Markov chains are super useful in statistics but it isn't obvious at first what problem they solve and how - some further reading that I found helpful:<p><a href="https://twiecki.io/blog/2015/11/10/mcmc-sampling/" rel="nofollow">https://twiecki.io/blog/2015/11/10/mcmc-sampling/</a><p>Note that the point of the markov chain is it's possible to compute <i>relative</i> probabilities between two given points in the posterior even when you don't have a closed form expression for the posterior.<p>Also, the reason behind separating the proposal distribution and the acceptance probability is that it's a convenient method to make the Markov process stationary, which isn't true in general. (Wikipedia page on MCMC is also useful here).