Bayesian Indirect Inference using a Parametric Auxiliary Model

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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 July 25, 214

2 Bayesian Indirect Likelihood Bayesian inference is based on posterior distribution p(θ Ý) p(ý θ)p(θ). Thus Bayesian inference requires the likelihood function p(ý θ) Many models have an intractable likelihood function Replace with some alternative tractable likelihood p(θ Ý) p(ý θ)p(θ). Referred to as Bayesian Indirect Likelihood (BIL)

3 BIL Methods Composite Likelihood (e.g. Pauli et al 211) Emulation (e.g. Gaussian process, Wilkinson 214) parametric BIL (uses the likelihood of an alternative parametric model, e.g. Gallant and McCullagh 29) non-parametric BIL (traditional ABC method recovered, e.g. Beaumont et al 22)

4 pbii methods pbii - Bayesian Indirect Inference methods that uses a p arametric auxiliary model in some way Requires parametric model with parameter φ with tractable likelihood, p A (y φ) Does not have to be an obvious connection between θ and φ Assume throughout that dim(φ) dim(θ) Two classes of pbii methods: Use auxiliary model to form summary statistics for ABC (ABC II) Use auxiliary model to form replacement likelihood, pbil

5 ABC II (Intro to ABC) ABC assumes that simulation from model is straightforward, Ü p(ý θ) Compares observed and simulated data on the basis of summary statistics, s( ) Based on the following target distribution p ǫ (θ Ý) p(θ)p ǫ (Ý θ), where ǫ is ABC tolerance and p ǫ (Ý θ) = p(ü θ)k ǫ ( s(ý) s(ü) )dx, Ü where K is a kernel weighting function. This can be estimated unbiasedly and this is enough (Andrieu and Roberts 29) If s( ) is sufficient and ǫ then p ǫ (θ Ý) p(θ Ý) (does not happen in practice) Thus two sources of error

6 ABC II (Intro to ABC Cont...) 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

7 ABC II Methods In some applications can propose an alternative parametric auxiliary model Use auxiliary model to form summary statistic (ABC II) Auxiliary parameter estimate as summary statistic (ABC IP and ABC IL) Score of auxiliary model as summary statistic (ABC IS)

8 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 φ(θ, Ü) = argmax φ p A(Ü φ). Compare with φ(ý) ρ(s(ü),s(ý)) = (φ(ü) φ(ý)) T Á(φ(Ý))(φ(Ü) φ(ý)). Efficient weighting of summary statistics using observed Fisher information Á(φ(Ý))

9 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 φ(ý) φ(θ, Ü) = argmax φ p A(Ü φ). ρ(s(ü),s(ý)) = logp A (Ý φ(ý)) logp A (Ý φ(ü)).

10 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 φ(ý), Ë (Ü,φ(Ý)) Note that Ë (Ý,φ(Ý)) =. Uses following discrepancy function ρ(s(ü),s(ý)) = Ë (Ü,φ(Ý)) T Á(φ(Ý)) 1 Ë (Ü,φ(Ý)). Very fast when scores are analytic (no fitting of auxiliary model)

11 ABC II Assumptions Assumption (ABC IP Assumptions) The estimator of the auxiliary parameter, φ(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 φ(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, φ(ý), is an interior point of the parameter space of φ and Â(φ(Ý)) is positive definite. The log-likelihood of the auxiliary model, logp A ( φ), is differentiable and the score, ËA(Ü,φ(Ý)), is unique for any Ü that may be drawn according to any θ that has positive prior support.

12 ABC II vs Traditional ABC for Summary Statistics Advantages Can check if auxiliary model fits data well Natural ABC discrepancy functions Control dimensionality of summary statistics via parsimonious auxiliary model (e.g. compare auxiliary models via AIC) Disadvantages ABC II can be expensive Restricted to applications where auxiliary model can be proposed

13 pdbil (Gallant and McCullogh 29, Reeves and Pettitt 25) Replaces true likelihood with auxiliary likelihood (full data level) Simulate n independent datasets from generative model Ü1:n iid p( θ) Estimate auxiliary parameter φ(θ, Ü1:n) = argmax φ p A(Ü1:n φ). Provides estimate of mapping φ(θ) Evaluate auxiliary likelihood p A (Ý φ(θ, Ü1:n)) Not ABC. No summary statistics or ABC tolerance Theoretically behaves very different to ABC II

14 pdbil (Cont...) Ultimate target of pdbil method p A (θ Ý) p A (Ý φ(θ))p(θ). Unfortunately binding function unknown, estimate via n iid simulations from generative model, φ θ,n = φ(θ, Ü1:n) Target distribution of the method becomes p A,n (θ Ý) p A,n (Ý θ)p(θ), where p A,n (Ý θ) = p A (Ý φ θ,n )p(ü1:n θ)d Ü1:n Ü1:n p A,n (Ý θ) can be estimated unbiasedly, which is enough (Andrieu and Roberts (29))

15 pdbil (Cont...) Comparing p A,n (θ Ý) with p A (θ Ý) Under suitable conditions p A,n (θ Ý) p A (θ Ý) as n In reality must choose finite n Unfortunately E[p A (Ý φ θ,n )] p A (Ý φ(θ)) thus p A,n (θ Ý) p A (θ Ý) for finite n Empirical evidence suggests loss of precision as n decreases Can anticipate better approximations for pbil for n large

