Embedded Probabilistic Programming

We propose to represent a probability distribution as a program in a general-purpose programming language rather than a special language built from scratch. This approach makes it easier for the probabilistic-reasoning and programming-language communities to share their work. To demonstrate that this representation is simple and efficient, we implement inference by variable elimination and importance sampling using the concepts of reflection, memoization, and continuations. The embedded approach to probabilistic programming is to build a library that extends a generalpurpose host language with probabilistic constructs. Seen from within the host language, these embedded constructs are ordinary functions with side effects. For example, the two most important constructs are dist, a function that maps a list of probability-value pairs to a randomly chosen value, and fail, a function that takes no input and never returns because it observes an impossible event. The following program, written in the general-purpose programming language OCaml, expresses the distribution over a pair of fair coin flips, conditional on at least one of them being true. let coin_flip () = dist [ (0.5, V true); (0.5, V false) ] in let x = coin_flip () in let y = coin_flip () in if x || y then (x,y) else fail () (1) The variables x and y above are host-language variables that represent random variables. The embedded approach has several advantages over building a probabilistic language from scratch. • It is easy to learn and implement, because host-language features are reused such as compilers, I/O libraries, database interfaces, timing facilities, data structures, and type checking. • It can compile stochastic models to machine code. Deterministic parts of the models run as fast as in the host language, without the overhead imposed by non-embedded interpreters. • It treats probability distributions as an abstract container data-type. This treatment makes it natural to describe distributions over distributions, which are useful for modeling multiagent interactions with imperfect information. For the library not to just perform naı̈ve rejection sampling, the host language needs certain features other than random-number generation [2]. We use the delimited control operators shift and reset in the host language OCaml [5]. Embedded probabilistic programming essentially builds a library of weighted nondeterminism that turns the host-language compiler into a probabilistic-language compiler, so any improvement in the host-language implementation results in more efficient inference. To illustrate how inference algorithms correspond to library implementation strategies and host-language concepts, we present two implementations: one performing exact inference by enumeration and variable elimination [1], and one performing approximate inference by importance sampling and evidence pushing [6]. The implementations are short (totaling 200 lines), expressive, and available online along with tests.