Variational Inference: A Review for Statisticians

References

• Ahmed, A. , Aly, M. , Gonzalez, J. , Narayanamurthy, S. , and Smola, A. (2012), “Scalable Inference in Latent Variable Models,” in International Conference on Web Search and Data Mining , pp. 123–132.
   Google Scholar
• Airoldi, E. , Blei, D. , Fienberg, S. , and Xing, E. (2008), “Mixed Membership Stochastic Blockmodels,” Journal of Machine Learning Research , 9, 1981–2014.
   PubMed Web of Science ®Google Scholar
• Amari, S. (1982), “Differential Geometry of Curved Exponential Families-Curvatures and Information Loss,” The Annals of Statistics , 10, 357–385.
   Web of Science ®Google Scholar
• ——— (1998), “Natural Gradient Works Efficiently in Learning,” Neural Computation , 10, 251–276.
   Web of Science ®Google Scholar
• Archambeau, C. , Cornford, D. , Opper, M. , and Shawe-Taylor, J. (2007a), “Gaussian Process Approximations of Stochastic Differential Equations,” Workshop on Gaussian Processes in Practice , 1, 1–16.
   Google Scholar
• Archambeau, C. , Opper, M. , Shen, Y. , Cornford, D. , and Shawe-Taylor, J. (2007b), “Variational Inference for Diffusion Processes,” in Neural Information Processing Systems , pp. 17–24.
   Google Scholar
• Armagan, A. , Clyde, M. , and Dunson, D. (2011), “Generalized Beta Mixtures of Gaussians,” in Neural Information Processing Systems , pp. 523–531.
   Google Scholar
• Armagan, A. , and Dunson, D. (2011), “Sparse Variational Analysis of Linear Mixed Models for Large Data Sets,” Statistics & Probability Letters , 81, 1056–1062.
   PubMed Web of Science ®Google Scholar
• Barber, D. (2012), Bayesian Reasoning and Machine Learning , Cambridge, UK : Cambridge University Press.
   Google Scholar
• Barber, D. , and Bishop, C. M. (1998), “Ensemble Learning in Bayesian Neural Networks,” in Generalization in Neural Networks and Machine Learning , ed. C. M. Bishop, New York : Springer Verlag, pp. 215–237.
   Google Scholar
• Barber, D. , and Chiappa, S. (2006), “Unified Inference for Variational Bayesian Linear Gaussian State-Space Models,” in Neural Information Processing Systems , pp. 81–88.
   Google Scholar
• Barber, D. , and de van Laar, P. (1999), “Variational Cumulant Expansions for Intractable Distributions,” Journal of Artificial Intelligence Research , 10, 435–455.
   Web of Science ®Google Scholar
• Barber, D. , and Wiegerinck, W. (1999), “Tractable Variational Structures for Approximating Graphical Models,” in Neural Information Processing Systems , pp. 183–189.
   Google Scholar
• Beal, M. , and Ghahramani, Z. (2003), “The Variational Bayesian EM Algorithm for Incomplete Data: With Application to Scoring Graphical Model Structures,” in Bayesian Statistics (Vol. 7), eds. J. Bernardo , M. Bayarri , J. Berger , A. Dawid , D. Heckerman , A. Smith , and M. West , Oxford, UK : Oxford University Press, pp. 453–464.
   Google Scholar
• Bernardo, J. , and Smith, A. (1994), Bayesian Theory , Chichester, UK : Wiley.
   Google Scholar
• Bickel, P. , Choi, D. , Chang, X. , and Zhang, H. (2013), “Asymptotic Normality of Maximum Likelihood and its Variational Approximation for Stochastic Blockmodels,” The Annals of Statistics , 41, 1922–1943.
   Web of Science ®Google Scholar
• Bishop, C. (2006), Pattern Recognition and Machine Learning , New York : Springer.
   Google Scholar
• Bishop, C. , Lawrence, N. , Jaakkola, T. , and Jordan, M. I. (1998), “Approximating Posterior Distributions in Belief Networks using Mixtures,” in Neural Information Processing Systems , pp. 416–422.
   Google Scholar
• Bishop, C. , and Winn, J. (2000), “Non-linear Bayesian Image Modelling,” in European Conference on Computer Vision , pp. 3–17.
   Google Scholar
• Blei, D. (2012), “Probabilistic Topic Models,” Communications of the ACM , 55, 77–84.
   Web of Science ®Google Scholar
• Blei, D. , and Jordan, M. I. (2006), “Variational Inference for Dirichlet Process Mixtures,” Journal of Bayesian Analysis , 1, 121–144.
   Web of Science ®Google Scholar
• Blei, D. , and Lafferty, J. (2007), “A Correlated Topic Model of Science,” Annals of Applied Statistics , 1, 17–35.
