4 research outputs found
Borel Kernels and their Approximation, Categorically
This paper introduces a categorical framework to study the exact and
approximate semantics of probabilistic programs. We construct a dagger
symmetric monoidal category of Borel kernels where the dagger-structure is
given by Bayesian inversion. We show functorial bridges between this category
and categories of Banach lattices which formalize the move from kernel-based
semantics to predicate transformer (backward) or state transformer (forward)
semantics. These bridges are related by natural transformations, and we show in
particular that the Radon-Nikodym and Riesz representation theorems - two
pillars of probability theory - define natural transformations.
With the mathematical infrastructure in place, we present a generic and
endogenous approach to approximating kernels on standard Borel spaces which
exploits the involutive structure of our category of kernels. The approximation
can be formulated in several equivalent ways by using the functorial bridges
and natural transformations described above. Finally, we show that for sensible
discretization schemes, every Borel kernel can be approximated by kernels on
finite spaces, and that these approximations converge for a natural choice of
topology.
We illustrate the theory by showing two examples of how approximation can
effectively be used in practice: Bayesian inference and the Kleene star
operation of ProbNetKAT.Comment: 17 pages, 4 figure
Joint Distributions in Probabilistic Semantics
Various categories have been proposed as targets for the denotational
semantics of higher-order probabilistic programming languages. One such
proposal involves joint probability distributions (couplings) used in Bayesian
statistical models with conditioning. In previous treatments, composition of
joint measures was performed by disintegrating to obtain Markov kernels,
composing the kernels, then reintegrating to obtain a joint measure.
Disintegrations exist only under certain restrictions on the underlying spaces.
In this paper we propose a category whose morphisms are joint finite measures
in which composition is defined without reference to disintegration, allowing
its application to a broader class of spaces. The category is symmetric
monoidal with a pleasing symmetry in which the dagger structure is a simple
transpose.Comment: 14 pages, MFPS 202
Reversing information flow: retrodiction in semicartesian categories
In statistical inference, retrodiction is the act of inferring potential
causes in the past based on knowledge of the effects in the present and the
dynamics leading to the present. Retrodiction is applicable even when the
dynamics is not reversible, and it agrees with the reverse dynamics when it
exists, so that retrodiction may be viewed as an extension of inversion, i.e.,
time-reversal. Recently, an axiomatic definition of retrodiction has been made
in a way that is applicable to both classical and quantum probability using
ideas from category theory. Almost simultaneously, a framework for information
flow in in terms of semicartesian categories has been proposed in the setting
of categorical probability theory. Here, we formulate a general definition of
retrodiction to add to the information flow axioms in semicartesian categories,
thus providing an abstract framework for retrodiction beyond classical and
quantum probability theory. More precisely, we extend Bayesian inference, and
more generally Jeffrey's probability kinematics, to arbitrary semicartesian
categories.Comment: 20.5 pages + references, some diagram