Compositional Semantics for Probabilistic Programs with Exact Conditioning

We define a probabilistic programming language for Gaussian random variables with a first-class exact conditioning construct. We give operational, denotational and equational semantics for this language, establishing convenient properties like exchangeability of conditions. Conditioning on equality of continuous random variables is nontrivial, as the exact observation may have probability zero; this is Borel's paradox. Using categorical formulations of conditional probability, we show that the good properties of our language are not particular to Gaussians, but can be derived from universal properties, thus generalizing to wider settings. We define the Cond construction, which internalizes conditioning as a morphism, providing general compositional semantics for probabilistic programming with exact conditioning.Comment: 16 pages, 5 figure

Formal verification of higher-order probabilistic programs

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 combined features raise fundamental challenges for program semantics and verification. Several recent works offer promising answers to these challenges, but their primary focus is on semantical issues. In this paper, we take a step further and we develop a set of program logics, named PPV, for proving properties of programs written in an expressive probabilistic higher-order language with continuous distributions and operators for conditioning distributions by real-valued functions. Pleasingly, our program logics retain the comfortable reasoning style of informal proofs thanks to 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 the expressiveness of our logics 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) TT-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

Design of Evolutionary Methods Applied to the Learning of Bayesian Network Structures

Bayesian Network, Ahmed Rebai (Ed.), ISBN: 978-953-307-124-4, pp. 13-38

The geometry of Bayesian programming

We give two geometry of interaction models for a typed λ-calculus with recursion endowed with operators for sampling from a continuous uniform distribution and soft conditioning, namely a paradigmatic calculus for higher-order Bayesian programming. The models are based on the category of measurable spaces and partial measurable functions, and the category of measurable spaces and s-finite kernels, respectively. The former is proved adequate with respect to both a distribution-based and a sampling-based operational semantics, while the latter is proved adequate with respect to a sampling-based operational semantics

Learning Bayesian network equivalence classes using ant colony optimisation

Bayesian networks have become an indispensable tool in the modelling of uncertain knowledge. Conceptually, they consist of two parts: a directed acyclic graph called the structure, and conditional probability distributions attached to each node known as the parameters. As a result of their expressiveness, understandability and rigorous mathematical basis, Bayesian networks have become one of the first methods investigated, when faced with an uncertain problem domain. However, a recurring problem persists in specifying a Bayesian network. Both the structure and parameters can be difficult for experts to conceive, especially if their knowledge is tacit.To counteract these problems, research has been ongoing, on learning both the structure and parameters of Bayesian networks from data. Whilst there are simple methods for learning the parameters, learning the structure has proved harder. Part ofthis stems from the NP-hardness of the problem and the super-exponential space of possible structures. To help solve this task, this thesis seeks to employ a relatively new technique, that has had much success in tackling NP-hard problems. This technique is called ant colony optimisation. Ant colony optimisation is a metaheuristic based on the behaviour of ants acting together in a colony. It uses the stochastic activity of artificial ants to find good solutions to combinatorial optimisation problems. In the current work, this method is applied to the problem of searching through the space of equivalence classes of Bayesian networks, in order to find a good match against a set of data. The system uses operators that evaluate potential modifications to a current state. Each of the modifications is scored and the results used to inform the search. In order to facilitate these steps, other techniques are also devised, to speed up the learning process. The techniques includeThe techniques are tested by sampling data from gold standard networks and learning structures from this sampled data. These structures are analysed using various goodnessof-fit measures to see how well the algorithms perform. The measures include structural similarity metrics and Bayesian scoring metrics. The results are compared in depth against systems that also use ant colony optimisation and other methods, including evolutionary programming and greedy heuristics. Also, comparisons are made to well known state-of-the-art algorithms and a study performed on a real-life data set. The results show favourable performance compared to the other methods and on modelling the real-life data

Nonparametric Hamiltonian Monte Carlo

Probabilistic programming uses programs to express generative models whose posterior probability is then computed by built-in inference engines. A challenging goal is to develop general purpose inference algorithms that work out-of-the-box for arbitrary programs in a universal probabilistic programming language (PPL). The densities defined by such programs, which may use stochastic branching and recursion, are (in general) nonparametric, in the sense that they correspond to models on an infinite-dimensional parameter space. However standard inference algorithms, such as the Hamiltonian Monte Carlo (HMC) algorithm, target distributions with a fixed number of parameters. This paper introduces the Nonparametric Hamiltonian Monte Carlo (NP-HMC) algorithm which generalises HMC to nonparametric models. Inputs to NP-HMC are a new class of measurable functions called "tree representable", which serve as a language-independent representation of the density functions of probabilistic programs in a universal PPL. We provide a correctness proof of NP-HMC, and empirically demonstrate significant performance improvements over existing approaches on several nonparametric examples.Comment: Updated plots (after fixing minor bugs in the implementation) compared to the published version in Proceedings of the 38th International Conference on Machine Learning, PMLR 139, 2021. The conclusions of the version published at ICML 2021 are not affecte