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This paper examines concepts of independence for full conditional probabilities; that is, for set-functions that encode conditional probabilities as primary objects, and that allow conditioning on events of probability zero. Full conditional probabilities have been used in economics, in philosophy, in statistics, in artificial intelligence. This paper characterizes the structure of full conditional probabilities under various concepts of independence; limitations of existing concepts are examined with respect to the theory of Bayesian networks. The concept of layer independence (factorization across layers) is introduced; this seems to be the first concept of independence for full conditional probabilities that satisfies the graphoid properties of Symmetry, Redundancy, Decomposition, Weak Union, and Contraction. A theory of Bayesian networks is proposed where full conditional probabilities are encoded using infinitesimals, with a brief discussion of hyperreal full conditional probabilities. We derive the structure of full conditional probabilities under independence.We introduce layer independence (factorization across layers).We show that layer independence satisfies all semi-graphoid properties.We discuss non-uniqueness of joint full conditional probabilities.We introduce a theory of Bayesian networks based on full distributions.

Some philosophers have claimed that it is meaningless or paradoxical to consider the probability of a probability. Others have however argued that second-order probabilities do not pose any particular problem. We side with the latter group. On condition that the relevant distinctions are taken into account, second-order probabilities can be shown to be perfectly consistent.May the same be said of an infinite hierarchy of higher-order probabilities? Is it consistent to speak of a probability of a probability, and of a probability of a probability of a probability, and so on, ad infinitum? We argue that it is, for it can be shown that there exists an infinite system of probabilities that has a model. In particular, we define a regress of higher-order probabilities that leads to a convergent series which determines an infinite-order probability value. We demonstrate the consistency of the regress by constructing a model based on coin-making machines. We show that an infinite hierarchy of probabilities of probabilities is consistent.The proof consists in a model involving coin-making machines.Weak conditions are given for the convergence of the infinite system.

The usual coherence criterion by de Finetti is extended both to many-valued events and to conditional probability. Special attention is paid to assessments in which the betting odds for conditioning events are zero. This case is treated by means of infinitesimal probabilities. We propose a rationality criterion, called stable coherence, which is stronger than coherence in the sense of no sure loss.

In this paper we deep in the formal properties of an already stated discrepancy measure between a conditional assessment and the class of unconditional probability distributions compatible with the assessment domain.

The personalist conception of probability is often explicated in terms of betting rates acceptable to an individual. A common approach, that of de Finetti for example, assumes that the individual is willing to take either side of the bet, so that the bet is “fair” from the individual’s point of view. This can sometimes be unrealistic, and leads to difficulties in the case of conditional probabilities or previsions. An alternative conception is presented in which it is only assumed that the collection of acceptable bets forms a convex cone, rather than a linear space. This leads to the more general conception of an upper conditional prevision. The main concerns of the paper are with the extension of upper conditional previsions. The main result is that any upper conditional prevision is the upper envelope of a family of additive conditional previsions.

This paper presents an approximate algorithm to obtain a posteriori intervals of probability, when available information is also given with intervals. The algorithm uses probability trees as a means of representing and computing with the convex sets of probabilities associated to the intervals.

In this paper, the modal logic interpretation of plausibility and belief measures on an arbitrary universe of discourse, as proposed by Harmanec et al., is further developed by employing notions from set-valued analysis. In a model of modal logic, a multivalued mapping is constructed from the accessibility relation and a mapping determined by the value assignment function. This multivalued mapping induces a plausibility measure and a belief measure expressed in terms of conditional probabilities of inverse and superinverse images, or equivalently, in terms of conditional probabilities of truth sets of possibilitations and necessitations. Restricting to a finite universe of discourse, multivalued interpretations of basic probability assignments and of commonality functions are also obtained, in terms of conditional probabilities of pure inverse and subinverse images, or equivalently, in terms of conditional probabilities of truth sets of particular logical expressions involving possibilitations and necessitations.

Computing marginal probabilities in Bayes networks is a hard problem. Deterministic anytime approximation schemes accumulate the probability mass in a small number of value assignments to the network variables. Under certain assumptions, the probability mass in the assignments is sufficient to obtain a good approximation. Such methods are especially useful for highly connected networks, where the topology makes the exact algorithms intractable. Bayes networks often possess a fine independence structure not evident from the topology, but apparent in local conditional distributions. Independence-based (IB) assignments, originally proposed as a theory of abduction, take advantage of such independence, and thus contain fewer assigned variables-and more probability mass. We present several algorithms that use IB assignments for approximating marginal probabilities. Experimental results suggest that this approach is feasible for highly connected belief networks.

Many researchers have felt uncomfortable with the precision of degrees of belief that seems to be demanded by the subjective Bayesian treatment of uncertainty. Various responses have been suggested. The most common one has been to incorporate higher order probabilities in systems that reason in beliefs. These probabilities concern statements of first-order probability. Thus a first-order probability (e.g., the probability of heads on the next toss of this coin is 1 2 ) is the subject of a second-order probability; for example, the probability is .9 that the probability of heads on the next toss of this coin is 1 2 . This approach is explored and is found to be epistemologically wanting, although there are important intuitions about beliefs that are captured by it. Furthermore, this approach may, in some circumstances, be computationally attractive. We also briefly explore a number of other approaches, including taking probabilities to be intervals and construing probability values as fuzzy sets.

This paper investigates four questions. What are the logical presuppositions underlying classical probability that have a role to play in David Lewis's proof of triviality concerning probabilities of conditionals and conditional probabilities? To what extent and how are they avoided in fuzzy logics when we treat semantic evaluations as the analogues of probability distributions? The introduction into the classical setting of conditional events (or assertions)—as opposed to implications—as a class of objects whose probabilities are equated with conditional probabilities has been the object of much recent investigation. To what extent, if any, can fuzzy logics accommodate the analogues of conditional events? How is triviality avoided in conditional event algebras?

The apparent failure of individual probabilistic expressions to distinguish between uncertainty and ignorance, and between certainty and confidence, have swayed researchers to seek alternative formalisms, where confidence measures are provided explicit notation. This paper summarizes how a causal networks formulation of probabilities facilitates the representation of confidence measures as an integral part of a knowledge system that does not require the use of higher order probabilities. We also examine whether Dempster-Shafer intervals represent confidence about probabilities.