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Efficient Bayesian Experimental Design for Implicit Models

Steven Kleinegesse, Michael Gutmann

arXiv:1810.09912v2stat.MLcs.LGstat.COstat.ME

TL;DR

Bayesian experimental design for implicit models is challenging because likelihoods are intractable and both posterior approximation and utility evaluation are difficult. The paper uses LFIRE to estimate mutual-information utility and Bayesian optimization to select designs. The authors report more efficient optimization that supports higher-dimensional designs.

  • Problem

    Bayesian experimental design has seen little work for implicit models, whose intractable likelihoods make posterior computation difficult and utility evaluation costly.

    Source: §1 · p. 2

  • Method

    The framework uses LFIRE density-ratio estimates to approximate posterior distributions and mutual information, then Bayesian optimization to maximize expected utility.

    Source: §1 · p. 2, §2.1 · p. 3, §2.2 · p. 3

  • Results

    Bayesian optimization yields similar expected utility to grid search while concentrating evaluations in high-utility regions, and the authors report greater efficiency and higher-dimensional designs.

    Source: §1 · p. 2, §3.2 · p. 5

  • Takeaways & Limitations

    Preliminary results suggest the framework extends to sequential designs, a setting barely explored for models with intractable likelihoods.

    Source: §4 · p. 8

    Applicability of high-dimensional Bayesian optimization in hundreds of dimensions remains uninvestigated, and likelihood-free posterior estimates are approximate.

    Source: §4 · p. 8

Abstract

from arXiv · show

Bayesian experimental design involves the optimal allocation of resources in an experiment, with the aim of optimising cost and performance. For implicit models, where the likelihood is intractable but sampling from the model is possible, this task is particularly difficult and therefore largely unexplored. This is mainly due to technical difficulties associated with approximating posterior distributions and utility functions. We devise a novel experimental design framework for implicit models that improves upon previous work in two ways. First, we use the mutual information between parameters and data as the utility function, which has previously not been feasible. We achieve this by utilising Likelihood-Free Inference by Ratio Estimation (LFIRE) to approximate posterior distributions, instead of the traditional approximate Bayesian computation or synthetic likelihood methods. Secondly, we use Bayesian optimisation in order to solve the optimal design problem, as opposed to the typically used grid search or sampling-based methods. We find that this increases efficiency and allows us to consider higher design dimensions.

1 INTRODUCTION

Bayesian experimental design allocates experiment resources to choose data-collection configurations that optimize cost and performance. For implicit models, the paper combines LFIRE-based mutual-information utility with Bayesian optimization to address inference and search challenges.

  • Experimental design allocates resources to choose configurations that may improve parameter estimation or compare competing models.Source: §1 · p. 1
  • Implicit models have intractable likelihoods and are specified by stochastic data-generating processes, yet comparatively little Bayesian experimental-design work addresses them.Source: §1 · p. 1
  • Mutual information is a principled utility for Bayesian design, but posterior-density difficulties have led prior work to use alternatives or approximations such as ABC, moment closure, and KDE.Source: §1 · p. 2
  • For implicit models, expected utility is unavailable in closed form, costly to evaluate, and lacks accessible gradients, while grid search becomes infeasible at high dimensions.Source: §1 · p. 2
  • The framework uses LFIRE rather than traditional ABC or synthetic likelihood to approximate posteriors and use mutual information between parameters and data as utility.Source: §1 · p. 2
  • Bayesian optimization solves the design optimization problem more efficiently and allows consideration of higher design dimensions.Source: §1 · p. 2

2 PROPOSED METHOD

The framework addresses utility evaluation and design optimisation as two computational steps for implicit models.

