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 :
P(a<X<b) = G(b) – G(a)
From there he says that we consider G monotonic increasing and differentiable. So basically using the fundamental theorem of calculus showing that :
P(a<X<b) &= G(b) – G(a)
&= \int_a^b g(x)dx
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 :
P(a<X<b) = \int_a^b g(x)dx
Independently of discrete or continuous cases.
Thanks for any input !!!
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.