Bayesian Indirect Inference using a Parametric Auxiliary Model

Transcription

1 Bayesian Indirect Inference using a Parametric Auxiliary Model Dr Chris Drovandi Queensland University of Technology, Australia c.drovandi@qut.edu.au Collaborators: Tony Pettitt and Anthony Lee February 12, 214

2 Intro to ABC (intractable likelihood) Bayesian inference in based on posterior distribution p(θ y) p(y θ)p(θ). Thus Bayesian inference requires the likelihood function p(y θ) Many models have an intractable likelihood function One possible avenue is approximate Bayesian computation (ABC)

3 Intro to ABC (ABC target) ABC assumes that simulation from model is straightforward, x p(y θ) Compares observed and simulated data on the basis of summary statistics, s( ) Based on the following target distribution p ǫ (θ y) p(θ) p(x θ)k ǫ ( s(y) s(x) )dx, x where ǫ is ABC tolerance and K is a kernel weighting function If s( ) is sufficient and ǫ then p ǫ (θ y) p(θ y) (does not happen in practice) Thus two sources of error

4 Intro to ABC (choosing summary statistics) ABC has connections with kernel density estimation (Blum, 29) Choice of summary statistics involve trade-off between dimensionality and information loss Non-parametric aspect wants summary as low-dimensional as possible But decreasing dimension means information loss Choice of summary statistics most crucial to ABC approximation

5 Intro to ABC (MCMC ABC) A number of algorithms to sample from ABC target (Rejection sampling (Beaumont 22), MCMC (Marjoram et al 23) and SMC (e.g. Sisson et al 27 and Drovandi and Pettitt 211)) In this work used MCMC ABC. Involves the following steps: Propose new parameter θ q( θ). Simulate a dataset, x p(y θ ) Accept with probability:... and repeat... α = p(θ )K ǫ ( s(y) s(x ) )q(θ θ ) p(θ)k ǫ ( s(y) s(x) )q(θ θ)

6 Introduction Approximate Bayesian Computation ABC becoming standard for dealing with intractable (generative) model p(ý θ) Choosing summary statistics most difficult aspect Bayesian Indirect Inference (BII) Propose another (tractable) auxiliary model p A (Ý φ) Use the auxiliary model to build summary statistics or replace likelihood Aim: Review and Compare

7 BII Methods Figure: BII Methods

8 ABC II Methods ABC target distribution p ǫ,n (θ Ý) p(θ) ÜÒ p(ü Ò θ)1(ρ(s(ü Ò ),s(ý)) ǫ)d Ü Ò, where Ý is observed data (assume N observations), Ü Ò is simulated data (of size nn) θ is parameter, p(θ) is prior, p(ü Ò θ) is likelihood, s( ) is summary statistic, ρ(, ) is discrepancy function Common approach in II is to take n > 1 Can be shown that ABC can be over-precise when n > 1 Therefore take n = 1 for ABC II

9 ABC IP (Drovandi et al 211) Parameter estimates of auxiliary model are summary statistics Simulate data from generative model Ü p( θ) Estimate auxiliary parameter ˆφ(θ, Ü) = argmax φ p A(Ü φ). Compare with ˆφ(Ý) ρ(s(ü),s(ý)) = (ˆφ(Ü) ˆφ(Ý)) T Á(ˆφ(Ý))(ˆφ(Ü) ˆφ(Ý)). Efficient weighting of summary statistics using observed Fisher information Á(ˆφ(Ý))

10 ABC IL (Gleim and Pigorsch 213) Same as ABC IP but uses likelihood in the discrepancy function Parameter estimates of auxiliary model are summary statistics Simulate data from generative model Ü p( θ) Estimate auxiliary parameter Compare with ˆφ(Ý) ˆφ(θ, Ü) = argmax φ p A(Ü φ). ρ(s(ü),s(ý)) = logp A (Ý ˆφ(Ý)) logp A (Ý ˆφ(Ü)).

11 ABC IS (Gleim and Pigorsch 213) Uses scores of auxiliary model as summary statistics ( logpa (Ý φ) Ë (Ý,φ) =,, logp ) T A(Ý φ), φ 1 φ dim(φ) Simulate data from generative model Ü p( θ) Evaluate scores of auxiliary model based on simulated data and ˆφ(Ý), Ë (Ü, ˆφ(Ý)) Note that Ë (Ý, ˆφ(Ý)) =. Uses following discrepancy function ρ(s(ü),s(ý)) = Ë (Ü, ˆφ(Ý)) T Á(ˆφ(Ý)) 1 Ë (Ü, ˆφ(Ý)). Very fast when scores are analytic (no fitting of auxiliary model)

12 ABC II Assumptions Assumption (ABC IP Assumptions) The estimator of the auxiliary parameter, ˆφ(θ, Ü), is unique for all θ with positive prior support. Assumption (ABC IL Assumptions) The auxiliary likelihood evaluated at the auxiliary estimate, p A (Ý ˆφ(Ü,θ)), is unique for all θ with positive prior support. Assumption (ABC IS Assumptions) The MLE of the auxiliary model fitted to the observed data, ˆφ(Ý), is an interior point of the parameter space of φ. The log-likelihood of the auxiliary model, logp A ( φ), is differentiable and the score, Ë (Ü, ˆφ(Ý)), is unique for Ü that may be drawn from any θ that has positive prior support.

