The overall goal of neuro-symbolic computation is to integrate high-level reasoning with low-level perception. We argue 1) that neuro-symbolic computation should integrate neural networks with the two most prominent methods for reasoning, that is, logic and probability, and 2) that neuro-symbolic integrated methods should have the pure neural, logical and probabilistic methods as special cases. We examine the state-of-the-art with regard to these claims and briefly position our own contribution DeepProbLog in this perspective.

Probabilistic programming provides a convenient lingua franca for writing succinct and rigorous descriptions of probabilistic models and inference tasks. Several probabilistic programming languages, including Anglican, Church or Hakaru, derive their expressiveness from a powerful combination of continuous distributions, conditioning, and higher-order functions. Although very important for practical applications, these features raise fundamental challenges for program semantics and verification. Several recent works offer promising answers to these challenges, but their primary focus is on foundational semantics issues. In this paper, we take a step further by developing a suite of logics, collectively named PPV, for proving properties of programs written in an expressive probabilistic higher-order language with continuous sampling operations and primitives for conditioning distributions. Our logics mimic the comfortable reasoning style of informal proofs using carefully selected axiomatizations of key results from probability theory. The versatility of our logics is illustrated through the formal verification of several intricate examples from statistics, probabilistic inference, and machine learning. We further show expressiveness by giving sound embeddings of existing logics. In particular, we do this in a parametric way by showing how the semantics idea of (unary and relational) ⊤⊤-lifting can be internalized in our logics. The soundness of PPV follows by interpreting programs and assertions in quasi-Borel spaces (QBS), a recently proposed variant of Borel spaces with a good structure for interpreting higher order probabilistic programs.

Programming languages allow us to express complex predicates, but existing inference methods are unable to condition probabilistic models on most of them. To support a broader class of predicates, we develop an inference procedure called predicate exchange, which softens predicates. A soft predicate quantifies the extent to which values of model variables are consistent with its hard counterpart. We substitute the likelihood term in the Bayesian posterior with a soft predicate, and develop a variant of replica exchange MCMC to draw posterior samples. We implement predicate exchange as a language agnostic tool which performs a nonstandard execution of a probabilistic program. We demonstrate the approach on sequence models of health and inverse rendering.

We introduce DeepProbLog, a probabilistic logic programming language that incorporates deep learning by means of neural predicates. We show how existing inference and learning techniques can be adapted for the new language. Our experiments demonstrate that DeepProbLog supports both symbolic and subsymbolic representations and inference, 1) program induction, 2) probabilistic (logic) programming, and 3) (deep) learning from examples. To the best of our knowledge, this work is the first to propose a framework where general-purpose neural networks and expressive probabilistic-logical modeling and reasoning are integrated in a way that exploits the full expressiveness and strengths of both worlds and can be trained end-to-end based on examples.      

This book explores the probabilistic approach to cognitive science, which models learning and reasoning as inference in complex probabilistic models. We examine how a broad range of empirical phenomena, including intuitive physics, concept learning, causal reasoning, social cognition, and language understanding, can be modeled using probabilistic programs (using the WebPPL language).

This document is designed to be a first-year graduate-level introduction to probabilistic programming. It not only provides a thorough background for anyone wishing to use a probabilistic programming system, but also introduces the techniques needed to design and build these systems. It is aimed at people who have an undergraduate-level understanding of either or, ideally, both probabilistic machine learning and programming languages.
We start with a discussion of model-based reasoning and explain why conditioning as a foundational computation is central to the fields of probabilistic machine learning and artificial intelligence. We then introduce a simple first-order probabilistic programming language (PPL) whose programs define static-computation-graph, finite-variable-cardinality models. In the context of this restricted PPL we introduce fundamental inference algorithms and describe how they can be implemented in the context of models denoted by probabilistic programs.
In the second part of this document, we introduce a higher-order probabilistic programming language, with a functionality analogous to that of established programming languages. This affords the opportunity to define models with dynamic computation graphs, at the cost of requiring inference methods that generate samples by repeatedly executing the program. Foundational inference algorithms for this kind of probabilistic programming language are explained in the context of an interface between program executions and an inference controller.
This document closes with a chapter on advanced topics which we believe to be, at the time of writing, interesting directions for probabilistic programming research; directions that point towards a tight integration with deep neural network research and the development of systems for next-generation artificial intelligence applications.

This book describes and implements models of rational agents for (PO)MDPs and Reinforcement Learning. One motivation is to create richer models of human planning, which capture human biases and bounded rationality.

Agents are implemented as differentiable functional programs in a probabilistic programming language based on Javascript. Agents plan by recursively simulating their future selves or by simulating their opponents in multi-agent games. Our agents and environments run directly in the browser and are easy to modify and extend.

The book assumes basic programming experience but is otherwise self-contained. It includes short introductions to “planning as inference”, MDPs, POMDPs, inverse reinforcement learning, hyperbolic discounting, myopic planning, and multi-agent planning.

We develop the operational semantics of an untyped probabilistic lambda-calculus with continuous distributions, as a foundation for universal probabilistic programming languages such as Church, Anglican, and Venture. Our first contribution is to adapt the classic operational semantics of lambda-calculus to a continuous setting via creating a measure space on terms and defining step-indexed approximations. We prove equivalence of big-step and small-step formulations of this distribution-based semantics. To move closer to inference techniques, we also define the sampling-based semantics of a term as a function from a trace of random samples to a value. We show that the distribution induced by integrating over all traces equals the distribution-based semantics. Our second contribution is to formalize the implementation technique of trace Markov chain Monte Carlo (MCMC) for our calculus and to show its correctness. A key step is defining sufficient conditions for the distribution induced by trace MCMC to converge to the distribution-based semantics. To the best of our knowledge, this is the first rigorous correctness proof for trace MCMC for a higher-order functional language.

Bayesian inference, of posterior knowledge from prior knowledge and observed evidence, is typically defined by Bayes’s rule, which says the posterior multiplied by the probability of an observation equals a joint probability. But the observation of a continuous quantity usually has probability zero, in which case Bayes’s rule says only that the unknown times zero is zero. To infer a posterior distribution from a zero-probability observation, the statistical notion of disintegration tells us to specify the observation as an expression rather than a predicate, but does not tell us how to compute the posterior. We present the first method of computing a disintegration from a probabilistic program and an expression of a quantity to be observed, even when the observation has probability zero. Because the method produces an exact posterior term and preserves a semantics in which monadic terms denote measures, it composes with other inference methods in a modular way—without sacrificing accuracy or performance.

We study the semantic foundation of expressive probabilistic programming languages, that support higher-order functions, continuous distributions, and soft constraints (such as Anglican, Church, and Venture). We define a metalanguage (an idealised version of Anglican) for probabilistic computation with the above features, develop both operational and denotational semantics, and prove soundness, adequacy, and termination. This involves measure theory, stochastic labelled transition systems, and functor categories, but admits intuitive computational readings, one of which views sampled random variables as dynamically allocated read-only variables. We apply our semantics to validate nontrivial equations underlying the correctness of certain compiler optimisations and inference algorithms such as sequential Monte Carlo simulation. The language enables defining probability distributions on higher-order functions, and we study their properties.