16.4 The posterior

Estimation of the posterior predictive distribution is really the hallmark of Bayesian inference, and is the crux of any applied analysis that uses this framework to test hypotheses in biology and ecology. The posterior predictive distribution (more commonly called the ‘posterior’) is the estimated probability distribution of unobserved events conditional on some set of observations related to that event.

The posterior distribution can be estimated as the product of our prior distribution and the corresponding likelihood by re-arranging Bayes theorem:

\[posterior \propto prior \cdot likelihood\]

In the simplest sense, the posterior distribution is a combination of our prior distribution and our data. So if you remember nothing else in the explanation, remember that.