Clarifying Bayes' Rule Looking through the literature dedicated to Bayes factors I noticed that giv

hawwend8u

hawwend8u

Answered question

2022-05-26

Clarifying Bayes' Rule
Looking through the literature dedicated to Bayes factors I noticed that given formulae are quite unclear from the point of Kolmogorov probability axioms.
E.g. consider the following expression from Wikipedia:
Pr ( M D ) = Pr ( D M ) Pr ( M ) Pr ( D )   ,
where D stands for our sampled data, M - for the model.
Well, D is a random variable. But how should I interpret M? In applications, e.g. solving a hierarchial probabilistic model I can consider Pr(D∣M) as the conditional density under given parameter. But the concept of Pr(M) seems rather misleading to me.
Are there any papers with formal definitions based on axiomatic approach?

Answer & Explanation

Alberto Duffy

Alberto Duffy

Beginner2022-05-27Added 5 answers

Since a model of the distribution of a data set isn't in any strict sence an event, then P ( M ) isn't strictly the "probabity that the model is true" in terms of a sigma-algebra interpretation. (The development of Bayes' probability theory preceeded measure theory.) The prior is more sensibly interpreted as a measure of "our expectation for the truth of the model", or "our confidence in the model."
Assigning values to the prior is somewhat problematic, which is why Bayes' factor is a comparator that cancels the term.

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