12 research outputs found
Uniquely determined uniform probability on the natural numbers
In this paper, we address the problem of constructing a uniform probability
measure on . Of course, this is not possible within the bounds of
the Kolmogorov axioms and we have to violate at least one axiom. We define a
probability measure as a finitely additive measure assigning probability to
the whole space, on a domain which is closed under complements and finite
disjoint unions. We introduce and motivate a notion of uniformity which we call
weak thinnability, which is strictly stronger than extension of natural
density. We construct a weakly thinnable probability measure and we show that
on its domain, which contains sets without natural density, probability is
uniquely determined by weak thinnability. In this sense, we can assign uniform
probabilities in a canonical way. We generalize this result to uniform
probability measures on other metric spaces, including .Comment: We added a discussion of coherent probability measures and some
explanation regarding the operator we study. We changed the title to a more
descriptive one. Further, we tidied up the proofs and corrected and
simplified some minor issue
Assessing forensic evidence by computing belief functions
We first discuss certain problems with the classical probabilistic approach
for assessing forensic evidence, in particular its inability to distinguish
between lack of belief and disbelief, and its inability to model complete
ignorance within a given population. We then discuss Shafer belief functions, a
generalization of probability distributions, which can deal with both these
objections. We use a calculus of belief functions which does not use the much
criticized Dempster rule of combination, but only the very natural
Dempster-Shafer conditioning. We then apply this calculus to some classical
forensic problems like the various island problems and the problem of parental
identification. If we impose no prior knowledge apart from assuming that the
culprit or parent belongs to a given population (something which is possible in
our setting), then our answers differ from the classical ones when uniform or
other priors are imposed. We can actually retrieve the classical answers by
imposing the relevant priors, so our setup can and should be interpreted as a
generalization of the classical methodology, allowing more flexibility. We show
how our calculus can be used to develop an analogue of Bayes' rule, with belief
functions instead of classical probabilities. We also discuss consequences of
our theory for legal practice.Comment: arXiv admin note: text overlap with arXiv:1512.01249. Accepted for
publication in Law, Probability and Ris
The infinite epistemic regress problem has no unique solution
In this article we analyze the claim that a probabilistic interpretation of the infinite epistemic regress problem leads to a unique solution, the so called “completion” of the regress. This claim is implicitly based on the assumption that the standard Kolmogorov axioms of probability theory are suitable for describing epistemic probability. This assumption, however, has been challenged in the literature, by various authors. One of the alternatives that have been suggested to replace the Kolmogorov axioms in case of an epistemic interpretation of probability, are belief functions, introduced by Shafer in 1976. We show that when one uses belief functions to describe the infinite epistemic regress problem, it is no longer the case that the solution is unique. We also argue that this complies with common sense
A Behavioral Interpretation of Belief Functions
Shafer’s belief functions were introduced in the seventies of the previous century as a mathematical tool in order to model epistemic probability. One of the reasons that they were not picked up by mainstream probability was the lack of a behavioral interpretation. In this paper, we provide such a behavioral interpretation and re-derive Shafer’s belief functions via a betting interpretation reminiscent of the classical Dutch Book Theorem for probability distributions. We relate our betting interpretation of belief functions to the existing literature