[Paper Review] On variance stabilisation by double Rao-Blackwellisation

Alessandra Iacobucci, Jean‐Michel Marin+1 more|ArXiv.org|Feb 25, 2008

Markov Chains and Monte Carlo Methods6 references3 citations

TL;DR

This paper proposes a double Rao-Blackwellisation technique for Population Monte Carlo (PMC) to improve variance stabilization and mode exploration in multimodal posterior distributions. By integrating over both the proposal and transition particles, the method reduces weight variance and enhances sampling efficiency, demonstrating superior mode detection in Gaussian mixture models compared to standard PMC and single Rao-Blackwellisation.

ABSTRACT

Population Monte Carlo has been introduced as a sequential importance sampling technique to overcome poor fit of the importance function. In this paper, we compare the performances of the original Population Monte Carlo algorithm with a modified version that eliminates the influence of the transition particle via a double Rao-Blackwellisation. This modification is shown to improve the exploration of the modes through an large simulation experiment on posterior distributions of mean mixtures of distributions.

Motivation & Objective

• To address the instability and poor mode exploration in standard Population Monte Carlo (PMC) when using random walk proposals.
• To reduce variance in importance weights by eliminating the influence of transition particles through a double Rao-Blackwellisation step.
• To improve the robustness and convergence of PMC in multimodal target distributions, especially with biased or complex data.
• To empirically validate the superiority of double Rao-Blackwellisation over single and original PMC in detecting multiple modes in Gaussian location mixtures.
• To reduce reliance on a large number of PMC iterations by enhancing early convergence through improved weight stability.

Proposed method

• The method applies double Rao-Blackwellisation by integrating over both the previous sample and the transition particle in the importance weight calculation.
• It modifies the standard PMC algorithm by replacing the actual proposal with a mixture proposal in the weight computation, leveraging conditional expectations to reduce variance.
• The importance weight is computed as the expectation of π(x)/f(x) given the previous sample and the transition kernel, using a mixture of proposal components.
• The algorithm uses sequential resampling and adaptive proposal updates based on past samples, with double Rao-Blackwellisation applied at each iteration.
• A large-scale Monte Carlo experiment is conducted on a Gaussian location mixture model with outliers to simulate multimodal posteriors.
• The performance is evaluated via detection rates of all modes and entropy criterion monotonicity over iterations.

Experimental results

Research questions

• RQ1 Does double Rao-Blackwellisation significantly reduce variance in importance weights compared to standard and single-Rao-Blackwellised PMC?
• RQ2 Can double Rao-Blackwellisation improve the detection of multiple modes in multimodal posterior distributions?
• RQ3 Does the double Rao-Blackwellised PMC converge faster and require fewer iterations than standard PMC?
• RQ4 How does the method perform under high-dimensional or complex target distributions with multiple modes?
• RQ5 To what extent does the double Rao-Blackwellisation reduce sensitivity to poor initial proposals?

Key findings

• The double Rao-Blackwellised PMC achieved a 10-20% higher average mode detection rate than the single Rao-Blackwellised version after 10 iterations, particularly for larger sample sizes.
• In the double Rao-Blackwellised case, the average detection rate reached 0.64 at n=1000 and p=0.10, compared to 0.54 in the single version.
• The double Rao-Blackwellisation reduced the variance of importance weights, leading to more stable entropy criterion evolution and faster convergence.
• The method showed robustness to poor initial proposals, with stable performance even with low initial sample sizes (N=2835).
• The double Rao-Blackwellised PMC maintained consistent mode detection across all tested values of p and n, with detection rates above 0.5 even at p=0.60 and n=1000.
• Figure 4 visually confirmed that double Rao-Blackwellisation led to better coverage of all modal basins and reduced ridge structures in non-target regions.