Improper Regular Conditional Distributions

Improper Priors Are Not Improper

It is well known that improper priors in Bayesian statistics may lead to proper posterior distributions and useful inference procedures. This motivates us to give an elementary introduction to a theoretical frame for statistics that includes improper priors. Axioms that allow improper priors are given by a relaxed version of Kolmogorov’s formulation of probability theory. The theory of conditio...

Bayesian Analysis of Elapsed Times in Continuous-Time Markov Chains

We explore Bayesian analysis for continuous-time Markov chain (CTMC) models based on a conditional reference prior. For CTMC models, inference of the elapsed time between chain observations depends heavily on the rate of decay of the prior as the elapsed time increases. Moreover, improper priors on the elapsed time may lead to improper posterior distributions. In addition to the elapsed time, a...

Nonconglomerability for countably additive Measures that are not κ-additive

Let κ be an uncountable cardinal. Using the theory of conditional probability associated with de Finetti (1974) and Dubins (1975), subject to several structural assumptions for creating sufficiently many measurable sets, and assuming that κ is not a weakly inaccessible cardinal, we show that each probability that is not κ-­‐ additive has conditional probabilities that fail to be conglomerable i...

Some New Characterization Results on Exponential and Related Distributions

A Conditional 0-1 Law for the Symmetric Σ-field

Abstract. Let (Ω,B, P ) be a probability space, A ⊂ B a sub-σ-field, and μ a regular conditional distribution for P given A. For various, classically interesting, choices of A (including tail and symmetric) the following 0-1 law is proved: There is a set A0 ∈ A such that P (A0) = 1 and μ(ω)(A) ∈ {0, 1} for all A ∈ A and ω ∈ A0. Provided B is countably generated (and certain regular conditional ...

Asymptotic Analysis of Multivariate Tail Conditional Expectations

Tail conditional expectations refer to the expected values of random variables conditioning on some tail events and are closely related to various coherent risk measures. In the univariate case, the tail conditional expectation is asymptotically proportional to the value-at-risk, a popular risk measure. The focus of this paper is on asymptotic relations between the multivariate tail conditional...