Clarifying Bayes' RuleLooking 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?

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Clarifying Bayes&#039; RuleLooking 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?

Alberto Duffy · Accepted Answer

Since a model of the distribution of a data set isn&#039;t in any strict sence an event, then       P    (      M    ) isn&#039;t strictly the &quot;probabity that the model is true&quot; in terms of a sigma-algebra interpretation.   (The development of Bayes&#039; probability theory preceeded measure theory.)   The prior is more sensibly interpreted as a measure of &quot;our expectation for the truth of the model&quot;, or &quot;our confidence in the model.&quot;Assigning values to the prior is somewhat problematic, which is why Bayes&#039; factor is a comparator that cancels the term.

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

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