# A tutorial on generalizing the default Bayesian t-test via posterior sampling and encompassing priors

## TL;DR

This paper presents a flexible, generalized approach to default Bayesian hypothesis testing, specifically extending the JZS t-test using posterior sampling and encompassing priors, enabling tailored hypothesis tests in applied research.

## Contribution

It introduces a novel, adaptable workflow for Bayesian hypothesis testing by combining the Savage-Dickey density ratio and encompassing priors with MCMC sampling in R.

## Key findings

- Demonstrates the use of the approach with illustrative examples
- Provides a comprehensive mathematical framework
- Enables customization of Bayesian hypothesis tests

## Abstract

With the advent of so-called default Bayesian hypothesis tests, scientists in applied fields have gained access to a powerful and principled method for testing hypotheses. However, such default tests usually come with a compromise, requiring the analyst to accept a one-size-fits-all approach to hypothesis testing. Further, such tests may not have the flexibility to test problems the scientist really cares about. In this tutorial, I demonstrate a flexible approach to generalizing one specific default test (the JZS t-test; Rouder et al., 2009) that is becoming increasingly popular in the social and behavioral sciences. The approach uses two theoretical results, the Savage-Dickey density ratio (Dickey and Lientz, 1980) and the technique of encompassing priors (Klugkist et al., 2005) in combination with MCMC sampling via an easy-to-use probabilistic modeling package for R called Greta. Through a comprehensive mathematical description of the techniques as well as illustrative examples, the reader is presented with a general, flexible workflow that can be extended to solve problems relevant to his or her own work.