Loss as the Inconsistency of a Probabilistic Dependency Graph: Choose Your Model, Not Your Loss Function

TL;DR

This paper presents a unified perspective on loss functions as measures of inconsistency in probabilistic dependency graphs, linking them to statistical divergences, regularizers, and variational inference, offering new insights and visual tools.

Contribution

It introduces the concept of PDG inconsistency as a unifying framework for understanding loss functions, divergences, and variational bounds in probabilistic modeling.

Findings

Many standard loss functions are derived as PDG inconsistencies.

The approach justifies the connection between regularizers and priors.

ELBO and its variants naturally emerge from modeling assumptions.

Abstract

In a world blessed with a great diversity of loss functions, we argue that that choice between them is not a matter of taste or pragmatics, but of model. Probabilistic depencency graphs (PDGs) are probabilistic models that come equipped with a measure of "inconsistency". We prove that many standard loss functions arise as the inconsistency of a natural PDG describing the appropriate scenario, and use the same approach to justify a well-known connection between regularizers and priors. We also show that the PDG inconsistency captures a large class of statistical divergences, and detail benefits of thinking of them in this way, including an intuitive visual language for deriving inequalities between them. In variational inference, we find that the ELBO, a somewhat opaque objective for latent variable models, and variants of it arise for free out of uncontroversial modeling assumptions -- as do simple graphical proofs of their corresponding bounds. Finally, we observe that inconsistency becomes the log partition function (free energy) in the setting where PDGs are factor graphs.