ABC) Methods Population Monte Carlo Notes on PMC After T iterations of PMCA, PMC estimator of Π(h) given by T N 1 ¯ N,T ΠP M C (h) = ¯ ωi,t h(xi,t ). T t=1 i=1 ¯ ωi,t means normalising over whole sequence of simulations 1 Qi,t ’s chosen arbitrarily under support constraint 2 Qi,t ’s may depend on whole sequence of simulations 3 alternatives to multinomial sampling reduce variance/preserve 4 “unbiasedness” [Kitagawa, 1996 / Carpenter, Cliﬀord & Fearnhead, 1997]
- 11. Approximative Bayesian Computation (ABC) Methods ABC The 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) , x ∼ f (x|θ′ ) , until the auxiliary variable x is equal to the observed value, x = y. [Pritchard et al., 1999]
- 12. Approximative Bayesian Computation (ABC) Methods ABC Population genetics example Tree of ancestors in a sample of genes
- 13. Approximative Bayesian Computation (ABC) Methods ABC A as approximative When y is a continuous random variable, equality x = y is replaced with a tolerance condition, ̺(x, y) ≤ ǫ where ̺ is a distance between summary statistics Output distributed from π(θ) Pθ {̺(x, y) < ǫ} ∝ π(θ|̺(x, y) < ǫ)
- 14. Approximative Bayesian Computation (ABC) Methods ABC ABC improvements 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 ǫ [Beaumont et al., 2002]
- 15. Approximative Bayesian Computation (ABC) Methods ABC ABC-MCMC Markov chain (θ(t) ) created via the transition function θ′ ∼ K(θ′ |θ(t) ) if x ∼ f (x|θ′ ) is such that x = y π(θ′ )K(θ(t) |θ′ ) (t+1) and u ∼ U(0, 1) ≤ π(θ(t) )K(θ′ |θ(t) ) , θ = (t) θ otherwise, has the posterior π(θ|y) as stationary distribution [Marjoram et al, 2003]
- 16. Approximative Bayesian Computation (ABC) Methods ABC ABC-PRC Another sequential version producing a sequence of Markov (t) (t) transition kernels Kt and of samples (θ1 , . . . , θN ) (1 ≤ t ≤ T ) ABC-PRC Algorithm (t−1) Pick a θ⋆ is selected at random among the previous θi ’s 1 (t−1) (1 ≤ i ≤ N ). with probabilities ωi Generate 2 (t) (t) θi ∼ Kt (θ|θ⋆ ) , x ∼ f (x|θi ) , Check that ̺(x, y) < ǫ, otherwise start again. 3 [Sisson et al., 2007]
- 17. Approximative Bayesian Computation (ABC) Methods ABC ABC-PRC weight (t) Probability ωi computed as (t) (t) (t) (t) ωi ∝ π(θi )Lt−1 (θ⋆ |θi ){π(θ⋆ )Kt (θi |θ⋆ )}−1 , where Lt−1 is an arbitrary transition kernel. In case Lt−1 (θ′ |θ) = Kt (θ|θ′ ) , all weights are equal under a uniform prior. Inspired from Del Moral et al. (2006), who use backward kernels Lt−1 in SMC to achieve unbiasedness
- 18. Approximative Bayesian Computation (ABC) Methods ABC ABC-PRC bias Lack of unbiasedness of the method Joint density of the accepted pair (θ(t−1) , θ(t) ) proportional to (t−1) (t) (t−1) (t) |y)Kt (θ |θ )f (y|θ ), π(θ For an arbitrary function h(θ), E[ωt h(θ(t) )] proportional to π(θ (t) )Lt−1 (θ (t−1) |θ (t) ) ZZ (t) (t−1) (t) (t−1) (t) (t−1) (t) ) |y)Kt (θ |θ )f (y|θ )dθ dθ h(θ π(θ π(θ (t−1) )Kt (θ (t) |θ (t−1) ) π(θ (t) )Lt−1 (θ (t−1) |θ (t) ) ZZ (t) (t−1) (t−1) ∝ ) )f (y|θ ) h(θ π(θ π(θ (t−1) )Kt (θ (t) |θ (t−1) ) (t) (t−1) (t) (t−1) (t) × Kt (θ |θ )f (y|θ )dθ dθ Z ﬀ Z (t) (t) (t−1) (t) (t−1) (t−1) (t) ∝ )π(θ |y) Lt−1 (θ |θ )f (y|θ )dθ dθ h(θ .
