No abstract

Bayesian data analysis is about more than just computing a posterior distribution, and Bayesian visualization is about more than trace plots of Markov chains. Practical Bayesian data analysis, like all data analysis, is an iterative process of model building, inference, model checking and evaluation, and model expansion. Visualization is helpful in each of these stages of the Bayesian workflow and it is indispensable when drawing inferences from the types of modern, high dimensional models that are used by applied researchers.

The usual definition of R 2 (variance of the predicted values divided by the variance of the data) has a problem for Bayesian fits, as the numerator can be larger than the denominator. We propose an alternative definition similar to one that has appeared in the survival analysis literature: the variance of the predicted values divided by the variance of predicted values plus the expected variance of the errors.

This tutorial provides a pragmatic introduction to specifying, estimating and interpreting single-level and hierarchical linear regression models in the Bayesian framework. We start by summarizing why one should consider the Bayesian approach to the most common forms of regression. Next we introduce the R package rstanarm for Bayesian applied regression modeling. An overview of rstanarm fundamentals accompanies step-by-step guidance for fitting a single-level regression model with the stan_glm function, and fitting hierarchical regression models with the stan_lmer function, illustrated with data from an experience sampling study on changes in affective states. Exploration of the results is facilitated by the intuitive and user-friendly shinystan package. Data and scripts are available on the Open Science Framework page of the project. For readers unfamiliar with R, this tutorial is self-contained to enable all researchers who apply regression techniques to try these methods with their own data. Regression modeling with the functions in the rstanarm package will be a straightforward transition for researchers familiar with their frequentist counterparts, lm (or glm) and lmer.

Importance weighting is a convenient general way to adjust for draws from the wrong distribution, but the resulting ratio estimate can be noisy when the importance weights have a heavy right tail, as routinely occurs when there are aspects of the target distribution not well captured by the approximating distribution. More stable estimates can be obtained by truncating the importance ratios. Here we present a new method for stabilizing importance weights using a generalized Pareto distribution fit to the upper tail of the distribution of the simulated importance ratios. The method includes stabilized effective sample estimates, Monte Carlo error estimates and convergence diagnostics.