Martingale Posteriors from Score Functions

TL;DR

This paper introduces a novel method for constructing martingale posteriors using score functions, which avoids MCMC and allows parallel implementation, providing theoretical insights and practical illustrations.

Contribution

The paper proposes a score function-based martingale posterior approach that does not rely on MCMC, enabling efficient parallel computation and broad applicability.

Findings

The method converges under certain regularity conditions.

It can be implemented efficiently in parallel.

Theoretical properties are established and demonstrated through examples.

Abstract

Uncertainty associated with statistical problems arises due to what has not been seen as opposed to what has been seen. Using probability to quantify the uncertainty the task is to construct a probability model for what has not been seen conditional on what has been seen. The traditional Bayesian approach is to use prior distributions for constructing the predictive distributions, though recently a novel approach has used density estimators and the use of martingales to establish convergence of parameter values. In this paper we reply on martingales constructed using score functions. Hence, the method only requires the computing of gradients arising from parametric families of density functions. A key point is that we do not rely on Markov Chain Monte Carlo (MCMC) algorithms, and that the method can be implemented in parallel. We present the theoretical properties of the score driven martingale posterior. Further, we present illustrations under different models and settings.