Explanations have probably been important to people since non-trivial speech came into existence. Explanations are important for decision and action so that others can understand the reasons for deciding and acting in a certain way.
Explanations fell from their central importance as science developed and the body of knowledge increased and became more complex and interconnected. Positivism put the emphasis on observation and description with the discouragement of the development of abstractions and explanation.
Counters to Positivism, such as critical rationalism enabled the importance of explanation to re-emerge. This is clear from Popper's writing but became actually central to the version espoused by David Deutsch in the Beginning of Infinity.
Explanation is the purpose of science. Agreement with experience is not eh purpose but provides a means of testing the explanation. Therefore good explanations are:
The use of Bayes' Formula in providing good explanations is controversial. Karl Popper was dismissive of its use because he claimed that the subjective nature of the prior linked it strongly to a subjective, idealist philosophy. However by considering both the prior and and the likelihood component to be conjectures to be corrected by experience there this a connection with critical rationalism.
Extension of critical rationalism
Critical rationalism works within classical logic. It also works with provability theory using a measure theory formulation and also a still controversial propensity interpretation. There is no use of of Bayes' Theorem. However if the elements of Bayes' Formula are considered be conjectural elements of an explanation then not only do we have a useful conceptual tool but also a mathematical formalism to work with.
Critical rationalism provides strong recommendations for how Bayes Theorem should be used in testing explanations. Part of testing explanations is to predict what would happen in a situation given the theory behind the explanation and a specific situation.