## Expectation propagation for exponential families (2005)

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@TECHREPORT{Seeger05expectationpropagation,

author = {Matthias Seeger},

title = {Expectation propagation for exponential families},

institution = {},

year = {2005}

}

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### Abstract

This is a tutorial describing the Expectation Propagation (EP) algorithm for a general exponential family. Our focus is on simplicity of exposition. Although the overhead of translating a specific model into its exponential family representation can be considerable, many apparent complications of EP can simply be sidestepped by working in this canonical representation. Note: This material is extracted from the Appendix of my PhD thesis (see www.kyb.tuebingen.mpg.de/bs/people/seeger/papers/thesis.html). 1 Exponential Families Definition 1 (Exponential Family) A set F of distributions with densities P (x|θ) = exp � θ T φ(x) − Φ(θ) � , θ ∈ Θ, Φ(θ) = log exp � θ T φ(x) � dµ(x) w.r.t. a base measure µ is called an exponential family. Here, θ are called natural parameters, Θ the natural parameter space, φ(x) the sufficient statistics, and Φ(θ) is the log partition function. Furthermore, η = Eθ[φ(x)] are called moment parameters, where Eθ[·]