   Web of Science ®Google Scholar
• Blei, D. , Ng, A. , and Jordan, M. I. (2003), “Latent Dirichlet Allocation,” Journal of Machine Learning Research , 3, 993–1022.
   Web of Science ®Google Scholar
• Braun, M. , and McAuliffe, J. (2010), “Variational Inference for Large-Scale Models of Discrete Choice,” Journal of the American Statistical Association , 105, 324–335.
   Web of Science ®Google Scholar
• Brown, L. (1986), Fundamentals of Statistical Exponential Families , Hayward, CA : Institute of Mathematical Statistics.
   Google Scholar
• Bugbee, B. , Breidt, F. , and van der Woerd, M. (2016), “Laplace Variational Approximation for Semiparametric Regression in the Presence of Heteroscedastic Errors,” Journal of Computational and Graphical Statistics , 25, 225–245.
   PubMed Web of Science ®Google Scholar
• Carbonetto, P. , and Stephens, M. (2012), “Scalable Variational Inference for Bayesian Variable Selection in Regression, and its Accuracy in Genetic Association Studies,” Bayesian Analysis , 7, 73–108.
   Web of Science ®Google Scholar
• Celisse, A. , Daudin, J.-J. , and Pierre, L. (2012), “Consistency of Maximum-Likelihood and Variational Estimators in the Stochastic Block Model,” Electronic Journal of Statistics , 6, 1847–1899.
   Web of Science ®Google Scholar
• Challis, E. , and Barber, D. (2013), “Gaussian Kullback-Leibler Approximate Inference,” The Journal of Machine Learning Research , 14, 2239–2286.
   Web of Science ®Google Scholar
• Chan, A. , and Vasconcelos, N. (2009), “Layered Dynamic Textures,” IEEE Transactions on Pattern Analysis and Machine Intelligence , 31, 1862–1879.
   PubMed Web of Science ®Google Scholar
• Cohen, S. , and Smith, N. (2010), “Covariance in Unsupervised Learning of Probabilistic Grammars,” The Journal of Machine Learning Research , 11, 3017–3051.
   Web of Science ®Google Scholar
• Cummins, M. , and Newman, P. (2008), “FAB-MAP: Probabilistic Localization and Mapping in the Space of Appearance,” The International Journal of Robotics Research , 27, 647–665.
   Web of Science ®Google Scholar
• de Freitas, N. D. , Højen-Sørensen, P. , Jordan, M. , and Russell, S. (2001), “Variational MCMC,” in Uncertainty in Artificial Intelligence , pp. 120–127.
   Google Scholar
• Damianou, A. , Titsias, M. , and Lawrence, N. (2011), “Variational Gaussian Process Dynamical Systems,” in Neural Information Processing Systems , pp. 2510–2518.
   Google Scholar
• Daunizeau, J. , Adam, V. , and Rigoux, L. (2014), “VBA: A Probabilistic Treatment of Nonlinear Models for Neurobiological and Behavioural Data,” PLoS Computational Biology , 10, e1003441.
   PubMed Web of Science ®Google Scholar
• Dempster, A. , Laird, N. , and Rubin, D. (1977), “Maximum Likelihood from Incomplete Data via the EM Algorithm,” Journal of the Royal Statistical Society , Series B, 39, 1–38.
   Google Scholar
• Deng, L. (2004), “Switching Dynamic System Models for Speech Articulation and Acoustics,” in Mathematical Foundations of Speech and Language Processing, eds. M. Johnson, S. P. Khudanpur, M. Ostendorf, and R. Rosenfeld, New York : Springer, pp. 115–133.
   Google Scholar
• Diaconis, P. , and Ylvisaker, D. (1979), “Conjugate Priors for Exponential Families,” The Annals of Statistics , 7, 269–281.
   Web of Science ®Google Scholar
• Du, L. , Lu, R. , Carin, L. , and Dunson, D. (2009), “A Bayesian Model for Simultaneous Image Clustering, Annotation and Object Segmentation,” in Neural Information Processing Systems , pp. 486–494.
   Google Scholar
• Ermis, B. , and Bouchard, G. (2014), “Iterative Splits of Quadratic Bounds for Scalable Binary Tensor Factorization,” in Uncertainty in Artificial Intelligence , pp. 192–199.
   Google Scholar
• Erosheva, E. A. , Fienberg, S. E. , and Joutard, C. (2007), “Describing Disability through Individual-Level Mixture Models for Multivariate Binary Data,” The Annals of Applied Statistics , 1, 346–384.
   PubMed Web of Science ®Google Scholar
• Flandin, G. , and Penny, W. (2007), “Bayesian fMRI Data Analysis with Sparse Spatial Basis Function Priors,” NeuroImage , 34, 1108–1125.
   PubMed Web of Science ®Google Scholar
• Foti, N. , Xu, J. , Laird, D. , and Fox, E. (2014), “Stochastic Variational Inference for Hidden Markov Models,” in Neural Information Processing Systems , pp. 3599–3607.
