Disjoint Events Research Articles

Inset information measures on open domains

In this paper we give a survey on all so-called inset information measures which depend not only upon k discrete complete probability distributions (without zero probabilities) of length n but also upon n pairwise disjoint (nonempty) events and which satisfy natural properties like the branching property or generalized additivity properties.

A theory of belief

A theory of belief is presented in which uncertainty has two dimensions. The two dimensions have a variety of interpretations. The article focusses on two of these interpretations. The first is that one dimension corresponds to probability and the other to “definiteness,” which itself has a variety of interpretations. One interpretation of definiteness is as the ordinal inverse of an aspect of uncertainty called “ambiguity” that is often considered important in the decision theory literature. (Greater ambiguity produces less definiteness and vice versa.) Another interpretation of definiteness is as a factor that measures the distortion of an individual's probability judgments that is due to specific factors involved in the cognitive processing leading to judgments. This interpretation is used to provide a new foundation for support theories of probability judgments and a new formulation of the “Unpacking Principle” of Tversky and Koehler. The second interpretation of the two dimensions of uncertainty is that one dimension of an event A corresponds to a function that measures the probabilistic strength of A as the focal event in conditional events of the form A| B, and the other dimension corresponds to a function that measures the probabilistic strength of A as the context or conditioning event in conditional events of the form C| A. The second interpretation is used to provide an account of experimental results in which for disjoint events A and B, the judge probabilities of A|( A∪ B) and B|( A∪ B) do not sum to 1. The theory of belief is axiomatized qualitatively in terms of a primitive binary relation ≿ on conditional events. ( A|B≿C|D is interpreted as “the degree of belief of A| B is greater than the degree of belief of C| D.”) It is shown that the axiomatization is a generalization of conditional probability in which a principle of conditional probability that has been repeatedly criticized on normative grounds may fail. Representation and uniqueness theorems for the axiomatization demonstrate that the resulting generalization is comparable in mathematical richness to finitely additive probability theory.

Random Response Analysis of Preisach Hysteretic Systems With Symmetric Weight Distribution

The present study is intended to develop a new method for analyzing nonlinear stochastic dynamic response of the Preisach hysteretic systems based on covariance and switching probability analysis of a nonlocal memory hysteretic constitutive model. A nonlinear algebraic covariance equation is formulated for the single-degree-of-freedom Preisach hysteretic system subjected to stationary Gaussian white noise excitation, from which the stationary mean square response of the system is obtained. The correlation coefficients of hysteretic restoring force with response in the covariance equation are evaluated by using the second moments and switching probabilities that are derived from the disjoint event probability and the mathematical machinery of an exit problem. In recognizing the symmetry of the classical Preisach weighting function, an approximation of equal “up” and “down” switching probabilities is introduced, which greatly simplifies the evaluation of the correlation coefficients. An example of the Preisach hysteretic system with Gaussian distribution weighting function is presented and the analytical results are compared with the digital simulation findings to verify the accuracy of the derived formulas. Computation results show that there exists a sharp drop in the mean square responses with the increase of a hysteresis parameter, and the mean square responses are affected only in a certain range of the Preisach weighting function.

Probabilistic datalog: Implementing logical information retrieval for advanced applications

In the logical approach to information retrieval (IR), retrieval is considered as uncertain inference. Whereas classical IR models are based on propositional logic, we combine Datalog (function-free Horn clause predicate logic) with probability theory. Therefore, probabilistic weights may be attached to both facts and rules. The underlying semantics extends the well-founded semantics of modularly stratified Datalog to a possible worlds semantics. By using default independence assumptions with explicit specification of disjoint events, the inference process always yields point probabilities. We describe an evaluation method and present an implementation. This approach allows for easy formulation of specific retrieval models for arbitrary applications, and classical probabilistic IR models can be implemented by specifying the appropriate rules. In comparison to other approaches, the possibility of recursive rules allows for more powerful inferences, and predicate logic gives the expressiveness required for multimedia retrieval. Furthermore, probabilistic Datalog can be used as a query language for integrated information retrieval and database systems.

