Bayesian explorations in mathematical psychology

How can we best understand and analyze data obtained from psychological experiments? Throughout this dissertation, I will argue that this is best done by means of formal mathematical modeling using Bayesian inference. The goal of mathematical modeling is to capture regularities in the data using parameters that represent separate statistical of psychological processes. Mathematical models can take a variety of forms and this dissertation mirrors this diversity; I focus on descriptive and cognitive process models of performance in two-choice RT tasks, on multinomial processing tree models for the analysis of categorical data as well as well-known statistical models, such as the t test, analysis of variance, (partial) correlations, structural equation models, and mediation analysis. I focus on two applications of Bayesian inference for mathematical models: parameter estimation and model selection. With respect to parameter estimation, I demonstrate that the Bayesian approach can be extremely valuable for hierarchical models where maximum likelihood estimation becomes practically difficult. With respect to model selection, I rely on Bayes factors to measure statistical evidence in the context of competing cognitive models as well as standard statistical tests.