   Google Scholar
• Furmston, T. , and Barber, D. (2010), “Variational Methods for Reinforcement Learning,” Artificial Intelligence and Statistics , 9, 241–248.
   Google Scholar
• Gelfand, A. , and Smith, A. (1990), “Sampling Based Approaches to Calculating Marginal Densities,” Journal of the American Statistical Association , 85, 398–409.
   Web of Science ®Google Scholar
• Geman, S. , and Geman, D. (1984), “Stochastic Relaxation, Gibbs Distributions and the Bayesian Restoration of Images,” IEEE Transactions on Pattern Analysis and Machine Intelligence , 6, 721–741.
   PubMed Web of Science ®Google Scholar
• Gershman, S. J. , Blei, D. M. , Norman, K. A. , and Sederberg, P. B. (2014), “Decomposing Spatiotemporal Brain Patterns into Topographic Latent Sources,” NeuroImage , 98, 91–102.
   PubMed Web of Science ®Google Scholar
• Ghahramani, Z. , and Jordan, M. I. (1997), “Factorial Hidden Markov Models,” Machine Learning , 29, 245–273.
   Web of Science ®Google Scholar
• Giordano, R. J. , Broderick, T. , and Jordan, M. I. (2015), “Linear Response Methods for Accurate Covariance Estimates from Mean Field Variational Bayes,” in Neural Information Processing Systems , pp. 1441–1449.
   Google Scholar
• Grimmer, J. (2011), “An introduction to Bayesian Inference via Variational Approximations,” Political Analysis , 19, 32–47.
   Web of Science ®Google Scholar
• Hall, P. , Ormerod, J. , and Wand, M. (2011a), “Theory of Gaussian Variational Approximation for a Poisson Mixed Model,” Statistica Sinica , 21, 369–389.
   Web of Science ®Google Scholar
• Hall, P. , Pham, T. , Wand, M. , and Wang, S. (2011b), “Asymptotic Normality and Valid Inference for Gaussian Variational Approximation,” Annals of Statistics , 39, 2502–2532.
   Web of Science ®Google Scholar
• Harrison, L. , and Green, G. (2010), “A Bayesian Spatiotemporal Model for very Large Data Sets,” Neuroimage , 50, 1126–1141.
   PubMed Web of Science ®Google Scholar
• Hastings, W. (1970), “Monte Carlo Sampling Methods using Markov Chains and their Applications,” Biometrika , 57, 97–109.
   Web of Science ®Google Scholar
• Hensman, J. , Fusi, N. , and Lawrence, N. (2013), “Gaussian Processes for Big Data,” in Proceedings of the Twenty-Ninth Conference on Uncertainty in Artificial Intelligence , Corvallis, OR : AUAI Press, pp. 282–290.
   Google Scholar
• Hensman, J. , Rattray, M. , and Lawrence, N. (2012), “Fast Variational Inference in the Conjugate Exponential Family,” in Neural Information Processing Systems , pp. 2888–2896.
   Google Scholar
• Hinton, G. , and Van Camp, D. (1993), “Keeping the Neural Networks Simple by Minimizing the Description Length of the Weights,” in Computational Learning Theory , pp. 5–13.
   Google Scholar
• Hoffman, M. , Blei, D. , and Mimno, D. M. (2012), “Sparse Stochastic Inference for Latent Dirichlet Allocation,” in Proceedings of the 29th International Conference on Machine Learning (ICML-12), eds. J. Langford and J. Pineau, New York: ACM, pp. 1599–1606.
   Google Scholar
• Hoffman, M. D. , Blei, D. , Wang, C. , and Paisley, J. (2013), “Stochastic Variational Inference,” Journal of Machine Learning Research , 14, 1303–1347.
   Web of Science ®Google Scholar
• Hoffman, M. D. , and Blei, D. M. (2015), “Structured Stochastic Variational Inference,” in Proceedings of the Eighteenth International Conference on Artificial Intelligence and Statistics (Vol. 38), eds. G. Lebanon and S. V. N. Vishwanathan, San Diego, CA: Proceedings of Machine Learning Research, pp. 361–369.
   Google Scholar
• Hoffman, M. D. , and Gelman, A. (2014), “The No-U-turn Sampler: Adaptively Setting Path Lengths in Hamiltonian Monte Carlo,” The Journal of Machine Learning Research , 15, 1593–1623.
   Web of Science ®Google Scholar
• Honkela, A. , Tornio, M. , Raiko, T. , and Karhunen, J. (2008), “Natural Conjugate Gradient in Variational Inference,” in Neural Information Processing, eds. J. C. Platt, D. Koller, Y. Singer, and S. T. Roweis, New York : Springer, pp. 305–314.
   Google Scholar
• Jaakkola, T. , and Jordan, M. I. (1996), “Computing Upper and Lower Bounds on Likelihoods in Intractable Networks,” in Uncertainty in Artificial Intelligence , pp. 340–348.
