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Efficient MCMC Sampling with Expensive-to-Compute and Irregular Likelihoods

Conor Rosato, Harvinder Lehal, Simon Maskell, Lee Devlin, Malcolm Strens

TL;DR

This work tackles Bayesian inference with expensive, irregular likelihoods by developing subset-MCMC methods that operate on data-science scenario batches. It introduces data-driven proxies (quadratic and nearest-neighbour) and a computation-cost aware adaptive MCMC framework to direct proposals and screening within subset samplers, notably integrating proxies into HINTS. Through extensive experiments on synthetic, tall-data, high-correlation, and disease-model tasks, the authors show that HINTS with a quadratic proxy delivers the best sampling accuracy under a fixed computation budget, while proxies enable efficient pre-screening of proposals and large, directed moves. The results demonstrate that subset evaluations can temper exploration cheaply and that a carefully designed proxy can yield exact or near-exact sampling in challenging likelihood landscapes. The work thus provides a practical, scalable pathway for exact Bayesian inference in models with costly, irregular likelihoods such as PMCMC-based epidemiological models.

Abstract

Bayesian inference with Markov Chain Monte Carlo (MCMC) is challenging when the likelihood function is irregular and expensive to compute. We explore several sampling algorithms that make use of subset evaluations to reduce computational overhead. We adapt the subset samplers for this setting where gradient information is not available or is unreliable. To achieve this, we introduce data-driven proxies in place of Taylor expansions and define a novel computation-cost aware adaptive controller. We undertake an extensive evaluation for a challenging disease modelling task and a configurable task with similar irregularity in the likelihood surface. We find our improved version of Hierarchical Importance with Nested Training Samples (HINTS), with adaptive proposals and a data-driven proxy, obtains the best sampling error in a fixed computational budget. We conclude that subset evaluations can provide cheap and naturally-tempered exploration, while a data-driven proxy can pre-screen proposals successfully in explored regions of the state space. These two elements combine through hierarchical delayed acceptance to achieve efficient, exact sampling.

Efficient MCMC Sampling with Expensive-to-Compute and Irregular Likelihoods

TL;DR

This work tackles Bayesian inference with expensive, irregular likelihoods by developing subset-MCMC methods that operate on data-science scenario batches. It introduces data-driven proxies (quadratic and nearest-neighbour) and a computation-cost aware adaptive MCMC framework to direct proposals and screening within subset samplers, notably integrating proxies into HINTS. Through extensive experiments on synthetic, tall-data, high-correlation, and disease-model tasks, the authors show that HINTS with a quadratic proxy delivers the best sampling accuracy under a fixed computation budget, while proxies enable efficient pre-screening of proposals and large, directed moves. The results demonstrate that subset evaluations can temper exploration cheaply and that a carefully designed proxy can yield exact or near-exact sampling in challenging likelihood landscapes. The work thus provides a practical, scalable pathway for exact Bayesian inference in models with costly, irregular likelihoods such as PMCMC-based epidemiological models.

Abstract

Bayesian inference with Markov Chain Monte Carlo (MCMC) is challenging when the likelihood function is irregular and expensive to compute. We explore several sampling algorithms that make use of subset evaluations to reduce computational overhead. We adapt the subset samplers for this setting where gradient information is not available or is unreliable. To achieve this, we introduce data-driven proxies in place of Taylor expansions and define a novel computation-cost aware adaptive controller. We undertake an extensive evaluation for a challenging disease modelling task and a configurable task with similar irregularity in the likelihood surface. We find our improved version of Hierarchical Importance with Nested Training Samples (HINTS), with adaptive proposals and a data-driven proxy, obtains the best sampling error in a fixed computational budget. We conclude that subset evaluations can provide cheap and naturally-tempered exploration, while a data-driven proxy can pre-screen proposals successfully in explored regions of the state space. These two elements combine through hierarchical delayed acceptance to achieve efficient, exact sampling.
Paper Structure (56 sections, 36 equations, 11 figures, 10 tables, 3 algorithms)

This paper contains 56 sections, 36 equations, 11 figures, 10 tables, 3 algorithms.

Table of Contents

  1. Introduction
  2. Irregular Likelihoods
  3. Scenarios & Subsets
  4. Motivating Example
  5. Subset Samplers
  6. Contribution & Paper Outline
  7. Subset Metropolis-Hastings MCMC
  8. Adaptive MCMC
  9. Proxies in Subset Samplers
  10. Subset Samplers
  11. Subsample MCMC
  12. Austerity Sampler
  13. Confidence Samplers
  14. Firefly Sampler
  15. Hierarchical Importance With Nested Training Samples (HINTS)
  16. ...and 41 more sections

Figures (11)

  • Figure 1: Simplified example of an irregular log likelihood.
  • Figure 2: Quadratic proxies fitted by least-squares: conditionals in each of the 4 dimensions.
  • Figure 3: (a)--(d): proportions of the data set evaluated on each step by existing samplers; full MCMC = 1. Each uses a different number of iterations (x axis) for an equivalent computation budget. Integration of a proxy enables HINTS to use (e) a large proposal scale and (f) much less computation per iteration.
  • Figure 4: Firefly quadratic lower bound for single scenario, 4D noise-free task. The bounds are tight when the smooth likelihood has near-quadratic shape.
  • Figure 6: Sample of 16 sequences from 2D disease model.
  • ...and 6 more figures