In this chapter, we introduce the basics of Bayesian data analysis. The key ingredients to a Bayesian analysis are the likelihood function, which reflects information about the parameters contained in the data, and the prior distribution, which quantifies what is known about the parameters before observing data. The prior distribution and likelihood can be easily combined to from the posterior distribution, which represents total knowledge about the parameters after the data have been observed. Simple summaries of this distribution can be used to isolate quantities of interest and ultimately to draw substantive conclusions. We illustrate each of these steps of a typical Bayesian analysis using three biomedical examples and briefly discuss more advanced topics, including prediction, Monte Carlo computational methods, and multilevel models.
