Using lower bounds to approximate integrals

Thomas Minka
2001 (revised 2015)

Variational lower bounds are a simple and efficient way to approximate Bayesian integrals. By bounding the integrand at every point, we obtain a bound on the integral value. Variational Bayes (Attias 1999) and convex duality methods (Jaakkola 1999a,b) are of this type, and---as shown in this paper---so is the approximation method of Cheeseman & Stutz (1996). In each case, EM is used to maximize the bound. The EM updates are derived for Gaussian mixtures and multinomial mixtures with conjugate priors. Experimental results show the lower bound method to be less accurate than Laplace's method, but often simpler.

Also see the talk "Variational bounds via reversing EM".

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