Overview.md

Inference in Probabilistic Models - Sampling Methods

Michael Habeck - Jena University Hospital - [email protected]

Wolfhart Feldmeier - Jena University Hospital - [email protected]

Dates and course organization

• Four weeks, two 2-hour lectures plus one 2-hour exercise session per week

• Lectures on Monday and Friday, exercises on Wednesday

• Timetable

Lecture	Date	Weekday	Time	Topic
1	Jan 15, 2024	Mon	10:15 - 11:45	Introduction / Direct Sampling Methods
Ex 1	Jan 17, 2024	Wed	10:15 - 11:45	Exercises for lecture 1
2	Jan 19, 2024	Fri	10:15 - 11:45	Rejection & Importance Sampling
3	Jan 22, 2024	Mon	10:15 - 11:45	Markov chains, MCMC
Ex 2	Jan 24, 2024	Wed	10:15 - 11:45	Exercises for lectures 2-3
4	Jan 26, 2024	Fri	10:15 - 11:45	The Metropolis-Hastings algorithm
5	Jan 19, 2024	Mon	10:15 - 11:45	Gibbs sampling
Ex 3	Jan 31, 2024	Wed	10:15 - 11:45	Exercises for lectures 4-5
6	Feb 02, 2024	Fri	10:15 - 11:45	Hamiltonian Monte Carlo
7	Feb 05, 2024	Mon	10:15 - 11:45	Hamiltonian Monte Calro II
Ex 4	Feb 07, 2024	Wed	10:15 - 11:45	Exercises for lectures 6-7
8	Feb 09, 2024	Fri	10:15 - 11:45	Practical aspects of HMC

Topics

Lecture 1: Introduction & Direct Sampling Methods

• Motivation
• Monte Carlo approximation
• An inefficient way of computing $\pi$
• Can we beat the curse of dimensionality?
• Random number generation
• Direct sampling by variable transformation methods

Lecture 2: Rejection and Importance Sampling

• More direct sampling methods
• Rejection sampling
• Importance sampling

Lecture 3: Markov chains

• Markov chains
• Some mathematical facts about Markov chains

Lecture 4: The Metropolis-Hastings Algorithm

• Fundamental theorem of Markov chains
• Metropolis-Hastings algorithm

Lecture 5: Gibbs sampling

• Combining Markov chains
• Gibbs sampling

Lecture 6: Hamiltonian Monte Carlo

• Auxiliary variable methods
• Hamiltonian Monte Carlo I

Lecture 7: Hamiltonian Monte Carlo

• Hamiltonian Monte Carlo II

Lecture 8: Hamiltonian Monte Carlo, Practical Issues

• Practical Issues (convergence, diagnostic checks)

Literature

• Matti Vihola: Lectures on Stochastic Simulation

• Radford Neal: Probabilistic Inference Using Markov Chain Monte Carlo Methods

• Chris Bishop: Pattern Recognition and Machine Learning, Chap. 11

• David MacKay: Information Theory, Inference, and Learning Algorithms, Chap. 29 + 30

• Iain Murray: Advances in Markov chain Monte Carlo methods

• Andrieu, de Freitas, Doucet, Jordan: An Introduction to MCMC for Machine Learning

• Charles Geyer: Introduction to Markov Chain Monte Carlo

• Jun S. Liu: Monte Carlo Strategies in Scientific Computing

• David A Levin, Yuval Peres: Markov chains and mixing times