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Bayesian inference is difficult because the likelihood involves a product of two Gaussian latent vectors. This work introduces variational Gaussian inference methods that make the variational lower bound available in closed form under non-trivial posterior constraints. The resulting bound is biconcave, enabling efficient optimization for mean-field approximations and extending to broader log-concave likelihoods beyond Poisson, delivering improved accuracy with modest computational cost.",{"@graph":69,"@context":114},[70,84,105],{"@type":71,"itemListElement":72},"BreadcrumbList",[73,77,79,82],{"item":74,"name":75,"@type":76,"position":8},"https://docshare.wps.com","Home","ListItem",{"item":78,"name":9,"@type":76,"position":14},"https://docshare.wps.com/document/",{"item":80,"name":40,"@type":76,"position":81},"https://docshare.wps.com/document/research-report/",3,{"item":83,"name":65,"@type":76,"position":19},"https://docshare.wps.com/document/variational-gaussian-inference-for-bilinear-models-of-count-data/128857/",{"url":83,"name":65,"@type":85,"image":86,"author":91,"headline":65,"publisher":94,"fileFormat":97,"inLanguage":63,"description":67,"dateModified":98,"datePublished":99,"encodingFormat":97,"isAccessibleForFree":100,"interactionStatistic":101},"DigitalDocument",{"url":87,"@type":88,"width":89,"height":90},"https://docshare.wps.com/thumbnails/variational-gaussian-inference-for-bilinear-models-of-count-data/128857.png","ImageObject",300,407,{"name":92,"@type":93},"Aria","Person",{"url":74,"name":95,"@type":96},"DocShare","Organization","application/pdf","2026-09-20","2026-08-06",true,{"@type":102,"interactionType":103,"userInteractionCount":44},"InteractionCounter",{"@type":104},"ViewAction",{"@type":106,"mainEntity":107},"FAQPage",[108],{"name":109,"@type":110,"acceptedAnswer":111},"How does the proposed method optimize inference efficiently?","Question",{"text":112,"@type":113},"It leverages the biconcavity of the lower bound to enable efficient optimization under mean-field Gaussian approximations, and it generalizes to larger log-concave likelihood families beyond Poisson.","Answer","https://schema.org",{"og:url":83,"og:type":116,"og:title":65,"og:site_name":95,"og:description":67},"article",{"robots":118,"canonical":83},"index,follow",{"doc_id":120,"site_id":62},128857,1786003942,{"code":4,"msg":5,"data":123},{"doc_id":120,"user_id":124,"nickname":92,"user_avatar":125,"doc_module":4,"category_id":39,"category_name":40,"doc_title":65,"doc_description":67,"doc_content":126,"file_id":127,"file_url":128,"file_type":129,"file_size":130,"view_count":44,"is_deleted":4,"is_public":8,"is_downloadable":8,"audit_status":8,"page_count":131,"language":132,"language_code":63,"site_id":62,"html_lang":63,"table_of_contents":133,"faqs":134,"seo_title":135,"seo_description":67,"update_tm":121,"read_time":136},2336474459895,"https://ap-avatar.wpscdn.com/avatar/22000baeef7a5ed0655?x-image-process=image/resize,m_fixed,w_180,h_180&k=1786071322749376916","CORE  Metadata, citation and similar [papers at core.ac.uk](papers at core.ac.uk)  \nProvided by Infoscience- École polytechnique fédérale de Lausanne  \nJMLR: Workshop and Conference Proceedings 39:330{343, 2014 ACML 2014  \nVariational Gaussian Inference for Bilinear Models  \nof Count Data  \nYoung-Jun Ko  \nMohammad Emtiyaz Khan  \n􀀓  \nEcole Polytechnique F􀀓ed􀀓erale de Lausanne, Switzerland  \n[youngjun.ko@epfl.ch](youngjun.ko@epfl.ch)[emtiyaz.khan@epfl.ch](emtiyaz.khan@epfl.ch)  \nEditor: Dinh Phung and Hang Li  \nAbstract  \nBilinear models of count data with Poisson distribution are popular in applications such as matrix factorization for recommendation systems, modeling of receptive 􀀌elds 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.  \nWe 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 e􀀎ciently optimized for mean-􀀌eld 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 tradeo􀀋.  \nKeywords: Variational Gaussian inference, bilinear models, Poisson likelihood, matrix factorization, latent Gaussian models  \n1. Introduction  \nLatent Gaussian factor models, such as probabilistic principal component analysis (PPCA) and factor analysis (FA), are very commonly used density models for continuous-valued data. They are extensively employed in various applications such as latent factor discovery, dimensionality reduction, missing-data imputation, and data fusion.  \nSuch latent factor models can be easily extended to handle non-Gaussian data using the generalized linear model framework (McCullagh and Nelder, 1989) where Gaussian likelihoods are replaced by other distributions (see e.g. Mohamed et al. (2008); Seeger and Bouchard (2012); Khan et al. (2010)) . In this paper, we focus on the modeling of count data using latent factor models with Poisson likelihoods. This is motivated by the fact, that counting the occurence of events is a natural mode of observing phenomena. Gathering such data leads to the problem of analysing non-negative and discrete-valued random processes, for which the use of Gaussian likelihoods for computational convenience can lead to inaccuracies, due to undesirable properties, such as symmetry and the distribution of mass over the whole real line. Therefore, latent Gaussian factor models endowed with Pois-  \n􀀍c 2014 Y.-J. Ko & M.E. Khan.  \nVG Inference for Count Data  \nson likelihoods have been successfully used in various real-world applications, examples of which are given in what follows. In neuroscience,(Park and Pillow, 2013) model the receptive 􀀌eld of a sensory neuron by a low-rank latent Gaussian 􀀌eld serving as the intensity function of an inhomogeneous Poisson process, and demonstrate, that Poisson likelihoods model neural spike counts, much more accurately than Gaussians. (Buesing et al. , 2012) analyse multi-electrode recordings of neural activity using a similar model for neural spike trains, where the latent Gaussian 􀀌eld captures temporal dependencies.. (Krishnamurthyet al. , 2010) analyse counts of newly infected individuals over space and time by describing the spatio-temporal dynamics as a latent Gaussian 􀀌eld. (Seeger and Bouchard, 2012; Zhou et al. , 2012) apply such models to the tas","cbCaihqC62aEsB4I","https://ap.wps.com/l/cbCaihqC62aEsB4I","pdf",387458,14,"English","# Abstract\n# Introduction\n## Latent Gaussian factor models and count data\n## Challenges of Bayesian inference with Poisson likelihoods\n## Variational Gaussian approach for bilinear count models","[{\"question\":\"How does the proposed method optimize inference efficiently?\",\"answer\":\"It leverages the biconcavity of the lower bound to enable efficient optimization under mean-field Gaussian approximations, and it generalizes to larger log-concave likelihood families beyond Poisson.\"}]","Variational Gaussian Inference for Bilinear Models of Count Data | PDF",35]