13 BIL (Gallant and McCullogh 29, Reeves and Pettitt 25) Replaces true likelihood with auxiliary likelihood Simulate data from generative model Ü Ò p( θ) Estimate auxiliary parameter ˆφ(θ, Ü Ò ) = argmax φ p A(Ü Ò φ). Evaluate auxiliary likelihood p A (Ý ˆφ(θ, Ü Ò )) Not ABC. No summary statistics or ABC tolerance Has the following target distribution p n (θ Ý) p(θ) p A (Ý ˆφ(θ, Ü Ò ))p(ü Ò θ)d Ü Ò, which depends on n. Theoretically behaves very different to ABC II ÜÒ

14 BIL Theory Assumption (BIL Assumptions) As n, p A (Ý ˆφ(Ü Ò,θ)) converges in probability to p A (Ý φ(θ)) for all θ with positive prior support. Result (BIL target as n ) The target distribution of BIL for n is p (θ Ý) p A (Ý φ(θ))p(θ). Thus BIL targets the correct posterior distribution provided that p A (Ý φ(θ)) p(ý θ) as a function of θ.

15 BIL Theory What about the target for finite n? p n and p have same target provided that E[p A (Ý ˆφ(θ, Ü Ò ))] = p A (Ý φ(θ)) (Andrieu and Roberts 29) In general p A (Ý ˆφ(θ, Ü Ò )) is a biased, noisy estimate of p A (Ý φ(θ)) Can anticipate better approximations for BIL for n large Contrasts with ABC II which needs n = 1

16 Synthetic Likeihood as a BIL Method Reduces data to set of summary statistics. Target distribution: p(θ s(ý)) p(s(ý) θ)p(θ). But p(s(y) θ) is unknown Synthetic likelihood approach Wood (21) assumes a parametric model, p A (s(ý) φ(θ)). Thus a BIL method. Wood takes p A to be multivariate normal N(µ(θ),Σ(θ)) For some θ, simulate n replicates of size N, Ü Ò = (1) (s(ü 1 ),...,s(ü(ò) 1 )), iid sample from p(s(ý) θ). Fit auxiliary model to this dataset: µ(ü Ò,θ) = 1 n Σ(Ü Ò,θ) = 1 n n i=1 s(ü ( ) 1 ), n ( ) ( ) (s(ü 1 ) µ(ü Ò,θ))(s(Ü 1 ) µ(ü Ò,θ)) T. i=1

17 ABC as BIL with Non-Parametric Auxiliary Model ABC is recovered as a BIL method by selecting a non-parametric auxiliary likelihood Full data (dabc): simulate n datasets Ü Ò = (1) (Ü 1,..., Ü(Ò) 1 ) and estimate likelihood 1 n n i=1 K ǫ (ρ(ý, Ü ( ) 1 )), Data reduced to summary statistic (sabc): 1 n n i=1 K ǫ (ρ( (Ý), (Ü ( ) 1 ))). ABC II is a special sabc method where summary statistics come from a parametric auxiliary model

18 Theoretical comparison of BIL and ABC II BIL exact when true model contained within auxiliary model (as n ). Suggests to choose flexible auxiliary model Even when true model is special case, ABC II statistics not sufficient in general When ABC II have sufficient statistics, BIL not exact in general, even when n Some Toy Examples Later...

19 Sampling from ABC II Target - MCMC ABC Algorithm 1 MCMC ABC algorithm of Marjoram et al 23. 1: Set θ 2: for i = 1 to T do 3: Draw θ q( θ i 1 ) 4: Simulate x p( θ ) 5: Compute r = p(θ )q(θ i 1 θ ) p(θ i 1 )q(θ θ i 1 ) 1(ρ(s(x ),s(y)) ǫ) 6: if uniform(,1) < r then 7: θ i = θ 8: else 9: θ i = θ i 1 1: end if 11: end for

20 Sampling from BIL Target - MCMC BIL Find increase in acceptance probability with increase in n Algorithm 1 MCMC BIL algorithm(see also Gallant and McCullogh 29). 1: Set θ 2: Simulate x n p( θ ) 3: Compute φ = argmax φ p A (x n φ) 4: for i = 1 to T do 5: Draw θ q( θ i 1 ) 6: Simulate x n p( θ ) 7: Compute ˆφ(x n ) = argmax φp A (x n φ) 8: Compute r = pa(y ˆφ(x n ))π(θ )q(θ i 1 θ ) pa(y φ i 1 )π(θ i 1 )q(θ θ i 1 ) 9: if uniform(,1) < r then 1: θ i = θ 11: φ i = ˆφ(x n ) 12: else 13: θ i = θ i 1 14: φ i = φ i 1 15: end if 16: end for

