Stochastic lambda calculus and monads of probability distributions

Abstract

Probability distributions are useful for expressing the meanings of probabilistic languages, which support formal modeling of and reasoning about uncertainty. Probability distributions form a monad, and the monadic definition leads to a simple, natural semantics for a stochastic lambda calculus, as well as simple, clean implementations of common queries. But the monadic implementation of the expectation query can be much less efficient than current best practices in probabilistic modeling. We therefore present a language of measure terms, which can not only denote discrete probability distributions but can also support the best known modeling techniques. We give a translation of stochastic lambda calculus into measure terms. Whether one translates into the probability monad or into measure terms, the results of the translations denote the same probability distribution.

Probabilistic modelling, inference and learning using logical theories

“…These come in the form of language extensions to the λ-calculus. In [51], which is closely related to IBAL, the authors show how probability distributions can be captured using monads. They also introduced a simple language called measure terms that allows certain operations on (discrete) distributions to be computed efficiently.…”

Modeling Genome Evolution with a DSEL for Probabilistic Programming

“…A monadic probability implementation is demonstrated by [14]. The authors show how probability distributions can be constructed using transitions similar to our own.…”

“…The observation that probability distributions form a monad is not new [6]. However, previous work was mainly concerned with extending languages by offering probabilistic expressions as primitives and defining suitable semantics [7,10,14,12]. Consider the case where we take a sum, roll a die and add its value to the sum.…”

Probabilistic -calculus and Quantitative Program Analysis

“…Other works have introduced probabilistic features into the λ-calculus, mainly motivated by the design and implementation of probabilistic languages (see [22] and [21] for recent examples). In this work, we aim to introduce a probabilistic semantics for the λ-calculus by extending the classical theory with a notion of probabilistic term.…”