xbeta
Bayes' theorem

Bayes’ theorem relates the conditional and marginal probabilities of two events $A$ and $B$, where $B$ has nonzero probability:

Here, $\mathrm{Pr}(A)$ is the prior probability of $A$ (in that it doesn’t take into account any information about $B$), $\mathrm{Pr}(A\mid B)$ is the posterior probability of $A$ given $B$, $\mathrm{Pr}(B\mid A)$ is the conditional probability of $B$ given $A$, and $\mathrm{Pr}(B)$ is the marginal probability of $B$ which acts here as a normalizing constant.

In the context of Bayesian inference about a parameter $\theta$ given $X$, this relationship can be written

which states that the posterior density of $\theta$ is proportional to the likelihood? of $X$ given $\theta$ times the prior density of $\theta$.