$$\theta$$ 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.