foundations of probability

Following my reading of a note by Gunnar Taraldsen and co-authors on improper priors, I checked the 1970 book of Rényi from the Library at Warwick. (First time I visited this library, where I got very efficient help in finding and borrowing this book!)

“…estimates of probability of an event made by different persons may be different and each such estimate is to a certain extent subjective.” (p.33)

The main argument from Rényi used by the above mentioned note (and an earlier paper in The American Statistician) is that “every probability is in reality a conditional probability” (p.34). Which may be a pleonasm as everything depends on the settings in which it is applied. And as such not particularly new since conditioning is also present in e.g. Jeffreys’ book. In this approach, the definition of the conditional probability is traditional, if restricted to condition on a subset of elements from the σ algebra. The interesting part in the book is rather that a measure on this subset can be derived from the conditionals. And extended to the whole σ algebra. And is unique up to a multiplicative constant. Interesting because this indeed produces a rigorous way of handling improper priors.

“Let the random point (ξ,η) be uniformly distributed over the whole (x,y) plane.” (p.83)

Rényi also defines random variables ξ on conditional probability spaces, with conditional densities. With constraints on ξ for those to exist. I have more difficulties to ingest this notion as I do not see the meaning of the above quote or of the quantity

P(a<ξ<b|c<ξ<d)

when P(a<ξ<b) is not defined. As for instance I see no way of generating such a ξ in this case. (Of course, it is always possible to bring in a new definition of random variables that only agrees with regular ones for finite measure.)

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