Bayesian statistics is conceptually very simple; we have the knows and the unknowns; we use Bayes' theorem to condition the latter on the former. If we are lucky, this process will reduce the uncertainty about the unknowns. Generally, we refer to the knowns as data and treat it like a constant, and the unknowns as parameters and treat them as probability distributions. In more formal terms, we assign probability distributions to unknown quantities. Then, we use Bayes' theorem to transform the prior probability distribution 
into a posterior distribution 
. Although conceptually simple, fully probabilistic models often lead to analytically intractable expressions. For many years, this was a real problem and was probably one of the main issues that hindered the wide adoption of Bayesian methods.
The arrival of the computational era and the development...