Bayesian Statistics: Computation

Abstract

• By Bayes theorem the posterior density of θ is given by

  $$\displaystyle{p(\theta \vert x) = \frac{f(x;\theta )p(\theta )} {f(x)} }$$

  where

  $$\displaystyle{f(x) =\int _{\Theta }f(x;\theta )p(\theta )dm(\theta )}$$

• The calculation of the posterior thus requires calculation of an integral of the likelihood weighted by the prior.

• Usually this integral can only be determined in closed form for conjugate priors.