An introduction to statistical modelling semantics with higher-order measure theory
Ohad Kammar 11-16 July 2022
The last few years have seen several breakthroughs in the semantic foundations of statistical modelling. In this course, I will introduce one of these approaches — quasi-Borel spaces. We will review and develop a semantic tool-kit for measure theory using higher-order functions. The course will be accompanied by exercises and tutorials, allowing you to develop a working knowledge and hands-on experience.
(updated with corrections)
1-2 (video unavailable unfortunately)
See the Marseille course for additional content and videos.
Context and background
Doing all of these exercises is too much for the course, the goal is to learn something new, or — if you already know this material — help someone else learn something new.
set basics: introductory exercises if you’ve never worked with Borel
sets, or looking for a refresher.
spaces and functions: exercises exploring the structure of
measurable spaces and measurable functions.
category theory: use these exercises to improve your category
theorem: consequences of and related concepts to Aumann’s theorem on
the inexistence of measurable function-spaces.
examples of higher-order measure theory using sequences.
spaces: first acquaintance with quasi-Borel spaces.
construction: space combinators — you may want to spread these
exercises over several sittings.
subspaces: measurable subsets in a quasi-Borel space.
spaces: practice the definition of the function space of two qbses,
with the Borel subsets and random element spaces.
structure: use type-formers to put spaces together and form more
Standard Borel spaces (under development): use measurability-by-type to
construct standard Borel spaces.
Measures: introductory exercises if you’ve never worked with measures
Lebesgue integration: the definition of the Lebesgue integral.
Randomisable measures: the class of measures we will work with.
Random variable spaces: definition and basic properties of the Lebesgue
Geometry of random variable spaces: basic geometric and topological
properties of Lebesgue spaces.
Conditional expectation: existence and almost-certain uniqueness of
Kolmogorov’s conditional expectation.