[Paper Review] Note on the computation of the Metropolis-Hastings ratio for Birth-or-Death moves in trans-dimensional MCMC algorithms for signal decomposition problems

Alireza Roodaki, Julien Bect+1 more|arXiv (Cornell University)|Nov 27, 2011

Markov Chains and Monte Carlo Methods45 references3 citations

TL;DR

This paper corrects a long-standing error in the Metropolis-Hastings-Green (MHG) ratio for Birth-or-Death moves within trans-dimensional MCMC algorithms used in signal decomposition. It provides a mathematically rigorous derivation of the correct MHG ratio by analyzing mixture proposal kernels and applies the correction to the seminal work of Andrieu and Doucet (1999), showing that prior erroneous implementations have skewed posterior model distributions, particularly favoring fewer components due to incorrect acceptance probabilities.

ABSTRACT

Reversible jump MCMC (RJ-MCMC) sampling techniques, which allow to jointly tackle model selection and parameter estimation problems in a coherent Bayesian framework, have become increasingly popular in the signal processing literature since the seminal paper of Andrieu and Doucet (IEEE Trans. Signal Process., 47(10), 1999). Crucial to the implementation of any RJ-MCMC sampler is the computation of the so-called Metropolis-Hastings-Green (MHG) ratio, which determines the acceptance probability for the proposed moves. It turns out that the expression of the MHG ratio that was given in the paper of Andrieu and Doucet for "Birth-or-Death" moves---the simplest kind of trans-dimensional move, used in virtually all applications of RJ-MCMC to signal decomposition problems---was erroneous. Unfortunately, this mistake has been reproduced in many subsequent papers dealing with RJ-MCMC sampling in the signal processing literature. This note discusses the computation of the MHG ratio, with a focus on the case where the proposal kernel can be decomposed as a mixture of simpler kernels, for which the MHG ratio is easy to compute. We provide sufficient conditions under which the MHG ratio of the mixture can be deduced from the MHG ratios of the elementary kernels of which it is composed. As an application, we consider the case of Birth-or-Death moves, and provide a corrected expression for the erroneous ratio in the paper of Andrieu and Doucet.

Motivation & Objective

• To identify and correct a persistent error in the MHG ratio for Birth-or-Death moves in trans-dimensional MCMC algorithms used in signal decomposition.
• To provide a general framework for computing MHG ratios when the proposal kernel is a mixture of simpler kernels.
• To demonstrate that the error in Andrieu and Doucet's (1999) formulation leads to biased posterior model selection, favoring fewer components.
• To offer a corrected, mathematically sound expression for the MHG ratio applicable to signal decomposition problems with unknown numbers of sinusoidal components.
• To prevent continued propagation of the error in subsequent signal processing papers relying on the flawed formulation.

Proposed method

• Derives the MHG ratio for mixture proposal kernels by establishing sufficient conditions under which the ratio of the mixture can be deduced from the ratios of its elementary components.
• Applies the general framework to Birth-or-Death moves in signal decomposition, where the proposal kernel is a mixture of kernels corresponding to inserting or removing a sinusoidal component.
• Considers three representations: unsorted vectors, sorted vectors, and point processes, showing equivalence in MHG ratio computation under exchangeability.
• Identifies the core error: in the original formulation, birth moves insert the new component at the end, but death moves randomly select any component—causing inconsistency and zero acceptance probability for non-last components.
• Derives the corrected MHG ratio (Equation 5) that accounts for the correct Jacobian and proposal density ratio, ensuring detailed balance and correct posterior targeting.
• Validates the correction via simulation, comparing posterior model frequencies using the correct vs. erroneous ratio under Poisson and accelerated Poisson priors for model dimension.

Experimental results

Research questions

• RQ1 Why does the MHG ratio in Andrieu and Doucet's (1999) paper for Birth-or-Death moves contain a fundamental error?
• RQ2 Under what conditions can the MHG ratio of a mixture proposal kernel be computed from the ratios of its constituent kernels?
• RQ3 How does the incorrect MHG ratio affect posterior model selection in signal decomposition problems?
• RQ4 What is the correct expression for the MHG ratio in Birth-or-Death moves when using unsorted or sorted sinusoidal component representations?
• RQ5 Why does the erroneous acceptance probability lead to a systematic shift toward fewer components in the posterior distribution of model dimension?

Key findings

• The MHG ratio in Andrieu and Doucet's (1999) paper is incorrect because it assumes the new component is always inserted at the end during birth moves, but allows arbitrary removal during death moves, violating detailed balance.
• The corrected MHG ratio (Equation 5) accounts for the correct Jacobian and proposal density ratio, ensuring the Markov chain targets the correct posterior distribution.
• Simulation results show that using the erroneous ratio shifts the posterior distribution of model dimension $k$ significantly to the left, favoring fewer components, especially under non-uniform priors like the accelerated Poisson.
• The error has been reproduced in over 14 subsequent signal processing papers, all of which require correction to ensure valid Bayesian inference.
• The corrected ratio ensures that death moves which attempt to remove a component other than the last one have zero acceptance probability, preserving consistency with the birth move mechanism.
• The study underscores that even small errors in MCMC acceptance ratios can lead to plausible-looking but incorrect posterior inferences, highlighting the importance of rigorous measure-theoretic foundations.