# Variational Method

This page contains resources about Variational Methods, Variational Bayesian Inference, Variational Bayesian Learning and Deterministic Approximate Inference.

## Subfields and Concepts

• Variational Calculus / Calculus of Variations
• Variational Analysis‎
• Variational free energy
• Free energy principle
• Conjugate Duality
• Exponential family
• Conjugate prior family
• Variance reduction techniques (VRT) in Monte Carlo Gradients
• Control variates
• Rao–Blackwellization
• By linear regression
• Reparameterization trick / Reparameterization Gradient / Coordinate Tranformation / Invertible Tranformation / Elliptical Standarization
• Importance Sampling
• Score Function (SF) Estimator
• Pathwise Derivative (PD) Estimator
• Evidence Lower Bound (ELBO) / Variational Lower Bound
• Structured Variational Inference
• Kullback–Leibler (KL) Divergence
• Variational Bayes
• Variational Bayesian EM (VBEM)
• Stochastic Variational Inference
• Stochastic Gradient Variational Bayes (SGVB) Estimator
• Deep Variational Bayes Filter (DVBF)
• Wake-Sleep Algorithm
• Auto-Encoding Variational Bayes (AEVB) Algorithm
• Variational Autoencoder (VAE)
• Hierarchical Variational Models
• Expectation Propagation
• Loopy Belief Propagation / Loopy Sum-Product Message Passing
• Assumed Density Filtering (ADF) / Moment Matching
• Kullback-Leibler (KL) Variational Inference / Mean field Variational Bayes
• Structured Mean field / Structured Variational Approximation
• Weighted Mean Field
• Tree-based reparameterizations
• Tree-reweighted belief propagation
• Bethe and Kikuchi free energy
• Generalized Belief Propagation
• Forwared KL divergence / Moment Projection (M-Projection)
• Reverse KL divergence / Information Projection (I-Projection)
• Online Bayesian Variational (OBV) Inference Algorithms
• Neural Variational Inference and Learning (NVIL)
• Non-conjugate Variational Inference
• Rejection Sampling Variational Inference (RSVI)
• Reinforced Variational Inference
• Generic and Automated Variation Inference
• Black-Box Variational Inference (BBVI)
• Automatic Variational Inference (AVI)
• Automatic Differentiation Variational Inference (ADVI)
• SGVB with the log derivative trick (LdGrad) / Score Function Method
• Overdispersed BBVI (O-BBVI)
• Stochastic Optimization
• Stochastic Approximation
• Robbins-Monro Algorithm (using noisy estimates of the gradient)
• Energy-Based Model (EBM)
• Free energy (i.e. the contrastive term)
• Regularization term
• Loss functionals or Loss functions or Energy functionals
• Energy Loss
• Generalized Perceptron Loss
• Generalized Margin Losses
• Negative Log-Likelihood Loss

## Books and Book Chapters

• Bengio, Y., Goodfellow, I. J., & Courville, A. (2016). "Chapter 19: Approximate Inference". Deep Learning. MIT Press.
• Theodoridis, S. (2015). "Chapter 13: Bayesian Learning: Approximate Inference and Nonparametric Models". Machine Learning: A Bayesian and Optimization Perspective. Academic Press.
• Murphy, K. P. (2012). "Chapter 21: Variational inference". Machine Learning: A Probabilistic Perspective. MIT Press.
• Barber, D. (2012). "Section 7.7: Variational Inference and Planning". Bayesian Reasoning and Machine Learning. Cambridge University Press.
• Barber, D. (2012). "Chapter 11: Learning with Hidden Variables". Bayesian Reasoning and Machine Learning. Cambridge University Press.
• Barber, D. (2012). "Chapter 28: Deterministic Approximate Inference". Bayesian Reasoning and Machine Learning. Cambridge University Press.
• Koller, D., & Friedman, N. (2009). "Chapter 11: Inference as Optimization". Probabilistic Graphical Models. MIT Press.
• Bishop, C. M. (2006). "Chapter 10: Approximate Inference". Pattern Recognition and Machine Learning. Springer.
• Smidl, V., & Quinn, A. (2006). The Variational Bayes Method in Signal Processing. Springer Science & Business Media.
• MacKay, D. J. (2003). "Chapter 33: Variational Methods" Information Theory, Inference and Learning Algorithms. Cambridge University Press.
• Opper, M., & Saad, D. (2001). Advanced mean field methods: Theory and practice. MIT press.

