40 research outputs found
A Convenient Category for Higher-Order Probability Theory
Higher-order probabilistic programming languages allow programmers to write
sophisticated models in machine learning and statistics in a succinct and
structured way, but step outside the standard measure-theoretic formalization
of probability theory. Programs may use both higher-order functions and
continuous distributions, or even define a probability distribution on
functions. But standard probability theory does not handle higher-order
functions well: the category of measurable spaces is not cartesian closed.
Here we introduce quasi-Borel spaces. We show that these spaces: form a new
formalization of probability theory replacing measurable spaces; form a
cartesian closed category and so support higher-order functions; form a
well-pointed category and so support good proof principles for equational
reasoning; and support continuous probability distributions. We demonstrate the
use of quasi-Borel spaces for higher-order functions and probability by:
showing that a well-known construction of probability theory involving random
functions gains a cleaner expression; and generalizing de Finetti's theorem,
that is a crucial theorem in probability theory, to quasi-Borel spaces
The Power of Convex Algebras
Probabilistic automata (PA) combine probability and nondeterminism. They can
be given different semantics, like strong bisimilarity, convex bisimilarity, or
(more recently) distribution bisimilarity. The latter is based on the view of
PA as transformers of probability distributions, also called belief states, and
promotes distributions to first-class citizens.
We give a coalgebraic account of the latter semantics, and explain the
genesis of the belief-state transformer from a PA. To do so, we make explicit
the convex algebraic structure present in PA and identify belief-state
transformers as transition systems with state space that carries a convex
algebra. As a consequence of our abstract approach, we can give a sound proof
technique which we call bisimulation up-to convex hull.Comment: Full (extended) version of a CONCUR 2017 paper, to be submitted to
LMC
De Finetti's construction as a categorical limit
This paper reformulates a classical result in probability theory from the
1930s in modern categorical terms: de Finetti's representation theorem is
redescribed as limit statement for a chain of finite spaces in the Kleisli
category of the Giry monad. This new limit is used to identify among
exchangeable coalgebras the final one.Comment: In proceedings of CMCS 202
Probabilistic call by push value
We introduce a probabilistic extension of Levy's Call-By-Push-Value. This
extension consists simply in adding a " flipping coin " boolean closed atomic
expression. This language can be understood as a major generalization of
Scott's PCF encompassing both call-by-name and call-by-value and featuring
recursive (possibly lazy) data types. We interpret the language in the
previously introduced denotational model of probabilistic coherence spaces, a
categorical model of full classical Linear Logic, interpreting data types as
coalgebras for the resource comonad. We prove adequacy and full abstraction,
generalizing earlier results to a much more realistic and powerful programming
language
Measurable Cones and Stable, Measurable Functions
We define a notion of stable and measurable map between cones endowed with
measurability tests and show that it forms a cpo-enriched cartesian closed
category. This category gives a denotational model of an extension of PCF
supporting the main primitives of probabilistic functional programming, like
continuous and discrete probabilistic distributions, sampling, conditioning and
full recursion. We prove the soundness and adequacy of this model with respect
to a call-by-name operational semantics and give some examples of its
denotations