Formally justified and modular Bayesian inference for probabilistic programs

Probabilistic modelling offers a simple and coherent framework to describe the real world in the face of uncertainty. Furthermore, by applying Bayes' rule it is possible to use probabilistic models to make inferences about the state of the world from partial observations. While traditionally probabilistic models were constructed on paper, more recently the approach of probabilistic programming enables users to write the models in executable languages resembling computer programs and to freely mix them with deterministic code. It has long been recognised that the semantics of programming languages is complicated and the intuitive understanding that programmers have is often inaccurate, resulting in difficult to understand bugs and unexpected program behaviours. Programming languages are therefore studied in a rigorous way using formal languages with mathematically defined semantics. Traditionally formal semantics of probabilistic programs are defined using exact inference results, but in practice exact Bayesian inference is not tractable and approximate methods are used instead, posing a question of how the results of these algorithms relate to the exact results. Correctness of such approximate methods is usually argued somewhat less rigorously, without reference to a formal semantics. In this dissertation we formally develop denotational semantics for probabilistic programs that correspond to popular sampling algorithms often used in practice. The semantics is defined for an expressive typed lambda calculus with higher-order functions and inductive types, extended with probabilistic effects for sampling and conditioning, allowing continuous distributions and unbounded likelihoods. It makes crucial use of the recently developed formalism of quasi-Borel spaces to bring all these elements together. We provide semantics corresponding to several variants of Markov chain Monte Carlo and Sequential Monte Carlo methods and formally prove a notion of correctness for these algorithms in the context of probabilistic programming. We also show that the semantic construction can be directly mapped to an implementation using established functional programming abstractions called monad transformers. We develop a compact Haskell library for probabilistic programming closely corresponding to the semantic construction, giving users a high level of assurance in the correctness of the implementation. We also demonstrate on a collection of benchmarks that the library offers performance competitive with existing systems of similar scope. An important property of our construction, both the semantics and the implementation, is the high degree of modularity it offers. All the inference algorithms are constructed by combining small building blocks in a setup where the type system ensures correctness of compositions. We show that with basic building blocks corresponding to vanilla Metropolis-Hastings and Sequential Monte Carlo we can implement more advanced algorithms known in the literature, such as Resample-Move Sequential Monte Carlo, Particle Marginal Metropolis-Hastings, and Sequential Monte Carlo squared. These implementations are very concise, reducing the effort required to produce them and the scope for bugs. On top of that, our modular construction enables in some cases deterministic testing of randomised inference algorithms, further increasing reliability of the implementation.Engineering and Physical Sciences Research Council, Cambridge Trust, Cambridge-Tuebingen programm

Automatic Alignment in Higher-Order Probabilistic Programming Languages

Probabilistic Programming Languages (PPLs) allow users to encode statistical inference problems and automatically apply an inference algorithm to solve them. Popular inference algorithms for PPLs, such as sequential Monte Carlo (SMC) and Markov chain Monte Carlo (MCMC), are built around checkpoints -- relevant events for the inference algorithm during the execution of a probabilistic program. Deciding the location of checkpoints is, in current PPLs, not done optimally. To solve this problem, we present a static analysis technique that automatically determines checkpoints in programs, relieving PPL users of this task. The analysis identifies a set of checkpoints that execute in the same order in every program run -- they are aligned. We formalize alignment, prove the correctness of the analysis, and implement the analysis as part of the higher-order functional PPL Miking CorePPL. By utilizing the alignment analysis, we design two novel inference algorithm variants: aligned SMC and aligned lightweight MCMC. We show, through real-world experiments, that they significantly improve inference execution time and accuracy compared to standard PPL versions of SMC and MCMC

A Compilation Target for Probabilistic Programming Languages

Forward inference techniques such as sequential Monte Carlo and particle Markov chain Monte Carlo for probabilistic programming can be implemented in any programming language by creative use of standardized operating system functionality including processes, forking, mutexes, and shared memory. Exploiting this we have defined, developed, and tested a probabilistic programming language intermediate representation language we call probabilistic C, which itself can be compiled to machine code by standard compilers and linked to operating system libraries yielding an efficient, scalable, portable probabilistic programming compilation target. This opens up a new hardware and systems research path for optimizing probabilistic programming systems.Comment: In Proceedings of the 31st International Conference on Machine Learning (ICML), 201

A New Approach to Probabilistic Programming Inference

We introduce and demonstrate a new approach to inference in expressive probabilistic programming languages based on particle Markov chain Monte Carlo. Our approach is simple to implement and easy to parallelize. It applies to Turing-complete probabilistic programming languages and supports accurate inference in models that make use of complex control flow, including stochastic recursion. It also includes primitives from Bayesian nonparametric statistics. Our experiments show that this approach can be more efficient than previously introduced single-site Metropolis-Hastings methods.Comment: Updated version of the 2014 AISTATS paper (to reflect changes in new language syntax). 10 pages, 3 figures. Proceedings of the Seventeenth International Conference on Artificial Intelligence and Statistics, JMLR Workshop and Conference Proceedings, Vol 33, 201

Practical probabilistic programming with monads

The machine learning community has recently shown a lot of interest in practical probabilistic programming systems that target the problem of Bayesian inference. Such systems come in different forms, but they all express probabilistic models as computational processes using syntax resembling programming languages. In the functional programming community monads are known to offer a convenient and elegant abstraction for programming with probability distributions, but their use is often limited to very simple inference problems. We show that it is possible to use the monad abstraction to construct probabilistic models for machine learning, while still offering good performance of inference in challenging models. We use a GADT as an underlying representation of a probability distribution and apply Sequential Monte Carlo-based methods to achieve efficient inference. We define a formal semantics via measure theory. We demonstrate a clean and elegant implementation that achieves performance comparable with Anglican, a state-of-the-art probabilistic programming system.The first author is supported by EPSRC and the Cambridge Trust.This is the author accepted manuscript. The final version is available from ACM via http://dx.doi.org/10.1145/2804302.280431

Functional Programming for Modular Bayesian Inference

We present an architectural design of a library for Bayesian modelling and inference in modern functional programming languages. The novel aspect of our approach are modular implementations of existing state-of- the-art inference algorithms. Our design relies on three inherently functional features: higher-order functions, inductive data-types, and support for either type-classes or an expressive module system. We provide a perfor- mant Haskell implementation of this architecture, demonstrating that high-level and modular probabilistic programming can be added as a library in sufficiently expressive languages. We review the core abstractions in this architecture: inference representations, inference transformations, and inference representation transformers. We then implement concrete instances of these abstractions, counterparts to particle filters and Metropolis-Hastings samplers, which form the basic building blocks of our library. By composing these building blocks we obtain state-of-the-art inference algorithms: Resample-Move Sequential Monte Carlo, Particle Marginal Metropolis-Hastings, and Sequential Monte Carlo Squared. We evaluate our implementation against existing probabilistic programming systems and find it is already com- petitively performant, although we conjecture that existing functional programming optimisation techniques could reduce the overhead associated with the abstractions we use. We show that our modular design enables deterministic testing of inherently stochastic Monte Carlo algorithms. Finally, we demonstrate using OCaml that an expressive module system can also implement our design