The Moment Problem for Finitely Additive Probabilities

We study the moment problem for finitely additive probabilities and show that the information provided by the moments is equivalent to the one given by the associated lower and upper distribution functions

Finitely additive extensions of distribution functions and moment sequences: The coherent lower prevision approach

We study the information that a distribution function provides about the finitely additive probability measure inducing it. We show that in general there is an infinite number of finitely additive probabilities associated with the same distribution function. Secondly, we investigate the relationship between a distribution function and its given sequence of moments. We provide formulae for the sets of distribution functions, and finitely additive probabilities, associated with some moment sequence, and determine under which conditions the moments determine the distribution function uniquely. We show that all these problems can be addressed efficiently using the theory of coherent lower previsions

Lower and upper covariance

We give a definition for lower and upper covariance in Walley's theory of imprecise probabilities (or coherent lower previsions) that is direct, i.e., does not refer to credal sets. It generalizes Walley's definition for lower and upper variance. Just like Walley's definition of lower and upper variance, our definition for lower and upper covariance is compatible with the credal set approach; i.e., we also provide a covariance envelope theorem. Our approach mirrors the one taken by Walley: we first reformulate the calculation of a covariance as an optimization problem and then generalize this optimization problem to lower and upper previsions. We also briefly discuss the still unclear meaning of lower and upper (co)variances and mention some ideas about generalizations to other central moments

ISIPTA'07: Proceedings of the Fifth International Symposium on Imprecise Probability: Theories and Applications

B

Perturbation bounds and degree of imprecision for uniquely convergent imprecise Markov chains

The effect of perturbations of parameters for uniquely convergent imprecise Markov chains is studied. We provide the maximal distance between the distributions of original and perturbed chain and maximal degree of imprecision, given the imprecision of the initial distribution. The bounds on the errors and degrees of imprecision are found for the distributions at finite time steps, and for the stationary distributions as well.Comment: 20 pages, 2 figure