Introduction Bayes Design Intractable Likelihoods Design Optimisation Example Ending
Bayesian Experimental Design for Models with
Intractable Likelihoods Using Indirect Inference
Dr Chris DrovandiQueensland University of Technology
Collaborators: Tony Pettitt, Caitriona RyanAcknowledgements: Australian Research Council Discovery
Grant and organisers of DEMA
December 14, 2015
Chris Drovandi DEMA 2015, Sydney
Introduction Bayes Design Intractable Likelihoods Design Optimisation Example Ending
Bayesian Experimental Design
Set-up: Have prior distribution p(θ) for parameter ofstatistical model, with likelihood p(y |θ,d ).Wish to design for next n observations. Design variable:d = (d1, . . . , dn).
Define (general) utility function u(d , y ,θ). y is future data.
u(d ) = Eθ,y [u(d , y , θ)] = ∫y ∫θ u(d , y , θ)p(y |d , θ)p(θ)dθdy ,Objective d∗ = argmaxd∈D u(d ).Too difficult to do directly
Chris Drovandi DEMA 2015, Sydney
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Bayesian utility functions
Parameter Estimation Design
Kullback-Leibler divergence u(d , y ) = KL(p(θ|y ,d )||p(θ)).Mutual info between θ and y .Concentration of Posterior distribution
Set u(d , y) as entropy of p(θ|y ,d)Posterior precision u(d , y) = 1/det(var(θ|y ,d))
These utility functions assume that the likelihood function is easilycomputable!!!
Chris Drovandi DEMA 2015, Sydney
Introduction Bayes Design Intractable Likelihoods Design Optimisation Example Ending
Dealing with Intractable Likelihood - Parameter Estimation
To estimate u(d , y ) need to obtain posterior distribution
p(θ|y ,d ) ∝ p(y |θ,d )p(θ).But what to do if p(y |θ,d ) is intractable?
Approximate Bayesian Computation (ABC)Bayesian Indirect Inference (II)
Chris Drovandi DEMA 2015, Sydney
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Approximate Bayesian Computation
simulation based method that does not involve likelihoodevaluations
involves a joint ‘approximate’ posterior distribution
p(θ, x |y , ǫ) ∝ g(y |x , ǫ)p(x |θ)p(θ)where g(y |x , ǫ) is a weighting functionpopular choice g(y |x , ǫ) = 1(ρ(y , x) ≤ ǫ)
how to choose ρ(y , x)?usually based on small set of summary statistics
choice of ǫ trade-off between accuracy and efficiency
Chris Drovandi DEMA 2015, Sydney
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Rejection ABC
1 generate θi ∼ p(θ) for i = 1, . . . ,N
2 simulate x i ∼ p(y |θi ,d ) for i = 1, . . . ,N
3 compute discrepancies ρi = ρ(y , x i ) for i = 1, . . . ,N, creatingparticles {θi , ρi}Ni=1
4 sort particle set via discrepancy ρ
5 calculate ǫ = ρ⌊αN⌋ (where ⌊·⌋ denotes the floor function)
ABC posterior samples = {θi |ρi ≤ ǫ}αNi=1
Steps 1 and 2 are independent of data and {θi , x i }Ni=1 can bestored.Discrepancy function
ρ(y , x) = D∑
i=1
|yi − xi |
stdp(θ)(xi )(1)
Chris Drovandi DEMA 2015, Sydney
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ABC Rejection in Design (Drovandi and Pettitt 2013)
Discretise design space (e.g. one (time) dimensiontmin, tmax , tinc )
Store the prior simulations at each design point in the designspace
Search over discretised design space for optimal design
For each proposed design d ∗ can use the relevant stored priorsimulations to sample from p(θ|y ,d ∗, ǫ)
Drawbacks:
Must search over discretised design space
Storage intensive (does not scale to increase in designdimension)
Choosing ǫ (here we end up having different ǫ for eachposterior approximation)
Require larger ǫ for larger number of observations
We use Bayesian indirect inference to overcome these drawbacksChris Drovandi DEMA 2015, Sydney
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Bayesian Indirect Inference
Assume that there exists an auxiliary model with a fully tractablelikelihood pa(y |φ,d ), so that approximately
p(y |θ,d ) ∝ pa(y |φ = g(θ),d ).Unfortunately the mapping function g(θ) is unknown but can beestimated via simulation, g(θ). Then we obtain the followingapproximate posterior:
pa(θ|y ,d ) ∝ pa(y |φ = g(θ), d)p(θ).
