Exploiting Chain Rule and Bayes' Theorem to Compare Probability Distributions

TL;DR

This paper introduces a novel conditional transport method leveraging the chain rule and Bayes' theorem to compare probability distributions, improving generative model training by balancing mode coverage and collapse resistance.

Contribution

It proposes a new conditional transport approach that enhances distribution comparison and generative modeling, outperforming traditional methods on benchmark datasets.

Findings

CT balances mode-covering and mode-seeking behaviors

Replacing standard statistical distances with CT improves GAN performance

CT resists mode collapse effectively

Abstract

To measure the difference between two probability distributions, referred to as the source and target, respectively, we exploit both the chain rule and Bayes' theorem to construct conditional transport (CT), which is constituted by both a forward component and a backward one. The forward CT is the expected cost of moving a source data point to a target one, with their joint distribution defined by the product of the source probability density function (PDF) and a source-dependent conditional distribution, which is related to the target PDF via Bayes' theorem. The backward CT is defined by reversing the direction. The CT cost can be approximated by replacing the source and target PDFs with their discrete empirical distributions supported on mini-batches, making it amenable to implicit distributions and stochastic gradient descent-based optimization. When applied to train a generative model, CT is shown to strike a good balance between mode-covering and mode-seeking behaviors and strongly resist mode collapse. On a wide variety of benchmark datasets for generative modeling, substituting the default statistical distance of an existing generative adversarial network with CT is shown to consistently improve the performance. PyTorch code is provided.