Remember Bayes Rule in Baysian Parameter Learning:
\begin{equation} P\qty(\theta | D) = \frac{P\qty(D | \theta) p \qty(\theta)}{\int_{\theta}P\qty(D | \theta) p \qty(\theta) \dd{\theta}} \end{equation}
we can’t actually easily compute the bottom without taking an analytic integral; instead we can sample from it.
If you want analytical form, you should hope that your likelihood function is a conjugate prior which allows us to analytically update prirors.