- 19. Approximative Bayesian Computation (ABC) Methods ABC A mixture example 0.8 0.8 0.8 0.8 0.8 0.4 0.4 0.4 0.4 0.4 0.0 0.0 0.0 0.0 0.0 −3 −1 123 −3 −1 123 −3 −1 123 −3 −1 123 −3 −1 123 θ θ θ θ θ 0.8 0.8 0.8 0.8 0.8 0.4 0.4 0.4 0.4 0.4 0.0 0.0 0.0 0.0 0.0 −3 −1 123 −3 −1 123 −3 −1 123 −3 −1 123 −3 −1 123 θ θ θ θ θ Comparison of τ = 0.15 and τ = 1/0.15 in Kt
- 20. Approximative Bayesian Computation (ABC) Methods ABC-PMC A PMC version Use of the same kernel idea as ABC-PRC but with IS correction Generate a sample at iteration t by N (t−1) (t−1) πt (θ(t) ) ∝ Kt (θ(t) |θj ˆ ωj ) j=1 modulo acceptance of the associated xt , and use an importance (t) weight associated with an accepted simulation θi (t) (t) (t) ωi ∝ π(θi ) πt (θi ) . ˆ c Still likelihood free [Beaumont et al., 2008, arXiv:0805.2256]
- 21. Approximative Bayesian Computation (ABC) Methods ABC-PMC The ABC-PMC algorithm Given a decreasing sequence of approximation levels ǫ1 ≥ . . . ≥ ǫT , 1. At iteration t = 1, For i = 1, ..., N (1) (1) Simulate θi ∼ π(θ) and x ∼ f (x|θi ) until ̺(x, y) < ǫ1 (1) Set ωi = 1/N (1) Take τ 2 as twice the empirical variance of the θi ’s 2. At iteration 2 ≤ t ≤ T , For i = 1, ..., N , repeat (t−1) (t−1) ⋆ Pick θi from the θj ’s with probabilities ωj (t) (t) 2 ⋆ ⋆ generate θi |θi ∼ N (θi , σt ) and x ∼ f (x|θi ) until ̺(x, y) < ǫt (t) (t) (t−1) (t) (t−1) N Set ωi ∝ π(θi )/ θi − θj ωj ϕ σt ) −1 j=1 (t) 2 Take τt+1 as twice the weighted empirical variance of the θi ’s
- 22. Approximative Bayesian Computation (ABC) Methods ABC-PMC A mixture example (0) Toy model of Sisson et al. (2007): if θ ∼ U(−10, 10) , x|θ ∼ 0.5 N (θ, 1) + 0.5 N (θ, 1/100) , then the posterior distribution associated with y = 0 is the normal mixture θ|y = 0 ∼ 0.5 N (0, 1) + 0.5 N (0, 1/100) restricted to [−10, 10]. Furthermore, true target available as π(θ||x| < ǫ) ∝ Φ(ǫ−θ)−Φ(−ǫ−θ)+Φ(10(ǫ−θ))−Φ(−10(ǫ+θ)) .
- 23. Approximative Bayesian Computation (ABC) Methods ABC-PMC A mixture example (2) Recovery of the target, whether using a ﬁxed standard deviation of τ = 0.15 or τ = 1/0.15, or a sequence of adaptive τt ’s.
- 24. Approximative Bayesian Computation (ABC) Methods ABC for model choice in GRFs ABC for model choice Introduction 1 Population Monte Carlo 2 ABC 3 ABC-PMC 4 ABC for model choice in GRFs 5 Gibbs random ﬁelds Model choice via ABC Illustrations
- 25. Approximative Bayesian Computation (ABC) Methods ABC for model choice in GRFs Gibbs random ﬁelds Gibbs random ﬁelds Gibbs distribution y = (y1 , . . . , yn ) is a Gibbs random ﬁeld associated with the graph G if 1 f (y) = exp − Vc (yc ) . Z c∈C where Z is the normalising constant, C is the set of cliques and Vc is any function also called potential (and U (y) = c∈C Vc (yc ) is the energy function) c Z is usually unavailable in closed form
- 26. Approximative Bayesian Computation (ABC) Methods ABC for model choice in GRFs Gibbs random ﬁelds Potts model Potts model Vc (y) is of the form Vc (y) = θS(y) =θ δyl =yi l∼i where l∼i denotes a neighbourhood structure In most realistic settings, summation exp{θT S(x)} Zθ = x∈X involves too many terms to be manageable and numerical approximations cannot always be trusted [Cucala, Marin, CPR & Titterington, 2009]
- 27. Approximative Bayesian Computation (ABC) Methods ABC for model choice in GRFs Model choice via ABC Bayesian Model Choice Comparing a model with potential S0 taking values in Rp0 versus a model with potential S1 taking values in Rp1 through the Bayes factor corresponding to the priors π0 and π1 on each parameter space T Bm0 /m1 (x) = exp{θ0 S0 (x)}/Zθ0 ,0 π0 (dθ0 ) T exp{θ1 S1 (x)}/Zθ1 ,1 π1 (dθ1 ) Use of Jeﬀreys’ scale to select most appropriate model
- 28. Approximative Bayesian Computation (ABC) Methods ABC for model choice in GRFs Model choice via ABC Neighbourhood relations 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.
