Gibbs sampling
(45 minutes to learn)
Summary
Gibbs sampling is a Markov Chain Monte Carlo (MCMC) algorithm where each random variable is iteratively resampled from its conditional distribution given the remaining variables. It's a simple and often highly effective approach for performing posterior inference in probabilistic models.
Context
This concept has the prerequisites:
- Markov chain Monte Carlo (Gibbs sampling is an MCMC algorithm.)
- conditional distributions (Gibbs sampling is defined in terms of conditional distributions.)
- Markov random fields (The structure of a graphical model shows why the Gibbs updates can be computed efficiently.)
Core resources (read/watch one of the following)
-Free-
→ Coursera: Probabilistic Graphical Models (2013)
An online course on probabilistic graphical models.
Location:
Lecture "Gibbs sampling"
Additional dependencies:
- Bayesian networks
Other notes:
- Click on "Preview" to see the videos.
→ Bayesian Reasoning and Machine Learning
A textbook for a graudate machine learning course.
Additional dependencies:
- Bayesian networks
- multivariate Gaussian distribution
-Paid-
→ Pattern Recognition and Machine Learning
A textbook for a graduate machine learning course, with a focus on Bayesian methods.
Location:
Section 11.3, pages 542-546
Additional dependencies:
- multivariate Gaussian distribution
→ Probabilistic Graphical Models: Principles and Techniques
A very comprehensive textbook for a graduate-level course on probabilistic AI.
Location:
Sections 12.3.1 (pages 505-507) and 12.3.3 (pages 512-515)
Additional dependencies:
- Bayesian networks
Supplemental resources (the following are optional, but you may find them useful)
-Free-
→ Information Theory, Inference, and Learning Algorithms
A graudate-level textbook on machine learning and information theory.
Additional dependencies:
- Metropolis-Hastings algorithm
→ Machine learning summer school: Markov chain Monte Carlo (2009)
-Paid-
→ Machine Learning: a Probabilistic Perspective
A very comprehensive graudate-level machine learning textbook.
Location:
Section 24.2-24.2.2, pages 838-839
See also
- Gibbs sampling can be viewed as a special case of Metropolis-Hastings .
- Naive Gibbs sampling is often very slow to mix. Some improved versions include:
- block Gibbs sampling , where we sample multiple variables at a time
- collapsed Gibbs sampling , where some of the variables are integrated out in closed form
- We can analyze the mixing rate using spectral graph theory.