The politician scenario is a much better example of MCMC (and the Metropolis algorithm specifically) than it is of Bayesian inference per se. For example, the scenario doesn’t really include any “data” and thus does not describe any model that could capture the data generating process (i.e., the likelihood). As a consequence, there are no prior beliefs about the unknown/uncertain components of the model (i.e., priors).
But you can apply the Metropolis algorithm to simple inference problems (though you should not use Metropolis for anything other than educational/illustrative purposes). I wrote up this notebook to illustrate exact and approximate approaches to inference in standard a coin-flipping example (Krushke’s Figure 7.4 is similar, but different).