Coalgebraic Semantics for Probabilistic Logic Programming

Probabilistic logic programming is increasingly important in artificial intelligence and related fields as a formalism to reason about uncertainty. It generalises logic programming with the possibility of annotating clauses with probabilities. This paper proposes a coalgebraic semantics on probabilistic logic programming. Programs are modelled as coalgebras for a certain functor F, and two semantics are given in terms of cofree coalgebras. First, the F-coalgebra yields a semantics in terms of derivation trees. Second, by embedding F into another type G, as cofree G-coalgebra we obtain a `possible worlds' interpretation of programs, from which one may recover the usual distribution semantics of probabilistic logic programming. Furthermore, we show that a similar approach can be used to provide a coalgebraic semantics to weighted logic programming

Bayesian Inversion by ω-complete cone duality

The process of inverting Markov kernels relates to the important subject of Bayesian modelling and learning. In fact, Bayesian update is exactly kernel inversion. In this paper, we investigate how and when Markov kernels (aka stochastic relations, or probabilistic mappings, or simply kernels) can be inverted. We address the question both directly on the category of measurable spaces, and indirectly by interpreting kernels as Markov operators: For the direct option, we introduce a typed version of the category of Markov kernels and use the so-called 'disintegration of measures'. Here, one has to specialise to measurable spaces borne from a simple class of topological spaces -e.g. Polish spaces (other choices are possible). Our method and result greatly simplify a recent development in Ref. [4]. For the operator option, we use a cone version of the category of Markov operators (kernels seen as predicate transformers). That is to say, our linear operators are not just continuous, but are required to satisfy the stronger condition of being ω-chain-continuous. 1 Prior work shows that one obtains an adjunction in the form of a pair of contravariant and inverse functors between the categories of L 1 - and L ∞ -cones [3]. Inversion, seen through the operator prism, is just adjunction.2 No topological assumption is needed. We show that both categories (Markov kernels and ω-chain-continuous Markov operators) are related by a family of contravariant functors T p for 1 ≤ p ≤∞. The T p 's are Kleisli extensions of (duals of) conditional expectation functors introduced in Ref. [3]. With this bridge in place, we can prove that both notions of inversion agree when both defined: if f is a kernel, and f † its direct inverse, then T ∞ (f) † = T ∞ (f † )