16 pdbil (Cont...) Comparing p A (θ Ý) with p(θ Ý) Under suitable conditions pdbil is exact as n if generative model special case of auxiliary model Suggests in this context auxiliary model needs to be flexible to mimic behaviour of generative model Empirical evidence suggests for good approximation auxiliary likelihood must do well in non-negligible posterior regions

17 Synthetic Likeihood as a psbil Method Reduces data to set of summary statistics. Target distribution: p(θ s(ý)) p(s(ý) θ)p(θ). But p(s(ý) θ) is unknown Synthetic likelihood approach Wood (21) assumes a parametric model, p A (s(ý) φ(θ)). Thus a pbil method. Wood takes p A to be multivariate normal N(µ(θ),Σ(θ)) For some θ, simulate n iid replicates, Ü1:n = (s(ü1),...,s(ün)), iid sample from p(s(ý) θ). Fit auxiliary model to this dataset: µ(ü1:n,θ) = 1 n s(üi), n Σ(Ü1:n,θ) = 1 n i=1 n (s(üi) µ(ü1:n,θ))(s(üi) µ(ü1:n,θ)) T, i=1

18 ABC as BIL with Non-Parametric Auxiliary Model (npbil) ABC is recovered as a BIL method by selecting a non-parametric auxiliary likelihood Full data (npdbil): Choose φ(θ, Ü1:n) = Ü1:n p A (Ý φ(θ, Ü1:n)) = p A (Ý Ü1:n) = 1 n n K ǫ (ρ(ý, Üi)), i=1 Data reduced to summary statistic (npsbil): 1 n n K ǫ (ρ( (Ý), (Üi))). i=1 ABC II is a special npsbil method where summary statistics come from a parametric auxiliary model

19 pbii Methods Figure: pbii Methods

20 BIL pbil npbil=abc pdbil psbil npdbil npsbil Key: BIL - B ayesian I ndirect L ikelihood pbil - p arametric BIL pdbil - pbil based on full d ata psbil - pbil based on s ummary statistic ABC - a pproximate B ayesian c omputation npbil - n on- p arametric BIL (equivalent to ABC) npdbil - npbil based on full d ata npsbil - npbil based on s ummary statistic ABC II - ABC I ndirect I nference ABC IP - ABC I ndirect P arameter ABC IL - ABC I ndirect L ikelihood ABC IS - ABC I ndirect S core ABC II ABC IP ABC IL ABC IS Figure: BIL framework

21 Theoretical Comarison pdbil 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, pdbil not exact in general, even when n Some Toy Examples Later... Choice of Auxiliary Model ABC II requires good summarisation of data (independent of specification of generative model) Auxiliary model for pdbil does not necessarily have to fit data well (if generative model mis-specified). Requires flexibility wrt chosen generative model If generative model (approx) correct and fits data well then same auxiliary model may be chosen Comparison of pdbil and ABC II

22 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

23 Sampling from pdbil Target - MCMC pdbil 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

24 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 pdbil as n approximates Poisson(λ) likelihood with Normal(λ, λ) likelihood. Not exact..8.6 n = 1 n = 1 n = true 4 true auxiliary.4.2 log likelihood λ Acceptance probabilities: n = 1 46%, n = 1 67%, n = 1 72% and n = 1 73% for increasing n

25 Toy Example 1 Results for pdbil 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 = log likelihood 32 true τ unspecified τ=16 τ= λ

26 Toy Example 2 True model t-distribution(µ, σ, ν = 1) and auxiliary model t-distribution(µ, σ, ν) Here pdbil 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=

27 Toy Example 2 cont true true 1.8 ABC IP ABC IP ABC IL ABC IL 1.6 ABC IS 2 ABC IS

28 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

29 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

30 pdbil Results n=1 n=1 3 n=5 posterior (a) a (b) b (c) g (d) k Acc Prob: 2.8% for n = 1, 13.1% for n = 1 and 2.8% for n = 5

31 ABC II (no regression adjustment) posterior ABC IP 15 3 ABC IL ABC IS (e) a (f) b (g) g (h) k

32 Compare all (note ABC has regression adjustment) 3 ABC ABC IS ABC IL ABC IP pdbil n=5 posterior (i) a (j) b (k) g (l) k

33 4 Component Auxiliary Mixture Model 4 3 ABC ABC IS ABC IL ABC IP pdbil n=5 posterior (m) a (n) b (o) g (p) k

34 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

35 Proporton of Matures Time

36 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

37 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 )

38 pdbil Results 15 n=1 n=2 n= x (q) ν (r) µ I (s) µ L (t) β

39 pbii results (ABC II no regression adjustment) ABC IS ABC IP ABC IL pdbil n= x 1 3 (u) ν 1 2 (v) µ I (w) µ L (x) β

40 pbii results (ABC has regression adjustment) 2 15 ABC IS REG ABC IP REG ABC IL REG pdbil n= x (y) ν (z) µ L

41 Discussion pdbil very different to ABC II theoretically pdbil 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

42 Key References Drovandi et al. (211). Approximate Bayesian Computation using Indirect Inference. JRSS C. Drovandi, Pettitt and Lee (214). Bayesian Indirect Inference using a Parametric Auxiliary Model. 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.