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.
All slides (updated with corrections)
Lectures 1-2 (video unavailable unfortunately)
Lectures 3-4 (video)
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.
Borel set basics: introductory exercises if you’ve never worked with Borel sets, or looking for a refresher.
Measurable spaces and functions: exercises exploring the structure of measurable spaces and measurable functions.
Basic category theory: use these exercises to improve your category theory.
Aumann’s theorem: consequences of and related concepts to Aumann’s theorem on the inexistence of measurable function-spaces.
Sequences: examples of higher-order measure theory using sequences.
Quasi-Borel spaces: first acquaintance with quasi-Borel spaces.
Qbs construction: space combinators — you may want to spread these exercises over several sittings.
Borel subspaces: measurable subsets in a quasi-Borel space.
Function spaces: practice the definition of the function space of two qbses, with the Borel subsets and random element spaces.
Type structure: use type-formers to put spaces together and form more complex spaces.
Standard Borel spaces (under development): use measurability-by-type to construct standard Borel spaces.
Measures: introductory exercises if you’ve never worked with measures before.
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 spaces.
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.