Mathematics / advanced
Markov chains, mixing, and sampling
Quantify convergence to equilibrium through irreducibility, coupling, reversibility, spectral gaps, and conductance.
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Inside this module
Module 01 advanced
Markov chains, mixing, and sampling
Quantify convergence to equilibrium through irreducibility, coupling, reversibility, spectral gaps, and conductance.
- 01
Transition kernels and communicating classes
Separate stochastic evolution, accessibility, closed classes and periods before asking about convergence.
28 min - 02
Recurrence, transience and hitting times
Distinguish returning almost surely from returning quickly, and solve stopped-chain equations with the correct boundary condition.
30 min - 03
Invariant measures and positive recurrence
Normalize balance measures only when their total mass is finite and connect stationary probabilities to return cycles.
30 min - 04
Detailed balance and time reversal
Reverse stationary paths with the correct support and distinguish pairwise flow symmetry from stationarity alone.
29 min - 05
Total variation and mixing times
Measure distinguishability from equilibrium with an explicit worst-case convention and compute an exact two-state mixing time.
30 min - 06
Coupling and coalescence bounds
Build joint chains with correct marginal transitions and turn persistent meetings into quantitative distributional guarantees.
31 min - 07
Spectral gaps and relaxation times
Use the weighted reversible spectrum while separating slow positive modes from nearly periodic negative modes.
33 min - 08
Conductance and Cheeger inequalities
Normalize stationary boundary flow correctly and connect bottlenecks with the ordinary reversible spectral gap.
32 min - 09
Glauber dynamics and Gibbs sampling
Construct a finite sampler from exact conditional laws and distinguish stationarity, uniqueness and rapid mixing.
32 min - 10
Cutoff phenomena and comparison techniques
Distinguish an asymptotically sharp family-level transition from mere slow mixing and transfer gap estimates through Dirichlet forms.
35 min