   Google Scholar
• ——— (1997), “A Variational Approach to Bayesian Logistic Regression Models and their Extensions,” in Artificial Intelligence and Statistics , pp. 1–12.
   Google Scholar
• ——— (2000), “Bayesian Parameter Estimation via Variational Methods,” Statistics and Computing , 10, 25–37.
   Web of Science ®Google Scholar
• Ji, C. , Shen, H. , and West, M. (2010), “Bounded Approximations for Marginal Likelihoods,” Technical Report, Duke University.
   Google Scholar
• Johnson, M. , and Willsky, A. (2014), “Stochastic Variational Inference for Bayesian Time Series Models,” in International Conference on Machine Learning , pp. 1854–1862.
   Google Scholar
• Jojic, N. , and Frey, B. (2001), “Learning Flexible Sprites in Video Layers,” in Computer Vision and Pattern Recognition , pp. 1–8.
   Google Scholar
• Jojic, V. , Jojic, N. , Meek, C. , Geiger, D. , Siepel, A. , Haussler, D. , and Heckerman, D. (2004), “Efficient Approximations for Learning Phylogenetic HMM Models from Data,” Bioinformatics , 20, 161–168.
   PubMed Web of Science ®Google Scholar
• Jordan, M. I. , Ghahramani, Z. , Jaakkola, T. , and Saul, L. (1999), “Introduction to Variational Methods for Graphical Models,” Machine Learning , 37, 183–233.
   Web of Science ®Google Scholar
• Khan, M. E. , Bouchard, G. , Murphy, K. P. , and Marlin, B. M. (2010), “Variational Bounds for Mixed-Data Factor Analysis,” in Neural Information Processing Systems , pp. 1108–1116.
   Google Scholar
• Kiebel, S. , Daunizeau, J. , Phillips, C. , and Friston, K. (2008), “Variational Bayesian Inversion of the Equivalent Current Dipole Model in EEG/MEG,” NeuroImage , 39, 728–741.
   PubMed Web of Science ®Google Scholar
• Kingma, D. , and Welling, M. (2014), “Auto-Encoding Variational Bayes,” in Proceedings of the 2nd International Conference on Learning Representations (ICLR) .
   Google Scholar
• Knowles, D. , and Minka, T. (2011), “Non-Conjugate Variational Message Passing for Multinomial and Binary Regression,” in Neural Information Processing Systems , pp. 1701–1709.
   Google Scholar
• Kucukelbir, A. , Ranganath, R. , Gelman, A. , and Blei, D. (2015), “Automatic Variational Inference in Stan,” in Neural Information Processing Systems , pp. 568–576.
   Google Scholar
• Kucukelbir, A. , Tran, D. , Ranganath, R. , Gelman, A. , and Blei, D. M. (2017), “Automatic Differentiation Variational Inference,” Journal of Machine Learning Research , 18, 1–45.
   Google Scholar
• Kullback, S. , and Leibler, R. (1951), “On Information and Sufficiency,” The Annals of Mathematical Statistics , 22, 79–86.
   Google Scholar
• Kurihara, K. , and Sato, T. (2006), “Variational Bayesian Grammar Induction for Natural Language,” in Grammatical Inference: Algorithms and Applications , New York : Springer, pp. 84–96.
   Google Scholar
• Kushner, H. , and Yin, G. (1997), Stochastic Approximation Algorithms and Applications, eds. Y. Sakakibara, S. Kobayashi, K. Sato, T. Nishino, and E. Tomita, New York : Springer.
   Google Scholar
• Lashkari, D. , Sridharan, R. , Vul, E. , Hsieh, P. , Kanwisher, N. , and Golland, P. (2012), “Search for Patterns of Functional Specificity in the Brain: A Nonparametric Hierarchical Bayesian Model for Group fMRI Data,” Neuroimage , 59, 1348–1368.
   PubMed Web of Science ®Google Scholar
• Lauritzen, S. , and Spiegelhalter, D. (1988), “Local Computations with Probabilities on Graphical Structures and their Application to Expert Systems,” Journal of the Royal Statistical Society , Series B, 50, 157–224.
   Google Scholar
• Le Cun, Y. , and Bottou, L. (2004), “Large Scale Online Learning,” in Neural Information Processing Systems , pp. 217–224.
   Google Scholar
• Leisink, M. , and Kappen, H. (2001), “A Tighter Bound for Graphical Models,” Neural Computation , 13, 2149–2171.
   PubMed Web of Science ®Google Scholar
• Liang, P. , Jordan, M. I. , and Klein, D. (2009), “Probabilistic Grammars and Hierarchical Dirichlet Processes,” in The Handbook of Applied Bayesian Analysis , eds. T. O’Hagan , and M. West , New York : Oxford University Press, pp. 776–819.
