Cumulative Distribution Functions As The Foundation For Probabilistic Models

This thesis discusses applications of probabilistic and connectionist models for constructing and training cumulative distribution functions (CDFs). First, it is shown how existing tools from the copula literature can be combined to build probabilistic models. It is found that this simple construction leads to numerical and scalability issues that make training and inference challenging. Next, several innovative ideas, combining neural networks, automatic differentiation and copula functions, introduce how to assemble black-box probabilistic models. The basic building block is a cumulative distribution function that is straightforward to construct, composed of arithmetic operations and nonlinear functions. There is no need to assume any specific parametric probability density function (PDF), making the model flexible and normalisation unnecessary. The only requirement is to design a computational graph that parameterises monotonically non-decreasing functions with a constrained range. Training can be then performed using standard tools from any neural network software library. Finally, factorial hidden Markov models (FHMMs) for sequential data are presented. It is shown how to leverage cumulative distribution functions in the form of the Gaussian copula and amortised stochastic variational method to encode hidden Markov chains coherently. This approach enables efficient learning and inference to model long sequences of high-dimensional data with long-range dependencies. Tackling such complex problems was impossible with the established FHMM approximate inference algorithm. It is empirically verified on several problems that some of the estimators introduced in this work can perform comparably or better than the currently popular models. Especially for tasks requiring tail-area or marginal probabilities that can be read directly from a cumulative distribution function

Posterior Regularization for Structured Latent Varaible Models

We present posterior regularization, a probabilistic framework for structured, weakly supervised learning. Our framework efficiently incorporates indirect supervision via constraints on posterior distributions of probabilistic models with latent variables. Posterior regularization separates model complexity from the complexity of structural constraints it is desired to satisfy. By directly imposing decomposable regularization on the posterior moments of latent variables during learning, we retain the computational efficiency of the unconstrained model while ensuring desired constraints hold in expectation. We present an efficient algorithm for learning with posterior regularization and illustrate its versatility on a diverse set of structural constraints such as bijectivity, symmetry and group sparsity in several large scale experiments, including multi-view learning, cross-lingual dependency grammar induction, unsupervised part-of-speech induction, and bitext word alignment

Conceptual Spaces for Cognitive Architectures: A Lingua Franca for Different Levels of Representation

During the last decades, many cognitive architectures (CAs) have been realized adopting different assumptions about the organization and the representation of their knowledge level. Some of them (e.g. SOAR [35]) adopt a classical symbolic approach, some (e.g. LEABRA[ 48]) are based on a purely connectionist model, while others (e.g. CLARION [59]) adopt a hybrid approach combining connectionist and symbolic representational levels. Additionally, some attempts (e.g. biSOAR) trying to extend the representational capacities of CAs by integrating diagrammatical representations and reasoning are also available [34]. In this paper we propose a reflection on the role that Conceptual Spaces, a framework developed by Peter G¨ardenfors [24] more than fifteen years ago, can play in the current development of the Knowledge Level in Cognitive Systems and Architectures. In particular, we claim that Conceptual Spaces offer a lingua franca that allows to unify and generalize many aspects of the symbolic, sub-symbolic and diagrammatic approaches (by overcoming some of their typical problems) and to integrate them on a common ground. In doing so we extend and detail some of the arguments explored by G¨ardenfors [23] for defending the need of a conceptual, intermediate, representation level between the symbolic and the sub-symbolic one. In particular we focus on the advantages offered by Conceptual Spaces (w.r.t. symbolic and sub-symbolic approaches) in dealing with the problem of compositionality of representations based on typicality traits. Additionally, we argue that Conceptual Spaces could offer a unifying framework for interpreting many kinds of diagrammatic and analogical representations. As a consequence, their adoption could also favor the integration of diagrammatical representation and reasoning in CAs

