Kolmogorov’s axiomatization of probability includes the familiar ratio formula for conditional probability: (RATIO) P(A | B) = P(A ∩ B) P (B) (P (B) > 0). Call this the ratio analysis of conditional probability. It has become so entrenched that it is often referred to as the definition of conditional probability. I argue that it is not even an adequate analysis of that concept. I prove what I c...

Introduction. Doob [3, p. 29],2 has shown that if ()= [U, 11, u] is a probability space (i.e. 11 is a cr-field of subsets of a set U and u is a countably additive non-negative measure defined on 11, normalized by the condition u(U) = l) and 1l0 is a o--subfield of 11, and y is an re-dimensional random variable mapping U into an re-dimensional Euclidean space X, then y possesses a conditional di...

This paper proves, in very general settings, that convex risk minimization is a procedure to select a unique conditional probability model determined by the classification problem. Unlike most previous work, we give results that are general enough to include cases in which no minimum exists, as occurs typically, for instance, with standard boosting algorithms. Concretely, we first show that any...

Dynamic update of information states is the dernier cri in logical semantics. And it is old hat in Bayesian probabilistic reasoning. This note brings the two perspectives together, and proposes a mechanism for updating probabilities while changing the informational state spaces. 1 Tree diagrams for probability Many textbooks use a perspicuous tree format for simple probability spaces. Branches ...

We study a new framework for property testing of probability distributions, by considering distribution testing algorithms that have access to a conditional sampling oracle. This is an oracle that takes as input a subset S ⊆ [N ] of the domain [N ] of the unknown probability distribution D and returns a draw from the conditional probability distribution D restricted to S. This new model allows ...

This paper argues in favor of the thesis that two different concepts of conditional interval probability are needed, in order to serve the huge variety of tasks conditional probability has in the classical setting of precise probabilities. We compare the commonly used intuitive concept of conditional interval probability with the canonical concept, and see, in particular, that the canonical con...