Distribution-Free Bayesian multivariate predictive inference

TL;DR

This paper presents a Bayesian multivariate predictive inference framework using hierarchical models based on finite Polya trees, enabling flexible modeling of unknown distributions and constructing conformal prediction sets with finite-sample guarantees.

Contribution

The paper introduces a novel hierarchical Bayesian model with finite Polya trees for multivariate prediction and an algorithm for conformal prediction sets with finite-sample guarantees.

Findings

Effective modeling of unknown multivariate distributions.

Successful implementation on simulated data.

Finite-sample probability assurances for predictions.

Abstract

We introduce a comprehensive Bayesian multivariate predictive inference framework. The basis for our framework is a hierarchical Bayesian model, that is a mixture of finite Polya trees corresponding to multiple dyadic partitions of the unit cube. Given a sample of observations from an unknown multivariate distribution, the posterior predictive distribution is used to model and generate future observations from the unknown distribution. We illustrate the implementation of our methodology and study its performance on simulated examples. We introduce an algorithm for constructing conformal prediction sets, that provide finite sample probability assurances for future observations, with our Bayesian model.