2.1 Computing the Utility

  • Mutual information between parameters and data serves as the utility, and LFIRE ratios let the method estimate it without explicit posterior and prior densities.LFIRE approximates the likelihood-to-marginal ratio through logistic regression on simulated data, which also provides the posterior-to-prior ratio.Source: §2.1 · pp. 2–3
  • For each design, prior parameter draws are paired with simulated data, LFIRE ratios are computed, and their aggregation approximates the expected utility.The stated computational cost lies mainly in LFIRE ratio computations when prior sampling and model simulation are not overly expensive.Source: §2.1 · p. 3

2.2 Optimising the Utility

  • Bayesian optimisation searches the costly expected-utility objective without requiring its gradients, using Gaussian Processes and Expected Improvement.The authors use a Gaussian kernel, while noting more scalable kernels for design dimensions above 10.Source: §2.2 · p. 3

2.3 Obtaining the Posterior

  • After selecting the optimal design, LFIRE ratios evaluated at the real observation weight prior samples, which are normalised and resampled to obtain posterior samples.The weights are the LFIRE ratios at the selected design and observed data.Source: §2.3 · p. 4

3 EXPERIMENTS

The experiments apply the design framework to two epidemiological implicit models, comparing measurement-time choices and posterior estimates. They also test non-myopic designs with eight measurement times.

3.1 Example Implicit Models

  • The Death Model tracks infection through susceptible-to-infected transitions, while the SIR Model also includes a recovered population; both experiments choose measurement times to estimate model parameters.The Death Model estimates infection rate b; the SIR Model estimates infection rate β and recovery rate γ.Source: §3.1 · p. 4, §3.2 · p. 4, §3.3 · p. 7

3.2 Death Model Results

  • For the Death Model, LFIRE utility closely matches the analytic utility near its peak, and BO and grid search yield similar utilities; BO concentrates evaluations near the peak.The analytic, grid-search LFIRE, and BO LFIRE optima are (1.40, 1.347), (1.10, 1.350), and (1.06, 1.359), respectively.Source: §3.2 · p. 5, Figure 1 · p. 5
  • Randomly selected Death Model designs produce fluctuating posterior densities; an unfavorable time such as τ = 4 can yield a wide posterior and uncertain parameter estimate.Figure 3 compares 50 random-design posterior densities with the BO mean posterior density.Source: §3.2 · p. 6, Figure 3 · p. 6
  • For the Death Model with eight design dimensions, BO converges after around 20 evaluations versus 76,904,685 for the stated four-dimensional grid; its utility gain over eight equidistant times is small.U(d*) = 1.27, compared with U(deq) = 1.21.Source: §3.2 · pp. 6–7, Figure 4 · p. 6

3.3 SIR Model Results

  • For the SIR Model, BO and grid search yield similar utilities, with optimal times τ* = 0.44 and τ* = 0.40, respectively; grid search is less resolved around the peak.The model has no tractable likelihood, so an analytic expected-utility comparison is unavailable.Source: §3.3 · p. 7, Figure 5 · p. 7
  • In the SIR Model with eight design dimensions, BO converges in around 15 utility evaluations, and both 95% credibility intervals contain the true parameter values.The posterior means are (0.13, 0.05), for β and γ respectively; the intervals are (0.02, 0.18) for β and (0.02, 0.07) for γ.Source: §3.3 · p. 8, Figure 7 · p. 8

4 CONCLUSIONS

The framework combines LFIRE-based mutual information with Bayesian optimisation for implicit models, and the authors report optimal designs in dimensions impossible for grid search. They identify limits in high-dimensional optimisation and approximate posterior estimation, while pointing to broader and sequential applications.

4 CONCLUSIONS

  • LFIRE makes mutual information usable as the utility for implicit models, and Bayesian optimisation finds designs in dimensions impossible for grid search.The approach also smooths noise from Monte Carlo approximations.Source: §4 · p. 8
  • High-dimensional Bayesian optimisation remains unvalidated in hundreds of dimensions, and the effects of approximate posterior estimates on mutual information need broader characterization.For the Death Model, utility functions were very similar and optimal designs barely changed despite LFIRE approximation.Source: §4 · p. 8
  • The framework is presented as applicable beyond the epidemiology examples, including models in neurobiology, cell biology, and physics.The authors identify experiment cost or time, beyond information gain, as a valuable consideration.Source: §4 · p. 8
  • Preliminary results suggest the framework extends to sequential designs that update parameter beliefs from experimental outcomes.The authors describe this setting as more realistic and little explored for models with intractable likelihoods.Source: §4 · p. 8

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