Decision Making under Uncertainty through Extending Influence Diagrams with Interval-valued Parameters

Influence Diagrams (IDs) are one of the most commonly used graphical and mathematical decision models for reasoning under uncertainty. In conventional IDs, both probabilities representing beliefs and utilities representing preferences of decision makers are precise point-valued parameters. However, it is usually difficult or even impossible to directly provide such parameters. In this paper, we extend conventional IDs to allow IDs with interval-valued parameters (IIDs), and develop a counterpart method of Copper’s evaluation method to evaluate IIDs. IIDs avoid the difficulties attached to the specification of precise parameters and provide the capability to model decision making processes in a situation that the precise parameters cannot be specified. The counterpart method to Copper’s evaluation method reduces the evaluation of IIDs into inference problems of IBNs. An algorithm based on the approximate inference of IBNs is proposed, extensive experiments are conducted. The experimental results indicate that the proposed algorithm can find the optimal strategies effectively in IIDs, and the interval-valued expected utilities obtained by proposed algorithm are contained in those obtained by exact evaluating algorithms

Proceedings of the Fifth European Workshop on Probabilistic Graphical Models

Peer reviewe

Efficient Maximum A-Posteriori Inference in Markov Logic and Application in Description Logics

Maximum a-posteriori (MAP) query in statistical relational models computes the most probable world given evidence and further knowledge about the domain. It is arguably one of the most important types of computational problems, since it is also used as a subroutine in weight learning algorithms. In this thesis, we discuss an improved inference algorithm and an application for MAP queries. We focus on Markov logic (ML) as statistical relational formalism. Markov logic combines Markov networks with first-order logic by attaching weights to first-order formulas. For inference, we improve existing work which translates MAP queries to integer linear programs (ILP). The motivation is that existing ILP solvers are very stable and fast and are able to precisely estimate the quality of an intermediate solution. In our work, we focus on improving the translation process such that we result in ILPs having fewer variables and fewer constraints. Our main contribution is the Cutting Plane Aggregation (CPA) approach which leverages symmetries in ML networks and parallelizes MAP inference. Additionally, we integrate the cutting plane inference (Riedel 2008) algorithm which significantly reduces the number of groundings by solving multiple smaller ILPs instead of one large ILP. We present the new Markov logic engine RockIt which outperforms state-of-the-art engines in standard Markov logic benchmarks. Afterwards, we apply the MAP query to description logics. Description logics (DL) are knowledge representation formalisms whose expressivity is higher than propositional logic but lower than first-order logic. The most popular DLs have been standardized in the ontology language OWL and are an elementary component in the Semantic Web. We combine Markov logic, which essentially follows the semantic of a log-linear model, with description logics to log-linear description logics. In log-linear description logic weights can be attached to any description logic axiom. Furthermore, we introduce a new query type which computes the most-probable 'coherent' world. Possible applications of log-linear description logics are mainly located in the area of ontology learning and data integration. With our novel log-linear description logic reasoner ELog, we experimentally show that more expressivity increases quality and that the solutions of optimal solving strategies have higher quality than the solutions of approximate solving strategies

Uncertainty-Aware Numerical Solutions of ODEs by Bayesian Filtering

Numerical analysis is the branch of mathematics that studies algorithms that compute approximations of well-defined, but analytically-unknown mathematical quantities. Statistical inference, on the other hand, studies which judgments can be made on unknown parameters in a statistical model. By interpreting the unknown quantity of interest as a parameter and providing a statistical model that relates it to the available numerical information (the `data'), we can thus recast any problem of numerical approximation as statistical inference. In this way, the field of probabilistic numerics introduces new 'uncertainty-aware' numerical algorithms that capture all relevant sources of uncertainty (including all numerical approximation errors) by probability distributions. While such recasts have been a decades-long success story for global optimization and quadrature (under the names of Bayesian optimization and Bayesian quadrature), the equally important numerical task of solving ordinary differential equations (ODEs) has been, until recently, largely ignored. With this dissertation, we aim to further shed light on this area of previous ignorance in three ways: Firstly, we present a first rigorous Bayesian model for initial value problems (IVPs) as statistical inference, namely as a stochastic filtering problem, which unlocks the employment of all Bayesian filters (and smoothers) to IVPs. Secondly, we theoretically analyze the properties of these new ODE filters, with a special emphasis on the convergence rates of Gaussian (Kalman) ODE filters with integrated Brownian motion prior, and explore their potential for (active) uncertainty quantification. And, thirdly, we demonstrate how employing these ODE filters as a forward simulator engenders new ODE inverse problem solvers that outperform classical 'uncertainty-unaware' ('likelihood-free') approaches. This core content is presented in Chapter 2. It is preceded by a concise introduction in Chapter 1 which conveys the necessary concepts and locates our work in the research environment of probabilistic numerics. The final Chapter 3 concludes with an in-depth discussion of our results and their implications

Tools and Algorithms for the Construction and Analysis of Systems

This open access two-volume set constitutes the proceedings of the 27th International Conference on Tools and Algorithms for the Construction and Analysis of Systems, TACAS 2021, which was held during March 27 – April 1, 2021, as part of the European Joint Conferences on Theory and Practice of Software, ETAPS 2021. The conference was planned to take place in Luxembourg and changed to an online format due to the COVID-19 pandemic. The total of 41 full papers presented in the proceedings was carefully reviewed and selected from 141 submissions. The volume also contains 7 tool papers; 6 Tool Demo papers, 9 SV-Comp Competition Papers. The papers are organized in topical sections as follows: Part I: Game Theory; SMT Verification; Probabilities; Timed Systems; Neural Networks; Analysis of Network Communication. Part II: Verification Techniques (not SMT); Case Studies; Proof Generation/Validation; Tool Papers; Tool Demo Papers; SV-Comp Tool Competition Papers

Information Geometry

This Special Issue of the journal Entropy, titled “Information Geometry I”, contains a collection of 17 papers concerning the foundations and applications of information geometry. Based on a geometrical interpretation of probability, information geometry has become a rich mathematical field employing the methods of differential geometry. It has numerous applications to data science, physics, and neuroscience. Presenting original research, yet written in an accessible, tutorial style, this collection of papers will be useful for scientists who are new to the field, while providing an excellent reference for the more experienced researcher. Several papers are written by authorities in the field, and topics cover the foundations of information geometry, as well as applications to statistics, Bayesian inference, machine learning, complex systems, physics, and neuroscience