- 29. Approximative Bayesian Computation (ABC) Methods ABC for model choice in GRFs Model choice via ABC Model index Formalisation via a model index M that appears as a new parameter with prior distribution π(M = m) and π(θ|M = m) = πm (θm ) Computational target: P(M = m|x) ∝ fm (x|θm )πm (θm ) dθm π(M = m) , Θm
- 30. Approximative Bayesian Computation (ABC) Methods ABC for model choice in GRFs Model choice via ABC Suﬃcient statistics By deﬁnition, if S(x) suﬃcient statistic for the joint parameters (M, θ0 , . . . , θM −1 ), P(M = m|x) = P(M = m|S(x)) . For each model m, own suﬃcient statistic Sm (·) and S(·) = (S0 (·), . . . , SM −1 (·)) also suﬃcient. 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 therefore also suﬃcient for the joint parameters [Speciﬁc to Gibbs random ﬁelds!]
- 31. Approximative Bayesian Computation (ABC) Methods ABC for model choice in GRFs Model choice via ABC ABC model choice Algorithm ABC-MC Generate m∗ from the prior π(M = m). ∗ Generate θm∗ from the prior πm∗ (·). Generate x∗ from the model fm∗ (·|θm∗ ). ∗ Compute the distance ρ(S(x0 ), S(x∗ )). Accept (θm∗ , m∗ ) if ρ(S(x0 ), S(x∗ )) < ǫ. ∗ Note When ǫ = 0 the algorithm is exact
- 32. Approximative Bayesian Computation (ABC) Methods ABC for model choice in GRFs Model choice via ABC ABC approximation to the Bayes factor Frequency ratio: ˆ P(M = m0 |x0 ) π(M = m1 ) BF m0 /m1 (x0 ) = × ˆ P(M = m1 |x0 ) π(M = m0 ) ♯{mi∗ = m0 } π(M = m1 ) × = , ♯{mi∗ = m1 } π(M = m0 ) replaced with 1 + ♯{mi∗ = m0 } π(M = m1 ) BF m0 /m1 (x0 ) = × 1 + ♯{mi∗ = m1 } π(M = m0 ) to avoid indeterminacy (also Bayes estimate).
- 33. Approximative Bayesian Computation (ABC) Methods ABC for model choice in GRFs Illustrations Toy example iid Bernoulli model versus two-state ﬁrst-order Markov chain, i.e. n {1 + exp(θ0 )}n , f0 (x|θ0 ) = exp θ0 I{xi =1} i=1 versus n 1 {1 + exp(θ1 )}n−1 , f1 (x|θ1 ) = exp θ1 I{xi =xi−1 } 2 i=2 with priors θ0 ∼ U(−5, 5) and θ1 ∼ U(0, 6) (inspired by “phase transition” boundaries).
- 34. Approximative Bayesian Computation (ABC) Methods ABC for model choice in GRFs Illustrations Toy example (2) 10 5 5 BF01 BF01 0 0 ^ ^ −10 −5 −5 −40 −20 0 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 ǫ corresponding to the 1% quantile on the distances.
- 35. Approximative Bayesian Computation (ABC) Methods ABC for model choice in GRFs Illustrations Protein folding Superposition of the native structure (grey) with the ST1 structure (red.), the ST2 structure (orange), the ST3 structure (green), and the DT structure (blue).
- 36. Approximative Bayesian Computation (ABC) Methods ABC for model choice in GRFs Illustrations Protein folding (2) % seq . Id. TM-score FROST score 1i5nA (ST1) 32 0.86 75.3 1ls1A1 (ST2) 5 0.42 8.9 1jr8A (ST3) 4 0.24 8.9 1s7oA (DT) 10 0.08 7.8 Characteristics of dataset. % seq. Id.: percentage of identity with the query sequence. TM-score.: similarity between predicted and native structure (uncertainty between 0.17 and 0.4) FROST score: quality of alignment of the query onto the candidate structure (uncertainty between 7 and 9).
- 37. Approximative Bayesian Computation (ABC) Methods ABC for model choice in GRFs Illustrations Protein folding (3) NS/ST1 NS/ST2 NS/ST3 NS/DT BF 1.34 1.22 2.42 2.76 P(M = NS|x0 ) 0.573 0.551 0.708 0.734 Estimates of the Bayes factors between model NS and models ST1, ST2, ST3, and DT, and corresponding posterior probabilities of model NS based on an ABC-MC algorithm using 1.2 106 simulations and a tolerance ǫ equal to the 1% quantile of the distances.

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