Characterizing the principle of minimum cross-entropy within a conditional-logical framework

AbstractThe principle of minimum cross-entropy (ME-principle) is often used as an elegant and powerful tool to build up complete probability distributions when only partial knowledge is available. The inputs it may be applied to are a prior distribution P and some new information R, and it yields as a result the one distribution P∗ that satisfies R and is closest to P in an information-theoretic sense. More generally, it provides a “best” solution to the problem “How to adjust P to R?”In this paper, we show how probabilistic conditionals allow a new and constructive approach to this important principle. Though popular and widely used for knowledge representation, conditionals quantified by probabilities are not easily dealt with. We develop four principles that describe their handling in a reasonable and consistent way, taking into consideration the conditional-logical as well as the numerical and probabilistic aspects. Finally, the ME-principle turns out to be the only method for adjusting a prior distribution to new conditional information that obeys all these principles.Thus a characterization of the ME-principle within a conditional-logical framework is achieved, and its implicit logical mechanisms are revealed clearly

Information as Distinctions: New Foundations for Information Theory

The logical basis for information theory is the newly developed logic of partitions that is dual to the usual Boolean logic of subsets. The key concept is a "distinction" of a partition, an ordered pair of elements in distinct blocks of the partition. The logical concept of entropy based on partition logic is the normalized counting measure of the set of distinctions of a partition on a finite set--just as the usual logical notion of probability based on the Boolean logic of subsets is the normalized counting measure of the subsets (events). Thus logical entropy is a measure on the set of ordered pairs, and all the compound notions of entropy (join entropy, conditional entropy, and mutual information) arise in the usual way from the measure (e.g., the inclusion-exclusion principle)--just like the corresponding notions of probability. The usual Shannon entropy of a partition is developed by replacing the normalized count of distinctions (dits) by the average number of binary partitions (bits) necessary to make all the distinctions of the partition

An information theory for preferences

Recent literature in the last Maximum Entropy workshop introduced an analogy between cumulative probability distributions and normalized utility functions. Based on this analogy, a utility density function can de defined as the derivative of a normalized utility function. A utility density function is non-negative and integrates to unity. These two properties form the basis of a correspondence between utility and probability. A natural application of this analogy is a maximum entropy principle to assign maximum entropy utility values. Maximum entropy utility interprets many of the common utility functions based on the preference information needed for their assignment, and helps assign utility values based on partial preference information. This paper reviews maximum entropy utility and introduces further results that stem from the duality between probability and utility

Logical Entropy: Introduction to Classical and Quantum Logical Information theory

Logical information theory is the quantitative version of the logic of partitions just as logical probability theory is the quantitative version of the dual Boolean logic of subsets. The resulting notion of information is about distinctions, differences and distinguishability and is formalized using the distinctions of a partition. All the definitions of simple, joint, conditional and mutual entropy of Shannon information theory are derived by a uniform transformation from the corresponding definitions at the logical level. The purpose of this paper is to give the direct generalization to quantum logical information theory that similarly focuses on the pairs of eigenstates distinguished by an observable, i.e., qudits of an observable. The fundamental theorem for quantum logical entropy and measurement establishes a direct quantitative connection between the increase in quantum logical entropy due to a projective measurement and the eigenstates that are distinguished by the measurement. Both the classical and quantum versions of logical entropy have simple interpretations as “two-draw” probabilities for distinctions. The conclusion is that quantum logical entropy is the simple and natural notion of information for quantum information theory focusing on the distinguishing of quantum states

Dependence of dissipation on the initial distribution over states

We analyze how the amount of work dissipated by a fixed nonequilibrium process depends on the initial distribution over states. Specifically, we compare the amount of dissipation when the process is used with some specified initial distribution to the minimal amount of dissipation possible for any initial distribution. We show that the difference between those two amounts of dissipation is given by a simple information-theoretic function that depends only on the initial and final state distributions. Crucially, this difference is independent of the details of the process relating those distributions. We then consider how dissipation depends on the initial distribution for a 'computer', i.e., a nonequilibrium process whose dynamics over coarse-grained macrostates implement some desired input-output map. We show that our results still apply when stated in terms of distributions over the computer's coarse-grained macrostates. This can be viewed as a novel thermodynamic cost of computation, reflecting changes in the distribution over inputs rather than the logical dynamics of the computation

Axiomatic approach to the cosmological constant

A theory of the cosmological constant Lambda is currently out of reach. Still, one can start from a set of axioms that describe the most desirable properties a cosmological constant should have. This can be seen in certain analogy to the Khinchin axioms in information theory, which fix the most desirable properties an information measure should have and that ultimately lead to the Shannon entropy as the fundamental information measure on which statistical mechanics is based. Here we formulate a set of axioms for the cosmological constant in close analogy to the Khinchin axioms, formally replacing the dependency of the information measure on probabilities of events by a dependency of the cosmological constant on the fundamental constants of nature. Evaluating this set of axioms one finally arrives at a formula for the cosmological constant that is given by Lambda = (G^2/hbar^4) (m_e/alpha_el)^6, where G is the gravitational constant, m_e is the electron mass, and alpha_el is the low energy limit of the fine structure constant. This formula is in perfect agreement with current WMAP data. Our approach gives physical meaning to the Eddington-Dirac large number hypothesis and suggests that the observed value of the cosmological constant is not at all unnatural.Comment: 7 pages, no figures. Some further references adde