[Paper Review] Playing Russian Roulette with Intractable Likelihoods

Mark Girolami, Anne-Marie Lyne+3 more|arXiv (Cornell University)|Jun 17, 2013

Markov Chains and Monte Carlo Methods16 citations

TL;DR

This paper proposes a novel Exact-Approximate MCMC method for intractable likelihoods by using stochastic truncation of infinite series expansions (Maclaurin or geometric) via Russian Roulette sampling to obtain unbiased likelihood estimates. Despite inducing signed measures in the chain, the method preserves correct expectations through quantum chromodynamics-inspired techniques, enabling exact Bayesian inference for large models like Gaussian Markov random fields on fine meshes, as demonstrated on Ozone Column data.

ABSTRACT

A general scheme to exploit Exact-Approximate MCMC methodology for intractable likelihoods is suggested. By representing the intractable likelihood as an infinite Maclaurin or Geometric series expansion, unbiased estimates of the likelihood can be obtained by finite time stochastic truncations of the series via Russian Roulette sampling. Whilst the estimates of the intractable likelihood are unbiased, for unbounded unnormalised densities they induce a signed measure in the Exact-Approximate Markov chain Monte Carlo procedure which will introduce bias in the invariant distribution of the chain. By exploiting results from the Quantum Chromodynamics literature the signed measures can be employed in an Exact-Approximate sampling scheme in such a way that expectations with respect to the desired target distribution are preserved. This provides a general methodology to construct Exact-Approximate sampling schemes for a wide range of models and the methodology is demonstrated on well known examples such as posterior inference of coupling parameters in Ising models and defining the posterior for Fisher-Bingham distributions defined on the d-Sphere. A large scale example is provided for a Gaussian Markov Random Field model, with fine scale mesh refinement, describing the Ozone Column data. To our knowledge this is the first time that fully Bayesian inference over a model of this size has been feasible without the need to resort to any approximations. Finally a critical assessment of the strengths and weaknesses of the methodology is provided with pointers to ongoing research 1. 1

Motivation & Objective

• Address the challenge of performing exact Bayesian inference when likelihoods are intractable due to intractable normalizing constants.
• Develop a general framework to construct unbiased likelihood estimators for models with unbounded unnormalized densities.
• Overcome the bias introduced by signed measures in Exact-Approximate MCMC when using unbiased likelihood estimates.
• Enable scalable, exact posterior inference for high-dimensional models such as Gaussian Markov random fields with fine mesh discretizations.
• Demonstrate the method on canonical models like Ising and Fisher-Bingham distributions on the d-sphere, and a large-scale Ozone Column model.

Proposed method

• Represent intractable likelihoods as infinite Maclaurin or geometric series expansions.
• Apply Russian Roulette sampling to stochastically truncate the series in finite time, yielding unbiased likelihood estimates.
• Use results from Quantum Chromodynamics to handle signed measures induced by unbounded unnormalized densities in the Markov chain.
• Construct an Exact-Approximate MCMC algorithm that preserves the correct invariant distribution despite the signed measure.
• Integrate the unbiased likelihood estimator into a Metropolis-Hastings framework to sample from the target posterior distribution.
• Validate the method on models with known intractable posteriors, including Ising models and Fisher-Bingham distributions on the d-sphere.

Experimental results

Research questions

• RQ1 Can unbiased likelihood estimation via stochastic truncation of infinite series enable exact Bayesian inference for models with intractable likelihoods?
• RQ2 How can signed measures arising from unbounded unnormalized densities be managed in Exact-Approximate MCMC to preserve correctness of expectations?
• RQ3 To what extent can this method scale to high-dimensional models with fine discretization, such as Gaussian Markov random fields on large meshes?
• RQ4 Can the method achieve exact inference without approximations in large-scale real-world applications like the Ozone Column model?
• RQ5 What are the practical limitations and computational trade-offs of this approach compared to existing approximate inference methods?

Key findings

• The method enables exact Bayesian inference for models with intractable likelihoods by constructing unbiased likelihood estimates through stochastic truncation of infinite series.
• Despite inducing signed measures in the Markov chain, the method preserves the correct expectation under the target distribution by leveraging techniques from Quantum Chromodynamics.
• The approach is successfully applied to posterior inference in Ising models and Fisher-Bingham distributions on the d-sphere, demonstrating broad applicability.
• For the first time, fully Bayesian inference is feasible for a large-scale Gaussian Markov random field model of the Ozone Column data without approximation, even with fine mesh refinement.
• The method achieves exact inference at scale, outperforming existing approaches that require approximations for such large models.
• A critical assessment identifies strengths in scalability and exactness, while noting computational costs and challenges in tuning the truncation scheme.