Once we have calculated the posterior, the question arises on how to estimate the parameters of the posterior.
When there is a lot of information, the parameters can be estimated directly and subjectively.
Multi-phase assignment estimating the parameters with a Hyper Prior which is typically chosen to be non-informative.
In practical terms, it is as if we were repeating the prior selection reasoning a second time — but not on \(X\) for a parameter \(\theta\), rather on the random variable \(\theta\) itself (because in Bayesian statistics the parameter is also a random variable) with density function equal to the prior \(\pi(\theta)\) for a hyperparameter \(\alpha\).
I choose the hyperparameters by calculating some moments and setting up a system of equations equating the theoretical moments to those estimated with the sample.
Therefore:
\(hp. \space \space m(x)= \int_{\Theta}{f(\underline{x} | \theta) \pi(\theta)} \space \space d\theta\)
\[Theoretical \space moments\\ E^f[x]= E[x|\theta] =\mu_f(\theta)\\ Var^f(x)=Var(x|\theta)=\sigma^2_f(\theta)\] \[Moments \space computed \space from \space the \space sample\\ E^m(x)= \int_{S_x} x \space m(x) \space \space dx = \mu_m \\ Var^m(x)= \int_{S_x} x^2 \space m(x) \space \space dx = \sigma^2_m\] \[System \space of \space equations \\ \begin{cases} \mu_f(\theta) = \mu_m \\ \sigma^2_f(\theta)= \sigma^2_m \end{cases} \]
In general, when using this method a sufficiently large sample is needed; furthermore, it is advisable to use accurate estimation methods, evaluating where possible the variability of the moments of the parameter(s) \(\theta\), for example with a Double-Bootstrap.