As mentioned earlier, we started with the coin flip examples because of the ease of determining the posterior distribution analytically-primarily because of the beta distribution's self-conjugacy with respect to the binomial likelihood function.

It turns out that most real-world Bayesian analyses require a more complicated solution. In particular, the hyper-parameters that define the posterior distribution are rarely known. What can be determined is the probability density in the posterior distribution for each parameter value. The easiest way to get a sense of the shape of the posterior is to sample from it many thousands of times. More specifically, we sample from all possible parameter values and record the probability ...