# Bayesian cumulative shrinkage for infinite factorizations

## TL;DR

This paper introduces a novel cumulative shrinkage prior for models with unknown dimensions, such as factor analysis, improving dimension recovery and model inference through theoretical and practical advantages.

## Contribution

It proposes the cumulative shrinkage process, a new increasing shrinkage prior that enhances dimension inference in over-complete models like factor analysis.

## Key findings

- Improved ability to recover the true model dimension.
- Demonstrated advantages over existing methods in simulations.
- Effective in real personality traits data analysis.

## Abstract

There is a wide variety of models in which the dimension of the parameter space is unknown. For example, in factor analysis the number of latent factors is typically not known and has to be inferred from the observed data. Although classical shrinkage priors are useful in these contexts, increasing shrinkage priors can provide a more effective option, which progressively penalizes expansions with growing complexity. In this article we propose a novel increasing shrinkage prior, named the cumulative shrinkage process, for the parameters controlling the dimension in over-complete formulations. Our construction has broad applicability, simple interpretation, and is based on a sequence of spike and slab distributions which assign increasing mass to the spike as model complexity grows. Using factor analysis as an illustrative example, we show that this formulation has theoretical and practical advantages over current competitors, including an improved ability to recover the model dimension. An adaptive Markov chain Monte Carlo algorithm is proposed, and the methods are evaluated in simulation studies and applied to personality traits data.