Piecewise Bounds for Logistic-Gaussian Integrals
By Benjamin Marlin, University of British Columbia
We propose the application of a class of piecewise linear and quadratic bounds for parameter estimation in models involving intractable logistic-Gaussian integrals. We build on past work for optimal piece-wise linear bounds to the log-sum-exp function by introducing an optimization framework for minimax fitting of piecewise quadratic bounds. We show that piecewise quadratic bounds can reduce the maximum error in the bound by more than a factor of ten relative to a piecewise linear bound with the same number of pieces. Finally, we show that the uniform minimax error guarantees that our bounds possess lead to much more accurate estimation of covariance parameters in binary latent Gaussian graphical models where existing variational quadratic bounds fail. This is joint work with Emtiyaz Khan and Kevin Murphy.
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