Adjusted Priors for Bayes Factors Involving Reparameterized Order Constraints

TL;DR

This paper emphasizes the importance of adjusting prior distributions for auxiliary parameters in reparameterized models with order constraints to accurately compute Bayes factors for model comparison.

Contribution

It derives adjusted priors for auxiliary parameters in reparameterized models, improving the accuracy of Bayes factor calculations in order-constrained hypotheses.

Findings

Uniform priors on auxiliary parameters can bias Bayes factors.

Adjusted priors lead to more accurate model evidence assessment.

Application to a multi-trial pair-clustering model demonstrates practical relevance.

Abstract

Many psychological theories that are instantiated as statistical models imply order constraints on the model parameters. To fit and test such restrictions, order constraints of the form $\theta_i \leq \theta_j$ can be reparameterized with auxiliary parameters $\eta\in [0,1]$ to replace the original parameters by $\theta_i = \eta\cdot\theta_j$. This approach is especially common in multinomial processing tree (MPT) modeling because the reparameterized, less complex model also belongs to the MPT class. Here, we discuss the importance of adjusting the prior distributions for the auxiliary parameters of a reparameterized model. This adjustment is important for computing the Bayes factor, a model selection criterion that measures the evidence in favor of an order constraint by trading off model fit and complexity. We show that uniform priors for the auxiliary parameters result in a Bayes factor that differs from the one that is obtained using a multivariate uniform prior on the order-constrained original parameters. As a remedy, we derive the adjusted priors for the auxiliary parameters of the reparameterized model. The practical relevance of the problem is underscored with a concrete example using the multi-trial pair-clustering model.