Compositional Models in Valuation-Based Systems

This is the author final draft. Copyright 2014 ElsevierCompositional models were initially described for discrete probability theory, and later extended for possibility theory and for belief functions in Dempster–Shafer (D–S) theory of evidence. Valuation-based system (VBS) is an unifying theoretical framework generalizing some of the well known and frequently used uncertainty calculi. This generalization enables us to not only highlight the most important theoretical properties necessary for efficient inference (analogous to Bayesian inference in the framework of Bayesian network), but also to design efficient computational procedures. Some of the specific calculi covered by VBS are probability theory, a version of possibility theory where combination is the product t-norm, Spohn’s epistemic belief theory, and D–S belief function theory. In this paper, we describe compositional models in the general framework of VBS using the semantics of no-double counting, which is central to the VBS framework. Also, we show that conditioning can be expressed using the composition operator. We define a special case of compositional models called decomposable models, again in the VBS framework, and demonstrate that for the class of decomposable compositional models, conditioning can be done using local computation. As all results are obtained for the VBS framework, they hold in all calculi that fit in the VBS framework. For the D–S theory of belief functions, the compositional model defined here differs from the one studied by Jiroušek, Vejnarová, and Daniel. The latter model can also be described in the VBS framework, but with a combination operator that is different from Dempster’s rule of combination. For the version of possibility theory in which combination is the product t-norm, the compositional model defined here reduces to the one studied by Vejnarová

Approximate model composition for explanation generation

This thesis presents a framework for the formulation of knowledge models to sup¬ port the generation of explanations for engineering systems that are represented by the resulting models. Such models are automatically assembled from instantiated generic component descriptions, known as modelfragments. The model fragments are of suffi¬ cient detail that generally satisfies the requirements of information content as identified by the user asking for explanations. Through a combination of fuzzy logic based evidence preparation, which exploits the history of prior user preferences, and an approximate reasoning inference engine, with a Bayesian evidence propagation mechanism, different uncertainty sources can be han¬ dled. Model fragments, each representing structural or behavioural aspects of a com¬ ponent of the domain system of interest, are organised in a library. Those fragments that represent the same domain system component, albeit with different representation detail, form parts of the same assumption class in the library. Selected fragments are assembled to form an overall system model, prior to extraction of any textual infor¬ mation upon which to base the explanations. The thesis proposes and examines the techniques that support the fragment selection mechanism and the assembly of these fragments into models. In particular, a Bayesian network-based model fragment selection mechanism is de¬ scribed that forms the core of the work. The network structure is manually determined prior to any inference, based on schematic information regarding the connectivity of the components present in the domain system under consideration. The elicitation of network probabilities, on the other hand is completely automated using probability elicitation heuristics. These heuristics aim to provide the information required to select fragments which are maximally compatible with the given evidence of the fragments preferred by the user. Given such initial evidence, an existing evidence propagation algorithm is employed. The preparation of the evidence for the selection of certain fragments, based on user preference, is performed by a fuzzy reasoning evidence fab¬ rication engine. This engine uses a set of fuzzy rules and standard fuzzy reasoning mechanisms, attempting to guess the information needs of the user and suggesting the selection of fragments of sufficient detail to satisfy such needs. Once the evidence is propagated, a single fragment is selected for each of the domain system compo¬ nents and hence, the final model of the entire system is constructed. Finally, a highly configurable XML-based mechanism is employed to extract explanation content from the newly formulated model and to structure the explanatory sentences for the final explanation that will be communicated to the user. The framework is illustratively applied to a number of domain systems and is compared qualitatively to existing compositional modelling methodologies. A further empirical assessment of the performance of the evidence propagation algorithm is carried out to determine its performance limits. Performance is measured against the number of frag¬ ments that represent each of the components of a large domain system, and the amount of connectivity permitted in the Bayesian network between the nodes that stand for the selection or rejection of these fragments. Based on this assessment recommenda¬ tions are made as to how the framework may be optimised to cope with real world applications

ISIPTA'07: Proceedings of the Fifth International Symposium on Imprecise Probability: Theories and Applications

B

Algebraic Uncertainty Theory : A Unifying Perspective on Reasoning under Uncertainty

The question of how to represent and process uncertainty is of fundamental importance to the scientific process, but also in everyday life. Currently there exist a lot of different calculi for managing uncertainty, each having its own advantages and disadvantages. Especially, almost all are defining the domain and structure of uncertainty values a priori, e.g., one real number, two real numbers, a finite domain, and so on, but maybe uncertainty is best measured by complex numbers, matrices or still another mathematical structure. This thesis investigates the notion of uncertainty from a foundational point of view, provides an ontology and axiomatic core system for uncertainty and derives and not defines the structure of uncertainty. The main result, the ring theorem, stating that uncertainty values are elements of the [0,1]-interval of a partially ordered ring, is used to derive a general decomposition theorem for uncertainty values, splitting them into a numerical interval and an ``interaction term''. In order to illustrate the unifying power of these results, the relationship to Dempster-Shafer theory is discussed and it is shown that all Dempster-Shafer measures over finite domains can be represented by ring-valued uncertainty measures. Finally, the historical development of approaches to modeling uncertainty which have led to the results of this thesis are reviewed

First IJCAI International Workshop on Graph Structures for Knowledge Representation and Reasoning (GKR@IJCAI'09)

International audienceThe development of effective techniques for knowledge representation and reasoning (KRR) is a crucial aspect of successful intelligent systems. Different representation paradigms, as well as their use in dedicated reasoning systems, have been extensively studied in the past. Nevertheless, new challenges, problems, and issues have emerged in the context of knowledge representation in Artificial Intelligence (AI), involving the logical manipulation of increasingly large information sets (see for example Semantic Web, BioInformatics and so on). Improvements in storage capacity and performance of computing infrastructure have also affected the nature of KRR systems, shifting their focus towards representational power and execution performance. Therefore, KRR research is faced with a challenge of developing knowledge representation structures optimized for large scale reasoning. This new generation of KRR systems includes graph-based knowledge representation formalisms such as Bayesian Networks (BNs), Semantic Networks (SNs), Conceptual Graphs (CGs), Formal Concept Analysis (FCA), CPnets, GAI-nets, all of which have been successfully used in a number of applications. The goal of this workshop is to bring together the researchers involved in the development and application of graph-based knowledge representation formalisms and reasoning techniques

Transforming Gaussian processes with normalizing flows

Gaussian Processes (GP) can be used as flexible, non-parametric function priors. Inspired by the growing body of work on Normalizing Flows, we enlarge this class of priors through a parametric invertible transformation that can be made input-dependent. Doing so also allows us to encode interpretable prior knowledge (e.g., boundedness constraints). We derive a variational approximation to the resulting Bayesian inference problem, which is as fast as stochastic variational GP regression (Hensman et al., 2013; Dezfouli and Bonilla, 2015). This makes the model a computationally efficient alternative to other hierarchical extensions of GP priors (Lázaro-Gredilla,2012; Damianou and Lawrence,2013). The resulting algorithm’s computational and inferential performance is excellent, and we demonstrate this on a range of data sets. For example, even with only 5 inducing points and an input-dependent flow, our method is consistently competitive with a standard sparse GP fitted using 100 inducing points