Bayesian Regression for a Dirichlet Distributed Response using Stan

TL;DR

This paper introduces a Bayesian regression model for compositional data using the Dirichlet distribution, implemented in Stan, to analyze how covariates influence response proportions.

Contribution

It provides a practical implementation of Dirichlet regression in Stan based on recent formulations, facilitating Bayesian analysis of compositional data.

Findings

Demonstrates the implementation of Dirichlet regression in Stan.

Provides a framework for Bayesian inference on compositional data.

Illustrates the model with practical examples.

Abstract

For an observed response that is composed by a set - or vector - of positive values that sum up to 1, the Dirichlet distribution (Bol'shev, 2018) is a helpful mathematical construction for the quantification of the data-generating mechanics underlying this process. In applications, these response-sets are usually denoted as proportions, or compositions of proportions, and by means of covariates, one wishes to manifest the underlying signal - by changes in the value of these covariates - leading to differently distributed response compositions. This article gives a brief introduction into this class of regression models, and based on a recently developed formulation (Maier, 2014), illustrates the implementation in the Bayesian inference framework Stan.