   Google Scholar
• Liang, P. , Petrov, S. , Klein, D. , and Jordan, M. I. (2007), “The Infinite PCFG using Hierarchical Dirichlet Processes,” in Empirical Methods in Natural Language Processing , pp. 688–697.
   Google Scholar
• Likas, A. , and Galatsanos, N. (2004), “A Variational Approach for Bayesian Blind Image Deconvolution,” IEEE Transactions on Signal Processing , 52, 2222–2233.
   Web of Science ®Google Scholar
• Logsdon, B. , Hoffman, G. , and Mezey, J. (2010), “A Variational Bayes Algorithm for Fast and Accurate Multiple Locus Genome-Wide Association Analysis,” BMC Bioinformatics , 11, 58.
   PubMed Web of Science ®Google Scholar
• MacKay, D. J. (1997), “Ensemble Learning for Hidden Markov Models,” unpublished manuscript, available at http://www.inference.eng.cam.ac.uk/mackay/ensemblePaper.pdf .
   Google Scholar
• Manning, J. R. , Ranganath, R. , Norman, K. A. , and Blei, D. M. (2014), “Topographic Factor Analysis: A Bayesian Model for Inferring Brain Networks from Neural Data,” PloS one , 9, e94914.
   PubMed Web of Science ®Google Scholar
• Marlin, B. M. , Khan, M. E. , and Murphy, K. P. (2011), “Piecewise Bounds for Estimating Bernoulli-Logistic Latent Gaussian Models,” in International Conference on Machine Learning , pp. 633–640.
   Google Scholar
• McGrory, C. A. , and Titterington, D. M. (2007), “Variational Approximations in Bayesian Model Selection for Finite Mixture Distributions,” Computational Statistics and Data Analysis , 51, 5352–5367.
   Web of Science ®Google Scholar
• Metropolis, N. , Rosenbluth, A. , Rosenbluth, M. , Teller, M. , and Teller, E. (1953), “Equations of State Calculations by Fast Computing Machines,” Journal of Chemical Physics , 21, 1087–1092.
   Web of Science ®Google Scholar
• Minka, T. P. (2001), “Expectation Propagation for Approximate Bayesian Inference,” in Uncertainty in Artificial Intelligence , pp. 362–369.
   Google Scholar
• ——— (2005), “Divergence Measures and Message Passing,” Technical Report, Microsoft Research.
   Google Scholar
• Minka, T. , Winn, J. , Guiver, J. , Webster, S. , Zaykov, Y. , Yangel, B. , Spengler, A. , and Bronskill, J. (2014), Infer.NET 2.6. Cambridge, MA: Microsoft Research.
   Google Scholar
• Naseem, T. , Chen, H. , Barzilay, R. , and Johnson, M. (2010), “Using Universal Linguistic Knowledge to Guide Grammar Induction,” in Empirical Methods in Natural Language Processing , pp. 1234–1244.
   Google Scholar
• Nathoo, F. , Babul, A. , Moiseev, A. , Virji-Babul, N. , and Beg, M. (2014), “A Variational Bayes Spatiotemporal Model for Electromagnetic Brain Mapping,” Biometrics , 70, 132–143.
   PubMed Web of Science ®Google Scholar
• Neal, R. M. , and Hinton, G. E. (1998), “A View of the EM Algorithm that Justifies Incremental, Sparse, and other Variants,” in Learning in Graphical Models , New York: Springer, pp. 355–368.
   Google Scholar
• Neville, S. , Ormerod, J. , and Wand, M. (2014), “Mean Field Variational Bayes for Continuous Sparse Signal Shrinkage: Pitfalls and Remedies,” Electronic Journal of Statistics , 8, 1113–1151.
   Web of Science ®Google Scholar
• Nott, D. J. , Tan, S. L. , Villani, M. , and Kohn, R. (2012), “Regression Density Estimation with Variational Methods and Stochastic Approximation,” Journal of Computational and Graphical Statistics , 21, 797–820.
   Web of Science ®Google Scholar
• Opper, M. , and Winther, O. (2005), “Expectation Consistent Approximate Inference,” The Journal of Machine Learning Research , 6, 2177–2204.
   Web of Science ®Google Scholar
• Ormerod, J. , You, C. , and Muller, S. (2014), “A Variational Bayes Approach to Variable Selection,” unpublished manuscript, available at http://www.maths.usyd.edu.au/u/jormerod/JTOpapers/Variab-leSelectionFinal.pdf .
   Google Scholar
• Paisley, J. , Blei, D. , and Jordan, M. I. (2012), “Variational Bayesian Inference with Stochastic Search,” in Proceedings of the 29th International Conference on International Conference on Machine Learning , Madison, WI: Omnipress, pp. 1363–1370.
   Google Scholar
• Parisi, G. (1988), Statistical Field Theory , Melville, NY: Addison-Wesley.
