5.6 Posterior distribution
Today’s posterior is tomorrow’s prior.
\(\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\) ~ Bayesian principle
The posterior (distribution) \(\pi(\theta|x)\) can be given by
\[\begin{equation} \pi(\theta|x) = \frac{\pi(\theta) L(\theta|x)}{\pi(x)} \tag{5.5} \end{equation}\]
where \(\pi(\theta)\) is the (a) priori distribution of \(\theta\), \(L(\theta|x)\) is the (function of) likelihood according to Eq. @ref(eq:func-vero ) and the predictive (distribution) of \(x\) is \[\begin{equation} \pi(x) = \int_{\theta} \pi(\theta) L(\theta|x) d\theta \tag{5.6} \end{equation}\]
