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We describe a mathematical structure that can give extensional denotational semantics to higher-order probabilistic programs. It is not limited to discrete probabilities, and it is compatible with integration in a way the models that have been proposed before are not. It is organised as a model of propositional linear logic in which all the connectives have intuitive probabilistic interpretations. In addition, it has least fixed points for all maps, so it can interpret recursion.
Higher-order recursion schemes are recursive equations defining new operations from given ones called terminals. Every such recursion scheme is proved to have a least interpreted semantics in every Scotts model of lambda-calculus in which the termina
In this paper we provide two new semantics for proofs in the constructive modal logics CK and CD. The first semantics is given by extending the syntax of combinatorial proofs for propositional intuitionistic logic, in which proofs are factorised in a
We present a modular semantic account of Bayesian inference algorithms for probabilistic programming languages, as used in data science and machine learning. Sophisticated inference algorithms are often explained in terms of composition of smaller pa
We develop a denotational semantics of Linear Logic with least and greatest fixed points in coherence spaces (where both fixed points are interpreted in the same way) and in coherence spaces with totality (where they have different interpretations).
We study weakest precondition reasoning about the (co)variance of outcomes and the variance of run-times of probabilistic programs with conditioning. For outcomes, we show that approximating (co)variances is computationally more difficult than approx