   Google Scholar
• Pearl, J. (1988), Probabilistic Reasoning in Intelligent Systems: Networks of Plausible Inference , San Francisco, CA: Morgan Kaufmann.
   Google Scholar
• Penny, W. , Kiebel, S. , and Friston, K. (2003), “Variational Bayesian Inference for fMRI Time Series,” NeuroImage , 19, 727–741.
   PubMed Web of Science ®Google Scholar
• Penny, W. , Trujillo-Barreto, N. , and Friston, K. (2005), “Bayesian fMRI Time Series Analysis with Spatial Priors,” Neuroimage , 24, 350–362.
   PubMed Web of Science ®Google Scholar
• Peterson, C. , and Anderson, J. (1987), “A Mean Field Theory Learning Algorithm for Neural Networks,” Complex Systems , 1, 995–1019.
   Google Scholar
• Raj, A. , Stephens, M. , and Pritchard, J. (2014), “fastSTRUCTURE: Variational Inference of Population Structure in Large SNP Data Sets,” Genetics , 197, 573–589.
   PubMed Web of Science ®Google Scholar
• Ramos, F. , Upcroft, B. , Kumar, S. , and Durrant-Whyte, H. (2012), “A Bayesian Approach for Place Recognition,” Robotics and Autonomous Systems , 60, 487–497.
   Web of Science ®Google Scholar
• Ranganath, R. , Gerrish, S. , and Blei, D. (2014), “Black Box Variational Inference,” in Artificial Intelligence and Statistics , pp. 814–822.
   Google Scholar
• Ranganath, R. , Tran, D. , and Blei, D. (2016), “Hierarchical Variational Models,” in International Conference on Machine Learning , pp. 324–333.
   Google Scholar
• Regier, J. , Miller, A. , McAuliffe, J. , Adams, R. , Hoffman, M. , Lang, D. , Schlegel, D. , and Prabhat (2015), “Celeste: Variational Inference for a Generative Model of Astronomical Images,” in International Conference on Machine Learning , pp. 2095–2103.
   Google Scholar
• Reyes-Gomez, M. , Ellis, D. , and Jojic, N. (2004), “Multiband Audio Modeling for Single-Channel Acoustic Source Separation,” in Acoustics, Speech, and Signal Processing , pp. 641–644.
   Google Scholar
• Rezende, D. J. , Mohamed, S. , and Wierstra, D. (2014), “Stochastic Backpropagation and Approximate Inference in Deep Generative Models,” in Proceedings of the 31st International Conference on Machine Learning (Vol. 32), eds. E. P. Xing and T. Jebara, Beijing, China: Proceedings of Machine Learning Research, pp. 1278–1286.
   Google Scholar
• Robbins, H. , and Monro, S. (1951), “A Stochastic Approximation Method,” The Annals of Mathematical Statistics , 22, 400–407.
   Google Scholar
• Robert, C. , and Casella, G. (2004), Monte Carlo Statistical Methods (Springer Texts in Statistics) , New York : Springer-Verlag.
   Google Scholar
• Roberts, S. , Guilford, T. , Rezek, I. , and Biro, D. (2004), “Positional Entropy During Pigeon Homing I: Application of Bayesian Latent State Modelling,” Journal of Theoretical Biology , 227, 39–50.
   PubMed Web of Science ®Google Scholar
• Roberts, S. , and Penny, W. (2002), “Variational Bayes for Generalized Autoregressive Models,” IEEE Transactions on Signal Processing , 50, 2245–2257.
   Web of Science ®Google Scholar
• Rohde, D. , and Wand, M. (2016), “Semiparametric Mean Field Variational Bayes: General Principles and Numerical Issues,” Journal of Machine Learning Research , 17, 1–47.
   Google Scholar
• Salimans, T. , Kingma, D. , and Welling, M. (2015), “Markov Chain Monte Carlo and Variational Inference: Bridging the Gap,” in International Conference on Machine Learning , pp. 1218–1226.
   Google Scholar
• Salimans, T. , and Knowles, D. (2014), “On using Control Variates with Stochastic Approximation for Variational Bayes,” arXiv preprint, arXiv:1401.1022. Available at https://arxiv.org/abs/1401.1022
   Google Scholar
• Sanguinetti, G. , Lawrence, N. , and Rattray, M. (2006), “Probabilistic Inference of Transcription Factor Concentrations and Gene-Specific Regulatory Activities,” Bioinformatics , 22, 2775–2781.
   PubMed Web of Science ®Google Scholar
• Sato, M. (2001), “Online Model Selection Based on the Variational Bayes,” Neural Computation , 13, 1649–1681.
   Web of Science ®Google Scholar
• Sato, M. , Yoshioka, T. , Kajihara, S. , Toyama, K. , Goda, N. , Doya, K. , and Kawato, M. (2004), “Hierarchical Bayesian Estimation for MEG Inverse Problem,” NeuroImage , 23, 806–826.
