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Publication# Variational Gaussian Inference for Bilinear Models of Count Data

Abstract

Bilinear models of count data with Poisson distribution are popular in applications such as matrix factorization for recommendation systems, modeling of receptive fields of sensory neurons, and modeling of neural-spike trains. Bayesian inference in such models remains challenging due to the product term of two Gaussian random vectors. In this paper, we propose new algorithms for such models based on variational Gaussian (VG) inference. We make two contributions. First, we show that the VG lower bound for these models, previously known to be intractable, is available in closed form under certain non-trivial constraints on the form of the posterior. Second, we show that the lower bound is biconcave and can be efficiently optimized for mean-field approximations. We also show that bi-concavity generalizes to the larger family of log-concave likelihoods, that subsume the Poisson distribution. We present new inference algorithms based on these results and demonstrate better performance on real-world problems at the cost of a modest increase in computation. Our contributions in this paper, therefore, provide more choices for Bayesian inference in terms of a speed-vs-accuracy tradeoff.

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Ontological neighbourhood

Bayesian inference

Bayesian inference (ˈbeɪziən or ˈbeɪʒən ) is a method of statistical inference in which Bayes' theorem is used to update the probability for a hypothesis as more evidence or information becomes available. Bayesian inference is an important technique in statistics, and especially in mathematical statistics. Bayesian updating is particularly important in the dynamic analysis of a sequence of data. Bayesian inference has found application in a wide range of activities, including science, engineering, philosophy, medicine, sport, and law.

Statistical inference

Statistical inference is the process of using data analysis to infer properties of an underlying distribution of probability. Inferential statistical analysis infers properties of a population, for example by testing hypotheses and deriving estimates. It is assumed that the observed data set is sampled from a larger population. Inferential statistics can be contrasted with descriptive statistics. Descriptive statistics is solely concerned with properties of the observed data, and it does not rest on the assumption that the data come from a larger population.

Inference

Inferences are steps in reasoning, moving from premises to logical consequences; etymologically, the word infer means to "carry forward". Inference is theoretically traditionally divided into deduction and induction, a distinction that in Europe dates at least to Aristotle (300s BCE). Deduction is inference deriving logical conclusions from premises known or assumed to be true, with the laws of valid inference being studied in logic. Induction is inference from particular evidence to a universal conclusion.

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