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Bayesian inference on non-Gaussian data is often non-analytic and requirescomputationally expensive approximations such as sampling or variationalinference. We propose an approximate inference framework primarily designed tobe computationally cheap while still achieving high approximation quality. Theconcept, which we call Laplace Matching, involves closed-form, approximate,bi-directional transformations between the parameter spaces of exponentialfamilies. These are constructed from Laplace approximations undercustom-designed basis transformations. The mappings can then be leveraged toeffectively turn a latent Gaussian distribution into an approximate conjugateprior to a rich class of observable variables. This allows us to train latentGaussian models such as Gaussian Processes on non-Gaussian data at nearly noadditional cost. The method can be thought of as a pre-processing step whichcan be implemented in <5 lines of code and runs in less than a second.Furthermore, Laplace Matching yields a simple way to group similar data pointstogether, e.g. to produce inducing points for GPs. We empirically evaluate themethod with experiments for four different exponential distributions, namelythe Beta, Gamma, Dirichlet and inverse Wishart, showing approximation qualitycomparable to state-of-the-art approximate inference techniques at a drasticreduction in computational cost.

Laplace Matching for fast Approximate Inference in Latent Gaussian Models