For example (the not so ugly likelihood), If your likelihood is Poisson \[p(y|\theta ) = \frac{{{e^{ - \theta }}{\theta ^y}}}{{y!}}\] The nice conjugate prior is Gamma , or $\theta \sim G(\alpha ,\beta )$ \[p(\theta ) = \frac{{{\theta ^{\alpha - 1}}{e^{ - \theta /\beta }}}}{{\Gamma (\alpha ){\beta ^\alpha }}}\] Therefore, \[p(\theta |y)