Probability updating using second order probabilities and conditional event algebra

The second order probability technique is a procedure whereby an incompletely specified (finite-argument) probability function is modeled as a random vector over the associated natural simplex of all possible probability functions of a fixed common number of arguments. Often, the distribution of the random vector is assumed to be a uniform one, as in the case of Goodman’s treatment of the probabilistic syllogism (or penguin triangle) problem (1998). But, when compatibility with a non-uniform prior distribution is sought, the uniform random vector assumption must be replaced by a different one. In a related, but distinct direction, conditional event algebra is a new mathematical tool which allows the direct incorporation of conditional relations expressed through conditional probabilities with other such conditional relations or unconditional events [I.R. Goodman, H.T. Nguyen, Journal of Uncertainty, Fuzziness, and Knowledge-Based Systems 3 (3) (1995) 247–339; I.R. Goodman, R.P. Mahler, H.T. Nguyen, Mathematics of Data Fusion, Kluwer Academic Publishers, Dordrecht, Holland, 1997]. The “Judy Benjamin” (JB) problem, as originally posed by Van Fraasen [British Journal of Philosophy of Science 32 (1981) 375–379], has been recently addressed by Grove and Halpern (G&H) [Proceedings of the 13th Conference Uncertainty in AI,1997, pp. 208–214], and has elements in it conducive to analysis via both second order probability and conditional event algebra: The problem involves the issue of how to update a particular event of interest having a known prior probability – part of a known uniform prior probability function – upon a known conditional probability statement – when the latter is given through a probability function otherwise unknown and distinct from the prior probability function. In the problem as stated, the event of interest is disjoint from the antecedent event in the conditional probability statement. G&H make use of a trust principle in order to be able to apply, in effect, the second order probability technique with an associated uniform distribution. G&H’s result differs sharply from that of Van Fraasen, who employs instead a minimal cross-entropy approach to the problem, in that G&H show no change in probability values between the prior and posterior assessments. This paper considers in a more general context the application of the second order probability technique, using the Dirichlet family generalization of the uniform distribution over a simplex, in updating an event upon a conditional probability statement when no such disjointness as in the JB problem holds and compatibility is sought with a non-uniform prior. When this is specialized to the disjoint event case, even with a non-uniform prior, the independence result of G&H extends to this context. The JB problem is also considered via a conditional event algebra viewpoint – which avoids the trust principle – in conjunction with the second order probability technique and general agreement with that of G&H’s independence result is established for the case of disjoint event of interest from the antecedent (and consequence) of the conditional form. These relations are also extended to a more general setting.

Electronic Energy Funnels in Cis−Trans Photoisomerization of Retinal Protonated Schiff Base

The photoinduced cis−trans isomerization of all-trans retinal protonated Schiff base is studied using the full multiple spawning method. Our model allows for two classes of electronic transitions: exciton migration and quenching by coupling of the electronic and nuclear degrees of freedom. We find that quenching and exciton transfer are two temporally disjoint events and that exciton transport is highly directed. The observed selectivity in photoproducts is interpreted in terms of an electronic energy “funnel”, which is attributed to the Schiff base. The limits of the concept of an electronic energy “funnel”, when electron−phonon coupling is strong, are discussed and it is argued that such funnels can induce selectivity in excitation migration even when thermodynamic equilibrium among the excited electronic states is not reached.

The enhancement effect in probability judgment

Research has shown that the judged probability of an event depends on the specificity with which the focal and alternative hypotheses are described. In particular, unpacking the components of the focal hypothesis generally increases the judged probability of the focal hypothesis, while unpacking the components of the alternative hypothesis decreases the judged probability of the focal hypothesis. As a consequence, the judged probability of the union of disjoint events is generally less than the sum of their judged probabilities. This article shows that the total judged probability of a set of mutually exclusive and exhaustive hypotheses increases with the degree to which the evidence is compatible with these hypotheses. This phenomenon, which we refer to as the enhancement eAect, is consistent with a descriptive account of subjective probability called support theory. #1997 John Wiley & Sons, Ltd.

The closeness of the range of a probability on a certain system of random events -- an elementary proof

The closeness of the range of a probability on a certain system of random events -- an elementary proof

Regret aversion or event-splitting effects? more evidence under risk and uncertainty

Recent experimental evidence has concluded that experimentally observed juxtaposition effects, as predicted by regret theory1, are largely attributable to “event-splitting effects” (ESEs) whereby the subjective decision weight attached to an outcome depends on the number of, as well as on the combined probability of, the disjoint events in which that outcome occurs. An experiment is reported that discriminates between juxtaposition effects and ESEs under conditions of both complete and incomplete information. The results confirm that juxtaposition effects are indeed largely due to ESEs and are robust over different informational conditions.

