PSD Representations for Effective Probability Models
This work addresses the need for versatile probability models in density estimation and inference, though it appears incremental as it builds on existing PSD models for non-negative functions.
The paper tackles the problem of modeling probability densities for probabilistic inference by proposing positive semi-definite (PSD) models, showing they offer strong theoretical guarantees for approximation and generalization and enable efficient closed-form operations for sum and product rules.
Finding a good way to model probability densities is key to probabilistic inference. An ideal model should be able to concisely approximate any probability while being also compatible with two main operations: multiplications of two models (product rule) and marginalization with respect to a subset of the random variables (sum rule). In this work, we show that a recently proposed class of positive semi-definite (PSD) models for non-negative functions is particularly suited to this end. In particular, we characterize both approximation and generalization capabilities of PSD models, showing that they enjoy strong theoretical guarantees. Moreover, we show that we can perform efficiently both sum and product rule in closed form via matrix operations, enjoying the same versatility of mixture models. Our results open the way to applications of PSD models to density estimation, decision theory and inference.