   PubMed Web of Science ®Google Scholar
• Saul, L. , and Jordan, M. I. (1996), “Exploiting Tractable Substructures in Intractable Networks,” in Neural Information Processing Systems , pp. 486–492.
   Google Scholar
• Saul, L. K. , Jaakkola, T. , and Jordan, M. I. (1996), “Mean Field Theory for Sigmoid Belief Networks,” Journal of Artificial Intelligence Research , 4, 61–76.
   Web of Science ®Google Scholar
• Spall, J. (2003), Introduction to Stochastic Search and Optimization: Estimation, Simulation, and Control , New York : Wiley.
   Google Scholar
• Stan Development Team (2015), Stan Modeling Language Users Guide and Reference Manual, Version 2.8.0. New York: Columbia University.
   Google Scholar
• Stegle, O. , Parts, L. , Durbin, R. , and Winn, J. (2010), “A Bayesian Framework to Account for Complex Non-Genetic Factors in Gene Expression Levels Greatly Increases Power in eqtl Studies,” PLoS Computational Biology , 6, e1000770.
   PubMed Web of Science ®Google Scholar
• Sudderth, E. B. , and Jordan, M. I. (2009), “Shared Segmentation of Natural Scenes using Dependent Pitman-Yor Processes,” in Neural Information Processing Systems , pp. 1585–1592.
   Google Scholar
• Sung, J. , Ghahramani, Z. , and Bang, Y. (2008), “Latent-Space Variational Bayes,” IEEE Transactions on Pattern Analysis and Machine Intelligence , 30, 2236–2242.
   PubMed Web of Science ®Google Scholar
• Sykacek, P. , Roberts, S. , and Stokes, M. (2004), “Adaptive BCI Based on Variational Bayesian Kalman Filtering: An Empirical Evaluation,” IEEE Transactions on Biomedical Engineering , 51, 719–727.
   PubMed Web of Science ®Google Scholar
• Tan, L. , and Nott, D. (2013), “Variational Inference for Generalized Linear Mixed Models using Partially Noncentered Parametrizations,” Statistical Science , 28, 168–188.
   Web of Science ®Google Scholar
• Tan, L. , and Nott, D. (2014), “A Stochastic Variational Framework for Fitting and Diagnosing Generalized Linear Mixed Models,” Bayesian Analysis , 9, 963–1004.
   Web of Science ®Google Scholar
• Tan, L. , and Nott, D. 2018). Gaussian variational approximation with sparse precision matrices. Statistics and Computing, 28, 259-275. https://doi.org/10.1007/s11222-017-9729-7
   Web of Science ®Google Scholar
• Tipping, M. , and Lawrence, N. (2005), “Variational Inference for Student-t models: Robust Bayesian Interpolation and Generalised Component Analysis,” Neurocomputing , 69, 123–141.
   Web of Science ®Google Scholar
• Titsias, M. , and Lawrence, N. (2010), “Bayesian Gaussian Process Latent Variable Model,” in Artificial Intelligence and Statistics , pp. 844–851.
   Google Scholar
• Titsias, M. , and Lázaro-Gredilla, M. (2014), “Doubly Stochastic Variational Bayes for Non-Conjugate Inference,” in International Conference on Machine Learning , pp. 1971–1979.
   Google Scholar
• Tran, D. , Ranganath, R. , and Blei, D. M. (2016), “The Variational Gaussian Process,” in International Conference on Learning Representations , pp. 1–4.
   Google Scholar
• Ueda, N. , and Ghahramani, Z. (2002), “Bayesian Model Search for Mixture Models Based on Optimizing Variational Bounds,” Neural Networks , 15, 1223–1241.
   PubMed Web of Science ®Google Scholar
• Van Den Broek, B. , Wiegerinck, W. , and Kappen, B. (2008), “Graphical Model Inference in Optimal Control of Stochastic Multi-Agent Systems,” Journal of Artificial Intelligence Research , 32, 95–122.
   Web of Science ®Google Scholar
• Vermaak, J. , Lawrence, N. D. , and Pérez, P. (2003), “Variational Inference for Visual Tracking,” in Computer Vision and Pattern Recognition , pp. 1–8.
   Google Scholar
• Villegas, M. , Paredes, R. , and Thomee, B. (2013), “Overview of the ImageCLEF 2013 Scalable Concept Image Annotation Subtask,” in CLEF Evaluation Labs and Workshop , pp. 308–328.
   Google Scholar
• Wainwright, M. J. , and Jordan, M. I. (2008), “Graphical Models, Exponential Families, and Variational Inference,” Foundations and Trends in Machine Learning , 1, 1–305.
   Google Scholar
• Wand, M. (2014), “Fully Simplified Multivariate Normal Updates in Non-Conjugate Variational Message Passing,” Journal of Machine Learning Research , 15, 1351–1369.
