\(\theta\) 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.

## Conjugate family

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
\(f(x|\theta)\).

## References

Wasserman, L. (2013). *All of statistics: A concise course in
statistical inference*. Springer Science & Business Media.

Young, G. A., & Smith, R. L. (2005). *Essentials of statistical
inference* (Vol. 16). Cambridge University Press.