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I was wondering how a bayesian statistician would approach the problem of defining a probability density function for a random variable.

In a measure theoretic sense, If the distribution of the random variable is absolutely continuous w.r.t the lesbegue measure we have the very convenient Radon-Nikodym theorem.

Jaynes in (The logic of science) derive the density function by considering $P(X \leq a$ and $P(X > a)$, and still applying basic rules. He consider then that thoses probabilities should be a function $G$ of the variable $a$ and derive later that : \begin{align} P(a<X<b) = G(b) - G(a) \end{align}

From there he says that we consider G monotonic increasing and differentiable. So basically using the fundamental theorem of calculus showing that : \begin{align} P(a<X<b) &= G(b) - G(a) &= \int_a^b g(x)dx \end{align}

And call g the density function. My questions are it seems more restrictive than the measure theoretic approach ( absolutely continuous vs differentiable ), so :

1) How does bayesian statistician treat this case ? 2) Or measure theoretic is just more broad ?

Also if you have any other derivation from a bayesian point of view I would be glad to hear/read about it.

One last question :

3) Can measure theoretic using lesbegue integral can be used in conjunction of bayesian statistics ? Measure theory require a lot of set up ( sample space, measures, measurable spaces, etc ...) So it is intimidating to start with a plausibility function that respect Cox's axioms and making them fit the measure theoretic framework. for example it would be convenient to define : \begin{align} P(a<X<b) = \int_a^b g(x)dx \end{align}

Independently of discrete or continuous cases.

Thanks for any input !!!

user149705
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    measure theory is more general than calculus, so its no surprise that Jayne's approach is more restricted. Its not Bayesian vs. Measure theory, its Calculus vs Measure Theory –  Oct 18 '14 at 00:56
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    Mmm I see. I'm still really confuse about the two approaches. Jaynes claims that his degree of belief approach is more general than frequency approach. He also claim that he doesn't need to use measure theoretic methods. I studied quite a lot measure theoretic books and I just can't seems to find the applications of all the measure theoretic theorems/results to a book like Jaynes. Like for example if Jaynes approach can only deal with continuous functions how does he handle measurable functions ? And if he can't why does he claim that his treatment is more general. – user149705 Oct 18 '14 at 01:59
  • good questions! Jayne's was both brilliant and somewhat of an ideologue, he really saw no point in frequentist statistics. I'd be cautious about relying on just one person, especially one so contrarian, to define a discipline. I'd suggest you read Feller, Efron, Tukey, and other major statisticians before buying into Jaynes' framework completely. –  Oct 18 '14 at 20:14
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    Yes I agree. I have now an extensive view on both (or more ) worlds. The thing that I just can't deal with is why so many approaches to solve the same problem. I am trying everyday to reconcile my knowledge to every problem I deal with. But it's just not happening, so you have at one point to choose, and Jaynes provide much more intuition in everyday problems that I tend to use his methods. But then BAMM, measurable function, lesbegue integral not available anymore. Really frustrating ... – user149705 Oct 18 '14 at 23:46
  • what can I say...that's life. Statistics and inference has a lot of grey areas. –  Oct 19 '14 at 01:35

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Kadane's book Principles of Uncertainty derives Bayesian probability from first principles and does use measure theory (in section 4.9). It is based on DeFinetti's framework of coherence instead of Cox's axioms.

Tom Minka
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