Testing Implication of Probabilistic Dependencies

Axiomatization has been widely used for testing logical implications. This paper suggests a non-axiomatic method, the chase, to test if a new dependency follows from a given set of probabilistic dependencies. Although the chase computation may require exponential time in some cases, this technique is a powerful tool for establishing nontrivial theoretical results. More importantly, this approach provides valuable insight into the intriguing connection between relational databases and probabilistic reasoning systems.Comment: Appears in Proceedings of the Twelfth Conference on Uncertainty in Artificial Intelligence (UAI1996

On Axiomatization of Probabilistic Conditional Independencies

This paper studies the connection between probabilistic conditional independence in uncertain reasoning and data dependency in relational databases. As a demonstration of the usefulness of this preliminary investigation, an alternate proof is presented for refuting the conjecture suggested by Pearl and Paz that probabilistic conditional independencies have a complete axiomatization.Comment: Appears in Proceedings of the Tenth Conference on Uncertainty in Artificial Intelligence (UAI1994

Mean-field theory of learning: from dynamics to statics

Using the cavity method and diagrammatic methods, we model the dynamics of batch learning of restricted sets of examples. Simulations of the Green's function and the cavity activation distributions support the theory well. The learning dynamics approaches a steady state in agreement with the static version of the cavity method. The picture of the rough energy landscape is reviewed.Comment: 13 pages, 5 figures, to appear in "Advanced Mean Field Methods - Theory and Practice", edited by M. Opper and D. Saad, MIT Pres

Contextual Weak Independence in Bayesian Networks

It is well-known that the notion of (strong) conditional independence (CI) is too restrictive to capture independencies that only hold in certain contexts. This kind of contextual independency, called context-strong independence (CSI), can be used to facilitate the acquisition, representation, and inference of probabilistic knowledge. In this paper, we suggest the use of contextual weak independence (CWI) in Bayesian networks. It should be emphasized that the notion of CWI is a more general form of contextual independence than CSI. Furthermore, if the contextual strong independence holds for all contexts, then the notion of CSI becomes strong CI. On the other hand, if the weak contextual independence holds for all contexts, then the notion of CWI becomes weak independence (WI) nwhich is a more general noncontextual independency than strong CI. More importantly, complete axiomatizations are studied for both the class of WI and the class of CI and WI together. Finally, the interesting property of WI being a necessary and sufficient condition for ensuring consistency in granular probabilistic networks is shown.Comment: Appears in Proceedings of the Fifteenth Conference on Uncertainty in Artificial Intelligence (UAI1999

Segment-wise Description of the Dynamics of Traffic Congestion

We compare the point-wise and segment-wise descriptions of the traffic system. Using real data from the Taiwan highway system with a tremendous volume of segment-wise data, we find that the segment-wise description is much more informative of the evolution of the system during congestion. Congestion is characterized by a loopy trajectory in the fundamental diagram. By considering the area enclosed by the loop, we find that there are two types of congestion dynamics -- moderate flow and serious congestion. They are different in terms of whether the area enclosed vanishes. Data extracted from the time delays of individual vehicles show that the area enclosed is a measure of the economic loss due to congestion. The use of the loss area in helping to understand various road characteristics is also explored.Comment: 13 pages, 12 figure

Critical Remarks on Single Link Search in Learning Belief Networks

In learning belief networks, the single link lookahead search is widely adopted to reduce the search space. We show that there exists a class of probabilistic domain models which displays a special pattern of dependency. We analyze the behavior of several learning algorithms using different scoring metrics such as the entropy, conditional independence, minimal description length and Bayesian metrics. We demonstrate that single link lookahead search procedures (employed in these algorithms) cannot learn these models correctly. Thus, when the underlying domain model actually belongs to this class, the use of a single link search procedure will result in learning of an incorrect model. This may lead to inference errors when the model is used. Our analysis suggests that if the prior knowledge about a domain does not rule out the possible existence of these models, a multi-link lookahead search or other heuristics should be used for the learning process.Comment: Appears in Proceedings of the Twelfth Conference on Uncertainty in Artificial Intelligence (UAI1996

Interval Structure: A Framework for Representing Uncertain Information

In this paper, a unified framework for representing uncertain information based on the notion of an interval structure is proposed. It is shown that the lower and upper approximations of the rough-set model, the lower and upper bounds of incidence calculus, and the belief and plausibility functions all obey the axioms of an interval structure. An interval structure can be used to synthesize the decision rules provided by the experts. An efficient algorithm to find the desirable set of rules is developed from a set of sound and complete inference axioms.Comment: Appears in Proceedings of the Eighth Conference on Uncertainty in Artificial Intelligence (UAI1992

Dynamical Mechanisms in Multi-agent Systems: Minority Games

We consider a version of large population games whose agents compete for resources using strategies with adaptable preferences. Diversity among the agents reduces their maladpative behavior. We find interesting scaling relations with diversity for the variance of decisions. When diversity increases, the scaling dynamics is modified by kinetic sampling and waiting mechanisms.Comment: 4 pages, 2 figures, added reference

Compatibility of Quantitative and Qualitative Representations of Belief

The compatibility of quantitative and qualitative representations of beliefs was studied extensively in probability theory. It is only recently that this important topic is considered in the context of belief functions. In this paper, the compatibility of various quantitative belief measures and qualitative belief structures is investigated. Four classes of belief measures considered are: the probability function, the monotonic belief function, Shafer's belief function, and Smets' generalized belief function. The analysis of their individual compatibility with different belief structures not only provides a sound b<msis for these quantitative measures, but also alleviates some of the difficulties in the acquisition and interpretation of numeric belief numbers. It is shown that the structure of qualitative probability is compatible with monotonic belief functions. Moreover, a belief structure slightly weaker than that of qualitative belief is compatible with Smets' generalized belief functions.Comment: Appears in Proceedings of the Seventh Conference on Uncertainty in Artificial Intelligence (UAI1991

Effects of payoff functions and preference distributions in an adaptive population

Adaptive populations such as those in financial markets and distributed control can be modeled by the Minority Game. We consider how their dynamics depends on the agents' initial preferences of strategies, when the agents use linear or quadratic payoff functions to evaluate their strategies. We find that the fluctuations of the population making certain decisions (the volatility) depends on the diversity of the distribution of the initial preferences of strategies. When the diversity decreases, more agents tend to adapt their strategies together. In systems with linear payoffs, this results in dynamical transitions from vanishing volatility to a non-vanishing one. For low signal dimensions, the dynamical transitions for the different signals do not take place at the same critical diversity. Rather, a cascade of dynamical transitions takes place when the diversity is reduced. In contrast, no phase transitions are found in systems with the quadratic payoffs. Instead, a basin boundary of attraction separates two groups of samples in the space of the agents' decisions. Initial states inside this boundary converge to small volatility, while those outside diverge to a large one. Furthermore, when the preference distribution becomes more polarized, the dynamics becomes more erratic. All the above results are supported by good agreement between simulations and theory