Information theory provides a mathematical foundation to measure uncertainty in belief. Belief is represented by a probability distribution that captures our understanding of an outcome's plausibility. Information measures based on Shannon's concept of entropy include realization information, Kullback-Leibler divergence, Lindley's information in experiment, cross entropy, and mutual information. We derive a general theory of information from first principles that accounts for evolving belief and recovers all of these measures. Rather than simply gauging uncertainty, information is understood in this theory to measure change in belief. We may then regard entropy as the information we expect to gain upon realization of a discrete latent random variable. This theory of information is compatible with the Bayesian paradigm in which rational belief is updated as evidence becomes available. Furthermore, this theory admits novel measures of information with well-defined properties, which we explore in both analysis and experiment. This view of information illuminates the study of machine learning by allowing us to quantify information captured by a predictive model and distinguish it from residual information contained in training data. We gain related insights regarding feature selection, anomaly detection, and novel Bayesian approaches.

According to Boutillier, Darwiche and Pearl and others, principles for iterated revision can be characterised in terms of changing beliefs about conditionals. For iterated contraction a similar formulation is not known. This is especially because for iterated belief change the connection between revision and contraction via the Levi and Harper identity is not straightforward, and therefore, characterisation results do not transfer easily between iterated revision and contraction. In this article, we develop an axiomatisation of iterated contraction in terms of changing conditional beliefs. We prove that the new set of postulates conforms semantically to the class of operators like the ones given by Konieczny and Pino Pérez for iterated contraction. 1 Introduction For the three main classes of theory change, revision, expansion and contraction, different characterisations are known [12], which are heavily supported by the correspondence between revision and contraction via the Levi and Harper identities [13, 17].

W e argue the case for Gaussian Belief Propagation (GBP) as a strong algorithmic framework for the distributed, generic and incremental probabilistic estimation we need in Spatial AI as we aim at high performance smart robots and devices which operate within the constraints of real products. Processor hardware is changing rapidly, and GBP has the right character to take advantage of highly distributed processing and storage while estimating global quantities, as well as great flexibility. W e present a detailed tutorial on GBP, relating to the standard factor graph formulation used in robotics and computer vision, and give several simulation examples with code which demonstrate its properties.

In most classical models of belief change, epistemic states are represented by theories (AGM) or formulas (Katsuno-Mendelzon) and the new pieces of information by formulas. The Representation Theorem for revision operators says that operators are represented by total preorders. This important representation is exploited by Darwiche and Pearl to shift the notion of epistemic state to a more abstract one, where the paradigm of epistemic state is indeed that of a total preorder over interpretations. In this work, we introduce a 3-valued logic where the formulas can be identified with a generalisation of total preorders of three levels: a ranking function mapping interpretations into the truth values. Then we analyse some sort of changes in this kind of structures and give syntactical characterizations of them.

The problem of belief tracking in the presence of stochastic actions and observations is pervasive and yet computationally intractable. In this work we show however that probabilistic beliefs can be maintained in factored form exactly and efficiently across a number of causally closed beams, when the state variables that appear in more than one beam obey a form of backward determinism . Since computing marginals from the factors is still computationally intractable in general, and variables appearing in several beams are not always backward-deterministic, the basic formulation is extended with two approximations: forms of belief propagation for computing marginals from factors, and sampling of non-backward-deterministic variables for making such variables backward-deterministic given their sampled history. Unlike, Rao-Blackwellized particle-filtering, the sampling is not used for making inference tractable but for making the factorization sound . The resulting algorithm involves sampling and belief propagation or just one of them as determined by the structure of the model.

--Belief Propagation (BP) is a message-passing algorithm for approximate inference over Probabilistic Graphical Models (PGMs), finding many applications such as computer vision, error-correcting codes, and protein-folding. While general, the convergence and speed of the algorithm has limited its practical use on difficult inference problems. As an algorithm that is highly amenable to parallelization, many-core Graphical Processing Units (GPUs) could significantly improve BP performance. Improving BP through many-core systems is nontrivial: the scheduling of messages in the algorithm strongly affects performance. We present a study of message scheduling for BP on GPUs. We demonstrate that BP exhibits a tradeoff between speed and convergence based on parallelism and show that existing message schedulings are not able to utilize this tradeoff. T o this end, we present a novel randomized message scheduling approach, Randomized BP (RnBP), which outperforms existing methods on the GPU. I NTRODUCTION Probabilistic Graphical Models (PGMs) are powerful, general machine learning models that encode distributions over random variables. PGM Inference, in which we seek to compute some probabilistic beliefs within the system modeled by the PGM, is in general an intractable problem, leading to dependence on approximate algorithms. Belief Propagation (BP) is a widely employed approximate inference algorithms for PGMs [1].

To coordinate with other systems, agents must be able to determine what the systems are currently doing and predict what they will be doing in the future---plan and goal recognition. There are many methods for plan and goal recognition, but they assume a passive observer that continually monitors the target system. Real-world domains, where information gathering has a cost (e.g., moving a camera or a robot, or time taken away from another task), will often require a more active observer. We propose to combine goal recognition with other observer tasks in order to obtain \emph{active goal recognition} (AGR). We discuss this problem and provide a model and preliminary experimental results for one form of this composite problem. As expected, the results show that optimal behavior in AGR problems balance information gathering with other actions (e.g., task completion) such as to achieve all tasks jointly and efficiently. We hope that our formulation opens the door for extensive further research on this interesting and realistic problem.

Memory in an agent system is a process of reasoning: it is the l earning process of strengthening a concept. The interaction between an agent and the environment can pla y an important role in constructing its memory and may affect its future behaviour. In fact, through memory an agent is potentially able to recall and to learn from experiences so that its beliefs and i ts future course of action are grounded in these experiences. In computational logic, [2] introduces DLEK (Dynamic Logic of Explicit beliefs and Knowledge) as a logical formalization of the short-term and long-term memory. The underlying idea is to represent reasoning about the formation of beliefs throu gh perception and inference in non-omniscient resource-bounded agents. DLEK has however no notion of time, while agents' actual perceptions are inherently timed and so are many of the inferences drawn from such perceptions. In this paper we present an extension of LEK/DLEK to T-LEK/T-DLEK ("Timed LE K" and "Timed DLEK") obtained by introducing a special function which associates to each b elief the arrival time and controls timed inferences. Through this function it is easier to keep the ev olution of the surrounding world under control and the representation is more complete. This abstr act is an evolution version of [3], where we have introduced explicit time instants and time intervals i n formulas, and it is extracted from [4].

In the literature there different kind of timed logical fram eworks exist, where time is specified directly using hybrid logics (cf., e.g., [2]), temporal epistemic lo gic (cf., e.g., [4]) or simply by using Linear Temporal Logic. We propose a temporal module which can be ado pted to "temporalize" many logical framework. This module is in practice a particular kind of fu nction that assigns a "timing" to atoms. We have exploited this T function in two different settings. The first one is the formalization of the reasoning on the formation of beliefs and the interaction wi th background knowledge in non-omniscient agents' memory.

Belief propagation (BP) can do exact inference in loop-free graphs, but its performance could be poor in graphs with loops, and the understanding of its solution is limited. This work gives an interpretable belief propagation rule that is actually minimization of a localized $\alpha$-divergence. We term this algorithm as $\alpha$ belief propagation ($\alpha$-BP). The performance of $\alpha$-BP is tested in MAP (maximum a posterior) inference problems, where $\alpha$-BP can outperform (loopy) BP by a significant margin even in fully-connected graphs.