This paper discusses belief revision under uncertain inputs in the framework of possibility theory. Revision can be based on two possible definitions of the conditioning operation, one based on min operator which requires a purely ordinal scale only, and another based on product, for which a richer structure is needed, and which is a particular case of Dempster's rule of conditioning. Besides, revision under uncertain inputs can be understood in two different ways depending on whether the input is viewed, or not, as a constraint to enforce. Moreover, it is shown that M.A. Williams' transmutations, originally defined in the setting of Spohn's functions, can be captured in this framework, as well as Boutilier's natural revision.

The study of belief change has been an active area in philosophy and AI. In recent years two special cases of belief change, belief revision and belief update, have been studied in detail. Roughly, revision treats a surprising observation as a sign that previous beliefs were wrong, while update treats a surprising observation as an indication that the world has changed. In general, we would expect that an agent making an observation may both want to revise some earlier beliefs and assume that some change has occurred in the world. We define a novel approach to belief change that allows us to do this, by applying ideas from probability theory in a qualitative setting. The key idea is to use a qualitative Markov assumption, which says that state transitions are independent. We show that a recent approach to modeling qualitative uncertainty using plausibility measures allows us to make such a qualitative Markov assumption in a relatively straightforward way, and show how the Markov assumption can be used to provide an attractive belief-change model.

A nonmonotonic logic of thresholded generalizations is presented. Given propositions A and B from a language L and a positive integer k, the thresholded generalization A=>B{k} means that the conditional probability P(B|A) falls short of one by no more than c*d^k. A two-level probability structure is defined. At the lower level, a model is defined to be a probability function on L. At the upper level, there is a probability distribution over models. A definition is given of what it means for a collection of thresholded generalizations to entail another thresholded generalization. This nonmonotonic entailment relation, called "entailment in probability", has the feature that its conclusions are "probabilistically trustworthy" meaning that, given true premises, it is improbable that an entailed conclusion would be false. A procedure is presented for ascertaining whether any given collection of premises entails any given conclusion. It is shown that entailment in probability is closely related to Goldszmidt and Pearl's System-Z^+, thereby demonstrating that the conclusions of System-Z^+ are probabilistically trustworthy.

Poole has shown that nonmonotonic logics do not handle the lottery paradox correctly. In this paper we will show that Pollock's theory of defeasible reasoning fails for the same reason: defeasible reasoning is incompatible with the skeptical notion of derivability.

The monitoring and control of any dynamic system depends crucially on the ability to reason about its current status and its future trajectory. In the case of a stochastic system, these tasks typically involve the use of a belief state- a probability distribution over the state of the process at a given point in time. Unfortunately, the state spaces of complex processes are very large, making an explicit representation of a belief state intractable. Even in dynamic Bayesian networks (DBNs), where the process itself can be represented compactly, the representation of the belief state is intractable. We investigate the idea of maintaining a compact approximation to the true belief state, and analyze the conditions under which the errors due to the approximations taken over the lifetime of the process do not accumulate to make our answers completely irrelevant. We show that the error in a belief state contracts exponentially as the process evolves. Thus, even with multiple approximations, the error in our process remains bounded indefinitely. We show how the additional structure of a DBN can be used to design our approximation scheme, improving its performance significantly. We demonstrate the applicability of our ideas in the context of a monitoring task, showing that orders of magnitude faster inference can be achieved with only a small degradation in accuracy.

This paper relates comparative belief structures and a general view of belief management in the setting of deductively closed logical representations of accepted beliefs. We show that the range of compatibility between the classical deductive closure and uncertain reasoning covers precisely the nonmonotonic 'preferential' inference system of Kraus, Lehmann and Magidor and nothing else. In terms of uncertain reasoning any possibility or necessity measure gives birth to a structure of accepted beliefs. The classes of probability functions and of Shafer's belief functions which yield belief sets prove to be very special ones.

The dependency graph is a data architecture that models all the dependencies between the different types of assets in the game. It depicts the dependency-based relationships between the assets of a game. For example, a player must construct an arsenal before he can build weapons. It is vital that the dependency graph of a game is designed logically to ensure a logical sequence of game play. However, a mere logical dependency graph is not sufficient in sustaining the players' enduring interests in a game, which brings the problem of game balancing into picture. The issue of game balancing arises when the players do not feel the chances of winning the game over their AI opponents who are more skillful in the game play. At the current state of research, the architecture of dependency graph is monolithic for the players. The sequence of asset possession is always foreseeable because there is only a single dependency graph. Game balancing is impossible when the assets of AI players are overwhelmingly outnumbering that of human players. This paper proposes a parallel architecture of dependency graph for the AI players and human players. Instead of having a single dependency graph, a parallel architecture is proposed where the dependency graph of AI player is adjustable with that of human player using a support dependency as a game balancing mechanism. This paper exhibits that the parallel dependency graph helps to improve game balancing.

The AGM theory of belief revision has become an important paradigm for investigating rational belief changes. Unfortunately, researchers working in this paradigm have restricted much of their attention to rather simple representations of belief states, namely logically closed sets of propositional sentences. In our opinion, this has resulted in a too abstract categorisation of belief change operations: expansion, revision, or contraction. Occasionally, in the AGM paradigm, also probabilistic belief changes have been considered, and it is widely accepted that the probabilistic version of expansion is conditioning. However, we argue that it may be more correct to view conditioning and expansion as two essentially different kinds of belief change, and that what we call constraining is a better candidate for being considered probabilistic expansion.

The dynamics of belief and knowledge is one of the major components of any autonomous system that should be able to incorporate new pieces of information. In this paper, we argue that to apply rationality result of belief dynamics theory to various practical problems, it should be generalized in two respects: first of all, it should allow a certain part of belief to be declared as immutable; and second, the belief state need not be deductively closed. Such a generalization of belief dynamics, referred to as base dynamics, is presented, along with the concept of a generalized revision algorithm for Horn knowledge bases. We show that Horn knowledge base dynamics has interesting connection with kernel change and abduction. Finally, we also show that both variants are rational in the sense that they satisfy certain rationality postulates stemming from philosophical works on belief dynamics.

We present a novel inference algorithm for arbitrary, binary, undirected graphs. Unlike loopy belief propagation, which iterates fixed point equations, we directly descend on the Bethe free energy. The algorithm consists of two phases, first we update the pairwise probabilities, given the marginal probabilities at each unit,using an analytic expression. Next, we update the marginal probabilities, given the pairwise probabilities by following the negative gradient of the Bethe free energy. Both steps are guaranteed to decrease the Bethe free energy, and since it is lower bounded, the algorithm is guaranteed to converge to a local minimum. We also show that the Bethe free energy is equal to the TAP free energy up to second order in the weights. In experiments we confirm that when belief propagation converges it usually finds identical solutions as our belief optimization method. However, in cases where belief propagation fails to converge, belief optimization continues to converge to reasonable beliefs. The stable nature of belief optimization makes it ideally suited for learning graphical models from data.