Neural distribution estimation as a two-part problem

Given a dataset of examples, distribution estimation is the task of approximating the assumed underlying probability distribution from which those samples were drawn. Neural distribution estimation relies on the powerful function approximation capabilities of deep neural networks to build models for this purpose, and excels when data is high-dimensional and exhibits complex, nonlinear dependencies. In this thesis, we explore several approaches to neural distribution estimation, and present a unified perspective for these methods based on a two-part design principle. In particular, we examine how many models iteratively break down the task of distribution estimation into a series of tractable sub-tasks, before fitting a multi-step generative process which combines solutions to these sub-tasks in order to approximate the data distribution of interest. Framing distribution estimation as a two-part problem provides a shared language in which to compare and contrast prevalent models in the literature, and also allows for discussion of alternative approaches which do not follow this structure. We first present the Autoregressive Energy Machine, an energy-based model which is trained by approximate maximum likelihood through an autoregressive decomposition. The method demonstrates the flexibility of an energy-based model over an explicitly normalized model, and the novel application of autoregressive importance sampling highlights the benefit of an autoregressive approach to distribution estimation which recursively transforms the problem into a series of univariate tasks. Next, we present Neural Spline Flows, a class of normalizing flow models based on monotonic spline transformations which admit both an explicit inverse and a tractable Jacobian determinant. Normalizing flows tackle distribution estimation by searching for an invertible map between the data distribution and a more tractable base distribution, and this map is typically constructed as the composition of a series of invertible building blocks. We demonstrate that spline flows can be used to enhance density estimation of tabular data, variational inference in latent variable models, and generative modeling of natural images. The third chapter presents Maximum Likelihood Training of Score-Based Diffusion Models. Generative models based on estimation of the gradient of the logarithm of the probability density---or score function---have recently gained traction as a powerful modeling paradigm, in which the data distribution is gradually transformed toward a tractable base distribution by means of a stochastic process. The paper illustrates how this class of models can be trained by maximum likelihood, resulting in a model which is functionally equivalent to a continuous normalizing flow, and which bridges the gap between two branches of the literature. We also discuss latent-variable generative models more broadly, of which diffusion models are a structured special case. Finally, we present On Contrastive Learning for Likelihood-Free Inference, a unifying perspective for likelihood-free inference methods which perform Bayesian inference using either density estimation or density-ratio estimation. Likelihood-free inference focuses on inference in stochastic simulator models where the likelihood of parameters given observations is computationally intractable, and traditional inference methods fall short. In addition to illustrating the power of normalizing flows as generic tools for density estimation, this chapter also gives us the opportunity to discuss likelihood-free models more broadly. These so-called implicit generative models form a large part of the distribution estimation literature under the umbrella of generative adversarial networks, and are distinct in how they treat distribution estimation as a one-part problem

Tools and Algorithms for the Construction and Analysis of Systems

This book is Open Access under a CC BY licence. The LNCS 11427 and 11428 proceedings set constitutes the proceedings of the 25th International Conference on Tools and Algorithms for the Construction and Analysis of Systems, TACAS 2019, which took place in Prague, Czech Republic, in April 2019, held as part of the European Joint Conferences on Theory and Practice of Software, ETAPS 2019. The total of 42 full and 8 short tool demo papers presented in these volumes was carefully reviewed and selected from 164 submissions. The papers are organized in topical sections as follows: Part I: SAT and SMT, SAT solving and theorem proving; verification and analysis; model checking; tool demo; and machine learning. Part II: concurrent and distributed systems; monitoring and runtime verification; hybrid and stochastic systems; synthesis; symbolic verification; and safety and fault-tolerant systems

On the semantics of fuzzy logic

AbstractThis paper presents a formal characterization of the major concepts and constructs of fuzzy logic in terms of notions of distance, closeness, and similarity between pairs of possible worlds. The formalism is a direct extension (by recognition of multiple degrees of accessibility, conceivability, or reachability) of the najor modal logic concepts of possible and necessary truth.Given a function that maps pairs of possible worlds into a number between 0 and 1, generalizing the conventional concept of an equivalence relation, the major constructs of fuzzy logic (conditional and unconditioned possibility distributions) are defined in terms of this similarity relation using familiar concepts from the mathematical theory of metric spaces. This interpretation is different in nature and character from the typical, chance-oriented, meanings associated with probabilistic concepts, which are grounded on the mathematical notion of set measure. The similarity structure defines a topological notion of continuity in the space of possible worlds (and in that of its subsets, i.e., propositions) that allows a form of logical “extrapolation” between possible worlds.This logical extrapolation operation corresponds to the major deductive rule of fuzzy logic — the compositional rule of inference or generalized modus ponens of Zadeh — an inferential operation that generalizes its classical counterpart by virtue of its ability to be utilized when propositions representing available evidence match only approximately the antecedents of conditional propositions. The relations between the similarity-based interpretation of the role of conditional possibility distributions and the approximate inferential procedures of Baldwin are also discussed.A straightforward extension of the theory to the case where the similarity scale is symbolic rather than numeric is described. The problem of generating similarity functions from a given set of possibility distributions, with the latter interpreted as defining a number of (graded) discernibility relations and the former as the result of combining them into a joint measure of distinguishability between possible worlds, is briefly discussed