   Web of Science ®Google Scholar
• Wand, M. , Ormerod, J. , Padoan, S. , and Fuhrwirth, R. (2011), “Mean Field Variational Bayes for Elaborate Distributions,” Bayesian Analysis , 6, 847–900.
   Web of Science ®Google Scholar
• Wang, B. , and Titterington, D. (2005), “Inadequacy of Interval Estimates Corresponding to Variational Bayesian Approximations,” in Artificial Intelligence and Statistics , pp. 373–380.
   Google Scholar
• Wang, B. , and Titterington, D. (2006), “Convergence Properties of a General Algorithm for Calculating Variational Bayesian Estimates for a Normal Mixture Model,” Bayesian Analysis , 1, 625–650.
   Web of Science ®Google Scholar
• Wang, C. , and Blei, D. (2013), “Variational Inference in Nonconjugate Models,” Journal of Machine Learning Research , 14, 1005–1031.
   Web of Science ®Google Scholar
• Wang, C. , and Blei, D. (2015), “A General Method for Robust Bayesian Modeling,” Journal of Machine Learning Research , 14, 1005–1031.
   Google Scholar
• Wang, P. , and Blunsom, P. (2013), “Collapsed Variational Bayesian Inference for Hidden Markov Models,” in Artificial Intelligence and Statistics , pp. 599–607.
   Google Scholar
• Wang, Y. , and Mori, G. (2009), “Human Action Recognition by Semilatent Topic Models,” IEEE Transactions on Pattern Analysis and Machine Intelligence , 31, 1762–1774.
   PubMed Web of Science ®Google Scholar
• Waterhouse, S. , MacKay, D. , and Robinson, T. (1996), “Bayesian Methods for Mixtures of Experts,” in Neural Information Processing Systems , pp. 351–357.
   Google Scholar
• Welling, M. , and Teh, Y. (2011), “Bayesian Learning via Stochastic Gradient Langevin Dynamics,” in International Conference on Machine Learning , pp. 681–688.
   Google Scholar
• Westling, T. , and McCormick, T. H. (2015), “Establishing Consistency and Improving Uncertainty Estimates of Variational Inference Through M-estimation,” arXiv preprint, arXiv:1510.08151. Available at https://arxiv.org/abs/1510.08151
   Google Scholar
• Wiggins, C. , and Hofman, J. (2008), “Bayesian Approach to Network Modularity,” Physical Review Letters , 100, 258701.
   PubMed Web of Science ®Google Scholar
• Wingate, D. , and Weber, T. (2013), “Automated Variational Inference in Probabilistic Programming,” arXiv preprint, arXiv:1301.1299. Available at https://arxiv.org/abs/1301.1299
   Google Scholar
• Winn, J. , and Bishop, C. (2005), “Variational Message Passing,” Journal of Machine Learning Research , 6, 661–694.
   Web of Science ®Google Scholar
• Wipf, D. , and Nagarajan, S. (2009), “A Unified Bayesian Framework for MEG/EEG Source Imaging,” NeuroImage , 44, 947–966.
   PubMed Web of Science ®Google Scholar
• Woolrich, M. , Behrens, T. , Beckmann, C. , Jenkinson, M. , and Smith, S. (2004), “Multilevel Linear Modeling for fMRI Group Analysis using Bayesian Inference,” Neuroimage , 21, 1732–1747.
   PubMed Web of Science ®Google Scholar
• Xing, E. , Wu, W. , Jordan, M. I. , and Karp, R. (2004), “Logos: A Modular Bayesian Model for de novo motif Detection,” Journal of Bioinformatics and Computational Biology , 2, 127–154.
   PubMedGoogle Scholar
• Yedidia, J. S. , Freeman, W. T. , and Weiss, Y. (2001), “Generalized Belief Propagation,” in Neural Information Processing Systems , pp. 689–695.
   Google Scholar
• Yogatama, D. , Wang, C. , Routledge, B. , Smith, N. A. , and Xing, E. (2014), “Dynamic Language Models for Streaming Text,” Transactions of the Association for Computational Linguistics , 2, 181–192.
   Google Scholar
• You, C. , Ormerod, J. , and Muller, S. (2014), “On Variational Bayes Estimation and Variational Information Criteria for Linear Regression Models,” Australian & New Zealand Journal of Statistics , 56, 73–87.
   Web of Science ®Google Scholar
• Yu, T. , and Wu, Y. (2005), “Decentralized Multiple Target Tracking using Netted Collaborative Autonomous Trackers,” in Computer Vision and Pattern Recognition , pp. 939–946.
   Google Scholar
• Zumer, J. , Attias, H. , Sekihara, K. , and Nagarajan, S. (2007), “A Probabilistic Algorithm Integrating Source Localization and Noise Suppression for MEG and EEG Data,” NeuroImage , 37, 102–115.
   PubMed Web of Science ®Google Scholar