Stochastic lambda calculus and monads of probability distributions

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.