## Scholarly Articles

• Laumann, F., & Shridhar, K. (2018). Bayesian Convolutional Neural Networks. arXiv preprint arXiv:1806.05978.
• Louizos, C., & Welling, M. (2017). Multiplicative Normalizing Flows for Variational Bayesian Neural Networks. In International Conference on Machine Learning (pp. 2218-2227).
• Kingma, D. P. (2017). Variational Inference & Deep Learning: A New Synthesis. PhD Diss. University of Amsterdam.
• Fortunato, M., Blundell, C., & Vinyals, O. (2017). Bayesian recurrent neural networks. arXiv preprint arXiv:1704.02798.
• Ruiz, F. J., Titsias, M. K., & Blei, D. M. (2016). The Generalized Reparameterization Gradient. arXiv preprint arXiv:1610.02287.
• Ruiz, F. J., Titsias, M. K., & Blei, D. M. (2016). Overdispersed Black-Box Variational Inference. arXiv preprint arXiv:1603.01140.
• Blei, D. M., Kucukelbir, A., & McAuliffe, J. D. (2016). Variational inference: A review for statisticians. arXiv preprint arXiv:1601.00670.
• Mandt, S., Hoffman, M. D., & Blei, D. M. (2016). A Variational Analysis of Stochastic Gradient Algorithms. arXiv preprint arXiv:1602.02666.
• Naesseth, C. A., Ruiz, F. J., Linderman, S. W., & Blei, D. M. (2016). Rejection Sampling Variational Inference. arXiv preprint arXiv:1610.05683.
• Kucukelbir, A., Tran, D., Ranganath, R., Gelman, A., & Blei, D. M. (2016). Automatic Differentiation Variational Inference. arXiv preprint arXiv:1603.00788.
• Gal, Y., & Ghahramani, Z. (2016). Dropout as a Bayesian approximation: Representing model uncertainty in deep learning. In International Conference on Machine Learning (pp. 1050-1059).
• Kucukelbir, A., Ranganath, R., Gelman, A., & Blei, D. (2015). Automatic variational inference in Stan. In Advances in Neural Information Processing Systems (pp. 568-576).
• Blundell, C., Cornebise, J., Kavukcuoglu, K., & Wierstra, D. (2015). Weight Uncertainty in Neural Network. In International Conference on Machine Learning (pp. 1613-1622).
• Schulman, J., Heess, N., Weber, T., & Abbeel, P. (2015). Gradient estimation using stochastic computation graphs. In Advances in Neural Information Processing Systems (pp. 3528-3536).
• Titsias, M., & Lazaro-Gredilla, M. (2015). Local expectation gradients for black box variational inference. In Advances in Neural Information Processing Systems (pp. 2638-2646).
• Kingma, D. P., Salimans, T., & Welling, M. (2015). Variational dropout and the local reparameterization trick. In Advances in Neural Information Processing Systems (pp. 2575-2583).
• Archer, E., Park, I. M., Buesing, L., Cunningham, J., & Paninski, L. (2015). Black box variational inference for state space models. arXiv preprint arXiv:1511.07367.
• Hoffman, M. D., & Blei, D. M. (2015). Structured stochastic variational inference. In Artificial Intelligence and Statistics.
• Kucukelbir, A., Ranganath, R., Gelman, A., & Blei, D. (2014). Fully automatic variational inference of differentiable probability models. In NIPS Workshop on Probabilistic Programming.
• Salimans, T., & Knowles, D. A. (2014). On using control variates with stochastic approximation for variational Bayes and its connection to stochastic linear regression. arXiv preprint arXiv:1401.1022.
• Ranganath, R., Gerrish, S., & Blei, D. M. (2014). Black Box Variational Inference. In AISTATS (pp. 814-822).
