Joint Bayesian Parameter and Model Order Estimation for Low-Rank Probability Mass Tensors

TL;DR

This paper introduces a Bayesian variational inference approach to jointly estimate the low-rank probability mass function tensor and its rank from data, eliminating the need for cross-validation and improving efficiency.

Contribution

It proposes a novel Bayesian framework with variational inference for automatic rank detection and accurate tensor-based PMF estimation in statistical modeling.

Findings

Improves estimation accuracy over existing methods

Automatically infers the tensor rank from data

Reduces computational resources needed for model selection

Abstract

Obtaining a reliable estimate of the joint probability mass function (PMF) of a set of random variables from observed data is a significant objective in statistical signal processing and machine learning. Modelling the joint PMF as a tensor that admits a low-rank canonical polyadic decomposition (CPD) has enabled the development of efficient PMF estimation algorithms. However, these algorithms require the rank (model order) of the tensor to be specified beforehand. In real-world applications, the true rank is unknown. Therefore, an appropriate rank is usually selected from a candidate set either by observing validation errors or by computing various likelihood-based information criteria, a procedure that could be costly in terms of computational time or hardware resources, or could result in mismatched models which affect the model accuracy. This paper presents a novel Bayesian framework for estimating the low-rank components of a joint PMF tensor and simultaneously inferring its rank from the observed data. We specify a Bayesian PMF estimation model and employ appropriate prior distributions for the model parameters, allowing the rank to be inferred without cross-validation.We then derive a deterministic solution based on variational inference (VI) to approximate the posterior distributions of various model parameters. Numerical experiments involving both synthetic data and real classification and item recommendation data illustrate the advantages of our VI-based method in terms of estimation accuracy, automatic rank detection, and computational efficiency.