See Smith (1993) for a classical version of this approach.
Chris Drovandi DEMA 2015, Sydney
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Estimating the Mapping Function
An estimate of the auxiliary parameter φ is found by
φ = g(θ) = argmaxφ∈Φ
pa(x |φ,dT ), x ∼ p(x |θ,dT ),Free to choose the training design dT (technically also adesign problem!).
To bring down variance of φ = g(θ) can consider simulatingm independent datasets at θ, x1:m = (x1, . . . , xm).
Do this for a collection of n samples across the prior space to
obtain the estimated mapping {θi , φi}ni=1. Store the mapping.
Chris Drovandi DEMA 2015, Sydney
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Approximating the Posterior
Consider a dataset y . Can obtain an importance samplingapproximation of the II posterior with weights
W i ∝ pa(y |φi,d ) for i = 1, . . . , n.
{W i ,θi}ni=1 can be used to estimate u(d , y ).Chris Drovandi DEMA 2015, Sydney
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Key Points of Indirect Inference Approach
Advantages:
Free to optimise over continuous design space
Requires storage of the mapping {θi , φi}ni=1 only. Shouldscale better with design dimension.
No need to choose ǫ.
Scale better with number of observations (see later)
Drawbacks:
User must select auxiliary model. Not an obvious choice in allapplications.
Chris Drovandi DEMA 2015, Sydney
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Design Optimisation
Once a strategy has been devised to estimate u(d , y ) anyoptmisation approach can be used to solve argmax u(d )where u(d ) = ∫y u(d , y )p(y |d )dy .Here we used the Muller (1999) algorithm. Involves MCMCsimulation from augmented probability model with marginalu(d )J then calculation of multivariate mode (very hard!)
Chris Drovandi DEMA 2015, Sydney
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Design Optimisation - Improvement to Muller
A large value of J is required to substantially sharpen theutility surface (computationally intensive)
Consider the modified utility
u(d , y ) = max{(u(d , y )− sup), 0}, (2)
where up is the utility calculated from the prior distribution;
up = 1/det(Var(θ)). (3)
Scaling factor s helps to ensure utility is positive.
Intuitively: how much utility do we obtain above and beyondthe prior.
u(d ) has same mode as u(d ) but u has greater curvature.
Chris Drovandi DEMA 2015, Sydney
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Sharpening the Utility Surface with u
0 100 200 3000
1
2
3
4
5
x 1011
Design (days)
u
0 100 200 3000
5
10
x 1010
Design (days)
u
Chris Drovandi DEMA 2015, Sydney
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Motivating Example - Macroparasite Immunity
estimate parameters of Markov process model explainingmacroparasite population development with host immunity
212 hosts (cats) i = 1, . . . , 212.Each cat injected with li juvenile Brugia pahangi larvae
at time ti host is sacrificed and number of mature parasitesrecorded
three variable problemM(t) matures, L(t) juveniles, I (t) immunity
only L(0) and M(ti ) observed for each host
[Durham et al (1972)]
Chris Drovandi DEMA 2015, Sydney
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0 200 400 600 800 1000 12000
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Time
Pro
porto
n of
Mat
ures
Chris Drovandi DEMA 2015, Sydney
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Trivariate Markov Process of Riley et al (2003)
M(t) L(t)
I(t)
Mature Parasites Juvenile Parasites
Immunity
Maturation
Gain of immunity Loss of immunity
Death due to im
munity
Natural death
Natural death
Invisible
Invisible
Invisible
γL(t)
νL(t) µI I (t)
βI(t)L
(t)
µLL(t)
µMM
(t)
Chris Drovandi DEMA 2015, Sydney
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Design Problem
How to choose the sacrifice times and initial juvenile injections togain most information about parameters (ν and µL)?