• Lazaro-Gredilla, M. (2014). Doubly stochastic variational Bayes for non-conjugate inference. In Proceedings of the 31st International Conference on Machine Learning (pp. 1971-1979).
• Mnih, A., & Gregor, K. (2014). Neural variational inference and learning in belief networks. arXiv preprint arXiv:1402.0030.
• Salimans, T., & Knowles, D. A. (2013). Fixed-form variational posterior approximation through stochastic linear regression. Bayesian Analysis8(4), 837-882.
• Hoffman, M. D., Blei, D. M., Wang, C., & Paisley, J. W. (2013). Stochastic variational inference.Journal of Machine Learning Research14(1), 1303-1347.
• Wingate, D., & Weber, T. (2013). Automated variational inference in probabilistic programming. arXiv preprint arXiv:1301.1299.
• Wang, C., & Blei, D. M. (2013). Variational inference in nonconjugate models. Journal of Machine Learning Research14(Apr), 1005-1031.
• Fox, C. W., & Roberts, S. J. (2012). A tutorial on variational Bayesian inference. Artificial intelligence review38(2), 85-95.
• Paisley, J., Blei, D., & Jordan, M. (2012). Variational Bayesian inference with stochastic search. arXiv preprint arXiv:1206.6430.
• Knowles, D. A., & Minka, T. (2011). Non-conjugate variational message passing for multinomial and binary regression. In Advances in Neural Information Processing Systems (pp. 1701-1709).
• Wainwright, M. J., & Jordan, M. I. (2008). Graphical models, exponential families, and variational inference. Foundations and Trends® in Machine Learning1(1-2), 1-305.
• Tzikas, D. G., Likas, A. C., & Galatsanos, N. P. (2008). The variational approximation for Bayesian inference. IEEE Signal Processing Magazine,25(6), 131-146.
• Wainwright, M., & Jordan, M. (2005). A variational principle for graphical models. New Directions in Statistical Signal Processing155.
• Yedidia, J. S., Freeman, W. T., & Weiss, Y. (2005). Constructing free-energy approximations and generalized belief propagation algorithms. IEEE Transactions on Information Theory51(7), 2282-2312.
• Beal, M. J. (2003). Variational algorithms for approximate Bayesian inference. Ph.D. Dissertation, University College London.
• Xing, E. P., Jordan, M. I., & Russell, S. (2003). A generalized mean field algorithm for variational inference in exponential families. In Proceedings of the Nineteenth conference on Uncertainty in Artificial Intelligence (pp. 583-591). Morgan Kaufmann Publishers Inc.
• Wainwright, M. J., & Jordan, M. I. (2003). Variational inference in graphical models: The view from the marginal polytope. In Proceeding of Annual Allerton Conference of Communication Control and Computing (Vol. 41, No. 2, pp. 961-971).
• Lawrence, N. D. (2001). Variational inference in probabilistic models. Ph.D. Dissertation, University of Cambridge.
• Minka, T. P. (2001). A family of algorithms for approximate Bayesian inference. Ph.D. Dissertation, Massachusetts Institute of Technology.
• Ghahramani, Z., & Beal, M. J. (2001). Propagation algorithms for variational Bayesian learning. In Advances in Neural Information Processing Systems, 507-513.
• Attias, H. (2000). A variational Bayesian framework for graphical models. In Advances in Neural Information Processing Systems, 209-215.
• Jordan, M. I., Ghahramani, Z., Jaakkola, T. S., & Saul, L. K. (1999). An introduction to variational methods for graphical models. Machine learning,37(2), 183-233.