# Generalized Cumulative Shrinkage Process Priors with Applications to Sparse Bayesian Factor Analysis

## TL;DR

This paper extends the cumulative shrinkage process (CUSP) prior with arbitrary stick-breaking representations and shows that exchangeable spike-and-slab priors can be viewed as generalized CUSP priors, enhancing sparse Bayesian factor analysis.

## Contribution

It introduces a generalized CUSP prior with arbitrary stick-breaking, and demonstrates that exchangeable spike-and-slab priors are finite generalized CUSP priors, facilitating flexible sparse Bayesian modeling.

## Key findings

- The generalized CUSP prior can be constructed from various stick-breaking distributions.
- Exchangeable spike-and-slab priors imply increasing shrinkage without explicit ordering.
- A new triple gamma-based spike-and-slab prior improves factor number estimation.

## Abstract

The paper discusses shrinkage priors which impose increasing shrinkage in a sequence of parameters. We review the cumulative shrinkage process (CUSP) prior of Legramanti et al. (2020), which is a spike-and-slab shrinkage prior where the spike probability is stochastically increasing and constructed from the stick-breaking representation of a Dirichlet process prior. As a first contribution, this CUSP prior is extended by involving arbitrary stick-breaking representations arising from beta distributions. As a second contribution, we prove that exchangeable spike-and-slab priors, which are popular and widely used in sparse Bayesian factor analysis, can be represented as a finite generalized CUSP prior, which is easily obtained from the decreasing order statistics of the slab probabilities. Hence, exchangeable spike-and-slab shrinkage priors imply increasing shrinkage as the column index in the loading matrix increases, without imposing explicit order constraints on the slab probabilities. An application to sparse Bayesian factor analysis illustrates the usefulness of the findings of this paper. A new exchangeable spike-and-slab shrinkage prior based on the triple gamma prior of Cadonna et al. (2020) is introduced and shown to be helpful for estimating the unknown number of factors in a simulation study.