Probability

Mark Burgin

Negative probability has found diverse applications in theoretical physics. Thus, construction of sound and rigorous mathematical foundations for negative probability is important for physics. There are different axiomatizations of conventional probability. So, it is natural that negative probability also has different axiomatic frameworks. In the previous publications (Burgin, 2009; 2010), negative probability was mathematically formalized and rigorously interpreted in the context of extended probability. In this work, the axiomatic system that synthesizes conventional probability and negative probability is constructed in the form of combined probability. In a mathematically rigorous way, both theoretical concepts - combined probability and extended probability - stretch conventional probability so that it can takes negative values. After introducing axioms for combined probability, we study its properties, as well as relations to extended probability and conventional probability.

AUGUST STERN

Philippe Réfrégier

Mark Burgin

In this paper, we give a frequency interpretation of negative probability, as well as of extended probability, demonstrating that to a great extent, these new types of probabilities, behave as conventional probabilities. Extended probability comprises both conventional probability and negative probability. The frequency interpretation of negative probabilities gives supportive evidence to the axiomatic system built in (Burgin, 2009; arXiv:0912.4767) for extended probability as it is demonstrated in this paper that frequency probabilities satisfy all axioms of extended probability.

Lawrence Sklar

The probability of an outcome was taken to be something closely related to the frequency with which that outcome could be expected to occur in a large number of identical repeated trials of some specified kind. A fundamental theorem of probability theory, Bayes' Theorem, relates the probability of a hypothesis on the basis of evidence to the conditional probability of the evidence, given the truth of the hypothesis and the initial probability that the hypothesis is true. "Nonpathological" means, essentially, that the initial probability distribution will give probability zero to any region of microconditions that have zero probability in the familiar, standard measure. Any non-pathological initial probability distribution for a mixing system will, as time goes into past infinity, also evolve into a probability distribution that is coarsely like the equilibrium distribution. The probability distribution is rationalized by showing that it is the uniquely stationary one that gives probability zero to regions given probability zero in that very natural measure.