Chris Drovandi DEMA 2015, Sydney
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Auxiliary Beta-Binomial model
The data show too much variation for Binomial
A Beta-Binomial model has an extra parameter to capturedispersion
p(mi |αi , βi ) =
(
li
mi
)
B(mi + αi , li −mi + βi )
B(αi , βi ),
Useful reparameterisation pi = αi/(αi + βi ) andξi = 1/(αi + βi )
Relate the proportion and over dispersion parameters to time,ti , and initial larvae, li , covariates
logit(pi ) = fp(ti , li) = β0 + β1 log(ti) + β2 log(ti)2,
log(ξi ) = η,
Four auxiliary parameters φ = (β0, β1, β2, η)
Chris Drovandi DEMA 2015, Sydney
Introduction Bayes Design Intractable Likelihoods Design Optimisation Example Ending
II design choices (Design just for sacrifice times)
For dT we consider 1000 randomly selected sacrifice times in (30,300) days where the initial number of larvae is set to 100 for everyhost in the training design. m = 1 and n = 10, 000
0.5 1 1.5 2 2.5
x 10−3
−1.5
−1
−0.5
0
ν
β 0
0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045−1.5
−1
−0.5
0
µL
β 0
Chris Drovandi DEMA 2015, Sydney
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Results - Sacrifice Times Only
method design (days) II utility ABC utility
II design 77.4 5.00 5.95ABC design 99 4.97 6.08
II design (69.7, 97.5) 5.38 7.20ABC design (71, 127) 5.35 7.18
II design (56.5, 82.8, 120.3) 5.73 7.39ABC design (95, 105, 231) 5.54 7.28
II design (50.5, 75.2, 98.2, 143.8) 6.10 7.28ABC design (79, 121, 231, 273) 5.61 7.07
Chris Drovandi DEMA 2015, Sydney
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Extend to Sacrifice Times and Initial Juveniles
Extend the auxiliary model (only one extra parameter)
logit(pi ) = fp(ti , li ) = β0 + β1 log(ti ) + β2 log(ti )2,
log(ξi ) = fξ(ti , li) = η0 + η1li ,dT : 1000 hosts in the training design, where the sacrifice times aresampled randomly in (30, 300) days and the initial larvae countsare randomly taken from the set of integers between 100 and 200inclusive.
Chris Drovandi DEMA 2015, Sydney
Introduction Bayes Design Intractable Likelihoods Design Optimisation Example Ending
Results with Initial Juveniles
One host: 69.7 days, 111 juveniles
Two hosts: 1st host 60.9 days, 108 juveniles. 2nd host 135.3 daysand 105 juveniles
Validation: consider 1 host 70 days, estimated utility with ABC for100, 150, 200 juvenilesThe expected utilities for 100, 150 and 200 initial larvae areroughly 6.05 × 1011, 5.32 × 1011 and 4.93 ×1011. Indeed 100juveniles appears optimal.
Chris Drovandi DEMA 2015, Sydney
Introduction Bayes Design Intractable Likelihoods Design Optimisation Example Ending
Future Work
Obtain training design dT optimally.
Smooth out the estimated mapping to obtain more precisemapping
Develop a Laplace approximation of II posterior (allow forhigh-dimensional design for which importance sampling is notsuitable)
Other motivating applications???
Chris Drovandi DEMA 2015, Sydney
Introduction Bayes Design Intractable Likelihoods Design Optimisation Example Ending
Key References
Ryan, Drovandi and Pettitt (2015). Optimal Bayesianexperimental design for models with intractable likelihoodsusing indirect inference applied to biological process models.Bayesian Analysis (to appear)
Drovandi and Pettitt (2013). Bayesian experimental design formodels with intractable likelihoods. Biometrics
Drovandi, Pettitt and Lee (2015). Bayesian indirect inferenceusing a parametric auxiliary model. Statistical Science.
Gallant and McCulloch (2009). On the determination ofgeneral scientific models with application to asset pricing.Journal of the American Statistical Association
Publication page:http://eprints.qut.edu.au/view/person/Drovandi,_Christopher
Chris Drovandi DEMA 2015, Sydney