\(\theta\) is not fixed, but a random variable. Thus, it is characterised by a prior \(\pi(\theta)\).
Accordingly, we now refer to the statistical model as \(f(x|\theta)\), instead of \(f(x; \theta)\), where \(x=(x_1,\dotsc,x_n)\).
\(\pi(\theta)\) and \(f(x|\theta)\) are related by Bayes rule
\[\pi(\theta|x) = \frac{f(x|\theta) \pi(\theta)}{\int_\theta f(x|\theta) \pi(\theta) d\theta}=k L(\theta)\pi(\theta)\]
The posterior \(\pi(\theta|x)\) has all the information.
A class of prior distributions is a conjugate family for the class \(f(x|\theta)\) if the posterior distributions belong to the class of the prior distributions for any \(f(x|\theta)\).