Expectation propagation

Method to approximate a probability distribution

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 $p(\mathbf {x} )$ with a tractable distribution $q(\mathbf {x} )$ . Expectation propagation achieves this approximation by minimizing the Kullback-Leibler divergence $\mathrm {KL} (p||q)$ .^[1] Variational Bayesian methods minimize $\mathrm {KL} (q||p)$ instead.^[1]

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

Applications

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

References

^ ^a ^b ^c ^d ^e ^f ^g Bishop, Christopher (2007). Pattern Recognition and Machine Learning. New York: Springer-Verlag New York Inc. ISBN 978-0387310732.

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)