21 Toy Example 1 (Drovandi and Pettitt 213) True model Poisson(λ) and auxiliary model Normal(µ, τ) ABC II gets lucky as ˆµ = ȳ, sufficient for λ BIL as n approximates Poisson(λ) likelihood with Normal(λ, λ) likelihood. Not exact..8.6 n = 1 n = 1 n = true Acceptance probabilities: n = 1 46%, n = 1 67%, n = 1 72% and n = 1 73% for increasing n

22 Toy Example 1 Results for BIL when auxiliary model mis-specified Normal(µ, 9) (underdispersed) Normal(µ, 49) (overdispersed) ABC II will still work by chance 1.8 true τ unspecified τ = 16 τ =

23 Toy Example 2 True model t-distribution(µ, σ, ν = 1) and auxiliary model t-distribution(µ, σ, ν) Here BIL exact as n since true is special case of auxiliary ABC II do not produce sufficient statistics (full set of order statistics are minimal sufficient) true true 1.8 n=1 n=1 n=1 n=1 1.6 n=1 2 n=

24 Toy Example 3 cont true true 1.8 ABC IP ABC IP ABC IL ABC IL 1.6 ABC IS 2 ABC IS

25 Quantile Distribution Example Model of Interest: g-and-k quantile distribution Defined in terms of its quantile function: ( Q(z(p);θ) = a+b 1+c 1 exp( gz(p)) ) (1+z(p) 2 ) k z(p). 1+exp( gz(p)) (1) p - quantile, z(p) - standard normal quantile, θ = (a,b,g,k), c =.8 (see Rayner and MacGillivray 23). Numerical likelihood evaluation possible Simulation easier via inversion method Data consists of 1 independent draws with a = 3, b = 1, g = 2 and k =.5

26 Quantile Distribution Example (Cont...) Auxiliary model is a 3-component normal mixture model Flexible and fits data well But breaks assumption of ABC IP (parameter estimates not unique).8 estimated density 3 component mixture density.6 density y

27 Quantile Distribution Example (Results) 3 ABC n=1 n=4 posterior (a) a (b) b (c) g (d) k Acc Prob: 1.5% for n = 1, 2.7% for n = 2 and 3.8% for n = 4

28 Quantile Distribution Example (Results) 3 ABC n=4 n=2 posterior (e) a (f) b (g) g (h) k MCMC acc prob: 6.5% for n = 1 and 8.4% for n = 2

29 Quantile Distribution Example (Results) ABC IS ABC IS REG 3 ABC IL BIL n=2 15 posterior (i) a (j) b (k) g (l) k

30 Macroparasite Immunity Example Estimate parameters of a Markov process model explaining macroparasite population development with host immunity 212 hosts (cats) i = 1,...,212. Each cat injected with l i juvenile Brugia pahangi larvae (approximately 1 or 2). At time t i host is sacrificed and the number of matures are recorded Host assumed to develop an immunity Three variable problem: M(t) matures, L(t) juveniles, I(t) immunity. Only L() and M(t i ) is observed for each host No tractable likelihood

31 Proporton of Matures Time

32 Trivariate Markov Process of Riley et al (23) Invisible Invisible Mature Parasites M(t) Maturation γl(t) Juvenile Parasites L(t) µ M M(t) Natural death Immunity µ L L(t) Natural death βi(t)l(t) Death due to immunity Gain of immunity νl(t) I(t) Loss of immunity µ I I(t) Invisible

33 Auxiliary Beta-Binomial model The data show too much variation for Binomial A Beta-Binomial model has an extra parameter to capture dispersion ( ) li B(mi +α i,l i m i +β i ) p(m i α i,β i ) =, B(α i,β i ) m i Useful reparameterisation p i = α i /(α i +β i ) and θ i = 1/(α i +β i ) Relate the proportion and over dispersion parameters to time, t i, and initial larvae, l i, covariates logit(p i ) = β +β 1 log(t i )+β 2 (log(t i )) 2, { η1, if l log(θ i ) = i 1 η 2, if l i 2, Five parameters φ = (β,β 1,β 2,η 1,η 2 )

34 Macroparasite Immunity Results - Compare BII 15 1 ABC IS ABC IP ABC IL BIL x (m) ν (n) µ I (o) µ L (p) β

35 Macroparasite Immunity Results - Regression Adjustment 2 15 ABC IS REG ABC IP REG ABC IL REG BIL n= x (q) ν (r) µ L

36 Macroparasite Immunity Results - Increase n 15 n=1 n=2 n= x (s) ν (t) µ I (u) µ L (v) β

37 Discussion BIL very different to ABC II theoretically BIL needs to have a good auxiliary model. ABC II might be useful is auxiliary model mis-specified ABC II more flexible; can incorporate other summary statistics

38 Key References Drovandi et al. (211). Approximate Bayesian Computation using Indirect Inference. JRSS C. Drovandi and Pettitt (213). Bayesian Indirect Inference. Gleim and Pigorsch (213). Approximate Bayesian Computation with Indirect Summary Statistics. Gallant and McCullogh (29). On the determination of general scientific models with application to asset pricing. JASA. Reeves and Pettitt (25). A theoretical framework for approximate Bayesian computation. 2th IWSM.