Supervisors: Pedro H. Azevedo de Amorim and Ruben van Belle
Probabilistic programming languages (PPLs) are a useful tool for statistical modelling, and semantics of PPLs are a highly active field of research. In the literature, two main approaches to give semantics to PPLs have emerged: Semantics based on categories of linear operators and semantics based on categories of Markov kernels, where a Markov kernel is a generalisation of a stochastic matrix. Both approaches have its strengths and weaknesses, and its weaknesses are complementary. Addressing these weaknesses is, traditionally, both highly technical and subtle.
Recent advance has been achieved by Azevedo de Amorim who defines a two-level calculus nicely combining both types of PPLs: One level can be interpreted by categories of linear operators and the other one using categories of Markov kernels, while a modality mediates between both levels, which is interpreted by a lax monoidal functor. We introduce the term Banach category to refer to models of this calculus.
Azevedo de Amorim's PPL has already been applied in the literature, but the theory of Banach categories remains underdeveloped, an issue we address in this thesis. Using string diagrams, a well-known calculus to reason about monoidal categories, we investigate how the structure of Markov kernels and the lax monoidal functor interact. Azevedo de Amorim further points out that the semantics of his calculus behave even better when the lax monoidal functor is full, motivating us to prove our main theorem: Subject to certain conditions, for any Banach category, one can find another Banach category - its fullification - such that the lax monoidal functor of the fullification is full and the original Banach category and its fullification are related in a natural way.