We present a unified logical framework for representing and reasoning about both probability quantitative and qualitative preferences in probability answer set programming, called probability answer set optimization programs. The proposed framework is vital to allow defining probability quantitative preferences over the possible outcomes of qualitative preferences. We show the application of probability answer set optimization programs to a variant of the well-known nurse restoring problem, called the nurse restoring with probability preferences problem. To the best of our knowledge, this development is the first to consider a logical framework for reasoning about probability quantitative preferences, in general, and reasoning about both probability quantitative and qualitative preferences in particular.

A number of writers have supposed that for the full specification of belief, higher order probabilities are required. Some have even supposed that there may be an unending sequence of higher order probabilities of probabilities of probabilities.... In the present paper we show that higher order probabilities can always be replaced by the marginal distributions of joint probability distributions. We consider both the case in which higher order probabilities are of the same sort as lower order probabilities and that in which higher order probabilities are distinct in character, as when lower order probabilities are construed as frequencies and higher order probabilities are construed as subjective degrees of belief. In neither case do higher order probabilities appear to offer any advantages, either conceptually or computationally.

Olson, Matthew A., Wyner, Abraham J.

A random forest is a popular tool for estimating probabilities in machine learning classification tasks. However, the means by which this is accomplished is unprincipled: one simply counts the fraction of trees in a forest that vote for a certain class. In this paper, we forge a connection between random forests and kernel regression. This places random forest probability estimation on more sound statistical footing. As part of our investigation, we develop a model for the proximity kernel and relate it to the geometry and sparsity of the estimation problem. We also provide intuition and recommendations for tuning a random forest to improve its probability estimates.

The original goal of this post was to explore the relationship between the softmax and sigmoid functions. In truth, this relationship had always seemed just out of reach: "One has an exponent in the numerator! One has a 1 in the denominator!" And of course, the two have different names. Once derived, I quickly realized how this relationship backed out into a more general modeling framework motivated by the conditional probability axiom itself.

Recently, Halpern and Leung suggested representing uncertainty by a set of weighted probability measures, and suggested a way of making decisions based on this representation of uncertainty: maximizing weighted regret. Their paper does not answer an apparently simpler question: what it means, according to this representation of uncertainty, for an event E to be more likely than an event E'. In this paper, a notion of comparative likelihood when uncertainty is represented by a set of weighted probability measures is defined. It generalizes the ordering defined by probability (and by lower probability) in a natural way; a generalization of upper probability can also be defined. A complete axiomatic characterization of this notion of regret-based likelihood is given.