# Idea In [[Bayesian statistics]], we update our [[prior]], $p(\theta)$ with observed data via [[Bayes theorem|Bayes's rule]] (see also [[sequential updating]]). Our updated data is the [[posterior]], which reflects updated knowledge about $\theta$, which is described by the [[posterior|posterior distribution]], denoted by $p(\theta | D)$, where $D$ indicates the observed data. The posterior distribution expresses the uncertainty about the value of $\theta$, quantifying the relative probability that each possible value is the true value. Often, [[Bayes theorem|Bayes's rule]] is written as the following, which tells us how the posterior distribution is a combination of what we knew before we saw the data, $p(\theta)$, and what we have learned from the data, $p(D|\theta)$: $ p(\theta \mid D) \propto p(D \mid \theta) p(\theta) $ We often use the posterior distribution to generate predictions, which gives us the [[posterior predictive distribution]]. # References