CAREL: computer aided reliability evaluator for distributed computing networks

An efficient method to compute the terminal reliability (the probability of communication between a pair of nodes) of a distributed computing system (DCS) is presented. It is assumed that the graph model G(V,E) for DCS is given and that the path and/or cut information for the network G(V,E) is available. Boolean algebraic concepts are used to define four operators: compare, reduce, combine, and generate. The proposed method, called CAREL, uses the four operators to generate exclusive and mutually disjoint events. CAREL has been implemented using bit vector representation on an Encore MULTIMAX 320 system. It is shown that CAREL solves large DCS networks (having a pathset on the order of 780 and a cutset on the order of 7300 or more) with a reasonable memory requirement. A comparison with other algorithms reveals the computational efficiency of the method. The proof of correctness of CAREL is included. >

Language, Metaphor, and Analogy in the Music Education Research Process

This article is guided by the belief that the purpose of research lies in the discovery of laws by which the workings of the universe may be explained. We describe seemingly disjoint events and phenomena, classify them according to observed recurrences of same events, and then organize them into patterns of causal relationships. From those patterns we formulate theories as to why the events and phenomena may have come about in the first place. We assume our theories to be valid when they allow for the prediction of same or similar phenomena in the future. As the description provides the evidence needed to test the theory, the latter is only as good as the descriptors underlying that theory are meaningful.1 The relationship between evidence and theory formation becomes of particular importance when-as is the case in music education-the nonverbal evidence, musical behavior, requires verbal description. By naming an observed phenomenon, researchers in music education select from their repertoire of words a label that appears to give meaning to the observed phenomenon. We handle the evidence in a way most appropriate to our own understanding of the phenomenon; in a sense, we manipulate that which we do not know.

Subjectively weighted linear utility

An axiomatized theory of nonlinear utility and subjective probability is presented in which assessed probabilities are allowed to depend on the consequences associated with events. The representation includes the expected utility model as a special case, but can accommodate the Ellsberg paradox and other types of ambiguity sensitive behavior, while retaining familiar properties of subjective probability, such as additivity for disjoint events and multiplication of conditional probabilities. It is an extension, to the states model of decision making under uncertainty, of Chew's weighted linear utility representation for decision making under risk.

Entropy and Uncertainty

This essay is, primarily, a discussion of four results about the principle of maximizing entropy (MAXENT) and its connections with Bayesian theory. Result1 provides a restricted equivalence between the two: where the Bayesian model for MAXENT inference uses an “a priori“ probability that is uniform, and where all MAXENT constraints are limited to 0–1 expectations for simple indicator-variables. The other three results report on an inability to extend the equivalence beyond these specialized constraints. Result2 established a sensitivity of MAXENT inference to the choice of the algebra of possibilities even though all empirical constraints imposed on the MAXENT solution are satisfied in each measure space considered. The resulting MAXENT distribution is not invariant over the choice of measure space. Thus, old and familiar problems with the Laplacian principle of Insufficient Reason also plague MAXENT theory. Result3 builds upon the findings of Friedman and Shimony (1971; 1973) and demonstrates the absence of an exchangeable, Bayesian model for predictive MAXENT distributions when the MAXENT constraints are interpreted according to Jaynes's (1978) prescription for his (1963) Brandeis Dice problem. Lastly, Result4 generalizes the Friedman and Shimony objection to cross-entropy (Kullback-information) shifts subject to a constraint of a new odds-ratio for two disjoint events.

A computerised system for deriving symbolic unreliability expression

The failure event of a network can be expressed as the simultaneous failure of all the paths from source to terminal. This event can be expanded into disjoint events by using three simple lemmas to obtain the symbolic unreliability expression [2]. This paper first describes an algorithm, illustrated with a manual example, to derive the minimal paths between source and terminal. This is followed by the description of a computerised system which first derives all the minimal paths and then the unreliability expression using Ahmad-Jamil lemmas [2]. The results of the program execution on three different networks along with a discussion on the results are also presented.

An analytical minefield evaluation model without space averages

AbstractThe paper describes an approach to the evaluation of the effectiveness of a minefield in terms of the number of mines that are detonated by a convoy of sweepers and ships and the corresponding number of vessels that are immobilized. The positions of the mines and the tracks of the vessels are assumed to be known, which means that the evaluation measures are dependent on a large number of disjoint events, each event being the immobilization of particular vessels by particular mines. This may render combinatorial methods computationally infeasible, but by introducing approximations in the assumptions, the difficulty can be overcome, specifically by modelling the arrival of each individual vessel in the neighborhood of a mine by an inhomogeneous Poisson stream for which the arrival rate is nonzero only over a short time interval. The plausibility of the approach is supported by results of a critical‐event simulation model.