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

Consistent Kernel Mean Estimation for Functions of Random Variables

We provide a theoretical foundation for non-parametrically estimating functions of random variables using kernel mean embeddings. We show that for any continuous function f, consistent estimators of the mean embedding of a random variable X lead to consistent estimators of the mean embedding of f(X). For Gaussian kernels and sufficiently smooth functions we also provide rates of convergence. Our results also apply for functions of multiple random variables. If the variables are dependent, we require an estimator of the mean embedding of their joint distribution as a starting point; if they are independent, it is sufficient to have separate mean embeddings of their marginal distributions. In either case, our results cover both mean embeddings expressed based on i.i.d. samples as well as reduced set expansions in terms of dependent expansion points. The latter serves as a justification for using such expansions to limit memory resources when we use the approach as a basis for probabilistic programming.Carl-Johann Simon-Gabriel is supported by a Google European Fellowship in Causal Inference

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