We introduce the fundamental tenets of Bayesian inference, which derive from two basic laws of probability theory. We cover the interpretation of probabilities, discrete and continuous versions of Bayes' rule, parameter estimation, and model comparison. Using seven worked examples, we illustrate these principles and set up some of the technical background for the rest of this special issue of Psychonomic Bulletin & Review. Supplemental material is available via https://osf.io/wskex/.

The editorial for a Special Issue of the journal Psychonomic Bulletin & Review.

People tend to slow down after they make an error. This phenomenon, generally referred to as post-error slowing, has been hypothesized to reflect perceptual distraction, time wasted on irrelevant processes, a priori bias against the response made in error, increased variability in a priori bias, or an increase in response caution. Although the response caution interpretation has dominated the empirical literature, little research has attempted to test this interpretation in the context of a formal process model. Here we used the drift diffusion model to isolate and identify the psychological processes responsible for post-error slowing. In a lexical decision data set comprised of 1,094,886 responses we found that post-error slowing was associated with an increase in response caution and-to a lesser extent-a change in response bias. In the present data set, we found no evidence that post-error slowing is caused by perceptual distraction or time wasted on irrelevant processes. These results support a response monitoring account of post-error slowing.

Hypothesis testing is a special form of model selection. Once a pair of competing models is fully defined, their definition immediately leads to a measure of how strongly each model supports the data. The ratio of their support is often called the likelihood ratio or the Bayes factor. Critical in the model selection endeavor is the specification of the models. In the case of hypothesis testing, it is of the greatest importance that we specify exactly what is meant by a "null" hypothesis as well as the alternative to which it is contrasted, and that these are suitable instantiations of theoretical positions. Here, we provide an overview of different instantiations of null and alternative hypotheses that can be useful in practice, while the underlying method of likelihood comparison is universal and identical in all cases. An associated app can be found via https://osf.io/mvp53/.

We demonstrate the use of three popular Bayesian software packages that enable researchers to estimate parameters in a broad class of models that are commonly used in psychological research. We focus on WinBUGS, JAGS, and Stan, and show how they can be interfaced from R and MATLAB. We illustrate the use of the packages through two fully worked examples; the examples involve a simple univariate linear regression and fitting a multinomial processing tree model to data from a classic false-memory experiment. We conclude with a comparison of the strengths and weaknesses of the packages. Our example code, data, and this text are available via https://osf.io/ucmaz/.