Expectation propagation

Expectation propagation (EP) is a technique in Bayesian machine learning.^[1]

EP finds approximations to a probability distribution.^[1] It uses an iterative approach that uses the factorization structure of the target distribution.^[1] It differs from other Bayesian approximation approaches such as variational Bayesian methods.^[1]

More specifically, suppose we wish to approximate an intractable probability distribution ${\displaystyle p(\mathbf {x} )}$ with a tractable distribution ${\displaystyle q(\mathbf {x} )}$. Expectation propagation achieves this approximation by minimizing the Kullback–Leibler divergence ${\displaystyle \mathrm {KL} (p||q)}$.^[1] Variational Bayesian methods minimize ${\displaystyle \mathrm {KL} (q||p)}$ instead.^[1]

If ${\displaystyle q(\mathbf {x} )}$ is a Gaussian ${\displaystyle {\mathcal {N}}(\mathbf {x} |\mu ,\Sigma )}$, then ${\displaystyle \mathrm {KL} (p||q)}$ is minimized with ${\displaystyle \mu }$ and ${\displaystyle \Sigma }$ being equal to the mean of ${\displaystyle p(\mathbf {x} )}$ and the covariance of ${\displaystyle p(\mathbf {x} )}$, respectively; this is called moment matching.^[1]

Expectation propagation via moment matching plays a vital role in approximation for indicator functions that appear when deriving the message passing equations for TrueSkill.

• Thomas Minka (August 2–5, 2001). "Expectation Propagation for Approximate Bayesian Inference". In Jack S. Breese, Daphne Koller (ed.). UAI '01: Proceedings of the 17th Conference in Uncertainty in Artificial Intelligence (PDF). University of Washington, Seattle, Washington, USA. pp. 362–369.{{cite book}}: CS1 maint: location missing publisher (link)

• Minka's EP papers
• List of papers using EP

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