This paper introduces a novel type theory and logic for probabilistic reasoning. Its logic is quantitative, with fuzzy predicates. It includes normalisation and conditioning of states. This conditioning uses a key aspect that distinguishes our probabilistic type theory from quantum type theory, namely the bijective correspondence between predicates and side-effect free actions (called instrument, or assert, maps). The paper shows how suitable computation rules can be derived from this predicate-action correspondence, and uses these rules for calculating conditional probabilities in two well-known examples of Bayesian reasoning in (graphical) models. Our type theory may thus form the basis for a mechanisation of Bayesian inference.  ACM Subject Classification A probabilistic program is understood (semantically) as a stochastic process. A key feature of probabilistic programs as studied in the 1980s and 1990s is the presence of probabilistic choice, for instance in the form of a weighted sum x + r y, where the number r ∈ [0, 1] determines the ratio of the contributions of x and y to the result. This can be expressed explicitly as a convex sum r · x + (1 − r) · y. Some of the relevant sources are [12, 13], and [11], and [15], and also [17] for the combination of probability and non-determinism. In the language of category theory, a probabilistic program is a map in the Kleisli category of the distribution monad D (in the discrete case) or of the Giry monad G (in the continuous case). In recent years, with the establishement of Bayesian machine learning as an important area of computer science, the meaning of probabilistic programming shifted towards conditional inference. The key feature is no longer probabilistic choice, but normalisation of distributions (states), see e.g. [3] . Interestingly, this can be done in basically the same underlying models, where a program still produces a distribution -discrete or continuous -over its output. This paper contributes to this latest line of work by formulating a novel type theory for probabilistic and Bayesian reasoning. We list the key features of our type theory.
Another related work is the one by Adams and Jacobs REF proposing a type theory for Bayesian inference.
A Type Theory for Probabilistic and Bayesian Reasoning
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