# Exact Dimensionality Selection for Bayesian PCA

## TL;DR

This paper introduces a Bayesian model selection method for accurately determining the intrinsic dimensionality of high-dimensional data using a novel probabilistic PCA formulation with a normal-gamma prior, providing a closed-form marginal likelihood for optimal component inference.

## Contribution

It presents a new Bayesian approach with a closed-form marginal likelihood for exact dimensionality selection in PCA, including a heuristic for hyperparameter tuning.

## Key findings

- Competitive with state-of-the-art methods on simulated data
- Effective in non-asymptotic settings
- Provides a closed-form solution for model evidence

## Abstract

We present a Bayesian model selection approach to estimate the intrinsic dimensionality of a high-dimensional dataset. To this end, we introduce a novel formulation of the probabilisitic principal component analysis model based on a normal-gamma prior distribution. In this context, we exhibit a closed-form expression of the marginal likelihood which allows to infer an optimal number of components. We also propose a heuristic based on the expected shape of the marginal likelihood curve in order to choose the hyperparameters. In non-asymptotic frameworks, we show on simulated data that this exact dimensionality selection approach is competitive with both Bayesian and frequentist state-of-the-art methods.