# Analysis of Probabilistic Systems

Lecture 1: Analysis of Probabilistic Systems I
Lecture 2: Analysis of Probabilistic Systems II
Lecture 3: Analysis of Probabilistic Systems III
Lecture 4: Analysis of Probabilistic Systems IV
Lecture 5: Analysis of Probabilistic Systems V

This series of talks is part of the Logical Structures in Computation Boot Camp. Videos for each talk area will be available through the links above.

Lecture 1: Logic, Probability and Semantics: Introduction and Motivation
Conditional probability as the analogue of logical inference, probabilistic programs as distribution transformers, overview of developments in programming languages for machine learning, probabilistic $\lambda$-calculi and probabilistic bisimulation, the need for measure theory.

Lecture 2: Background in Measure Theory and Integration
$\Sigma$-algebras, measures, extension theorems, Lebesgue integration, the Radon-Nikodym theorem, conditional expectation on continuous spaces, disintegration.

Lecture 3: The Lawvere-Giry Monad and Probabilistic Relations
The category of measure spaces, probabilistic mappings, the Lawvere-Giry monad, its Kleisli category, Markov kernels, Kozen's semantics for a language with while loops.

Lecture 4: Markov Processes, Bisimulation, and Logical Characterization
Probabilistic bisimulation, the logical characterization of bisimulation; this will include some descriptive set theory: analytic spaces, the unique structure theorem and smooth equivalence relations.

Lecture 5: Metrics for Markov Processes
Metrics between Markov processes, the Kantorovich metric aka the Wasserstein metric, lifting the metric from distributions to processes; metrics from a real-valued modal logic.