[Paper Review] Bootstrap Markov chain Monte Carlo and optimal solutions for the Law of Categorical Judgment (Corrected)

Greg Kochanski, Burton S. Rosner|arXiv (Cornell University)|Aug 9, 2010

Blind Source Separation Techniques23 references3 citations

TL;DR

This paper introduces a bootstrap Markov chain Monte Carlo (MCMC) algorithm that accelerates convergence for symmetric, convex posterior distributions—such as multivariate Gaussians—by using adaptive bootstrap resampling to generate efficient proposal steps. Applied to the Law of Categorical Judgment, the method successfully recovers correct parameters from practical-sized simulated rating data under Full Signal Detection Theory and its special cases.

ABSTRACT

A novel procedure is described for accelerating the convergence of Markov chain Monte Carlo computations. The algorithm uses an adaptive bootstrap technique to generate candidate steps in the Markov Chain. It is efficient for symmetric, convex probability distributions, similar to multivariate Gaussians, and it can be used for Bayesian estimation or for obtaining maximum likelihood solutions with confidence limits. As a test case, the Law of Categorical Judgment (Corrected) was fitted with the algorithm to data sets from simulated rating scale experiments. The correct parameters were recovered from practical-sized data sets simulated for Full Signal Detection Theory and its special cases of standard Signal Detection Theory and Complementary Signal Detection Theory.

Motivation & Objective

• To improve convergence speed of Markov chain Monte Carlo (MCMC) methods in Bayesian inference for symmetric, convex likelihoods.
• To develop a computationally efficient algorithm suitable for large-scale or high-dimensional posterior estimation.
• To validate the method on the Law of Categorical Judgment using realistic simulated rating scale data.
• To demonstrate accurate recovery of true parameters under Full Signal Detection Theory and its submodels.
• To provide confidence limits and maximum likelihood estimates using the proposed MCMC framework.

Proposed method

• Employs an adaptive bootstrap technique to generate candidate steps in the Markov chain, improving proposal efficiency.
• Uses resampled data from the current posterior to inform proposal distributions, reducing random walk behavior.
• Applies the method to symmetric, convex probability distributions such as multivariate Gaussians.
• Integrates the bootstrap-MCMC procedure into Bayesian estimation and maximum likelihood inference pipelines.
• Optimizes proposal variance dynamically based on resampled posterior estimates to enhance mixing and convergence.
• Validates performance on simulated rating data under the Law of Categorical Judgment with known ground-truth parameters.

Experimental results

Research questions

• RQ1 Can bootstrap-adaptive MCMC significantly accelerate convergence in symmetric, convex posterior distributions?
• RQ2 How accurately can the algorithm recover true parameters of the Law of Categorical Judgment from practical-sized simulated datasets?
• RQ3 Does the method maintain accuracy across different signal detection theory models, including Full, Standard, and Complementary Signal Detection Theory?
• RQ4 Can the algorithm reliably produce confidence intervals and maximum likelihood estimates in high-dimensional settings?
• RQ5 How does the bootstrap-MCMC approach compare to standard MCMC in terms of convergence speed and parameter recovery?

Key findings

• The bootstrap-MCMC algorithm achieves faster convergence than standard MCMC for symmetric, convex posterior distributions.
• True parameters of the Law of Categorical Judgment were successfully recovered from simulated rating scale data under Full Signal Detection Theory.
• The method accurately estimated parameters in both standard and complementary signal detection theory variants.
• The algorithm produced reliable confidence limits and maximum likelihood estimates using the same computational framework.
• The approach demonstrated robustness and efficiency on practical-sized datasets, suggesting applicability to real-world psychological and psychophysical data.
• The use of adaptive bootstrap resampling significantly improved proposal efficiency, reducing random walk behavior in the Markov chain.