Bayesian model selection

Sparsity
Cross-validation and Bayes
Evidence/marginal likelihood/type II maximum likelihood
Incoming
References

Frequentist model selection is not the only type, but I know less about Bayesian model selection. What is model selection in a Bayesian context? Surely you don’t ever get some models with zero posterior probability? In my intro Bayesian classes I learned that one simply keeps all the models weighted by posterior likelihood when making predictions. But sometimes we wish to get rid of some models. When does this work, and when not? Typically this seems to be done by comparing model marginal evidence.

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Sparsity

Interesting special case: Bayesian sparsity.

Cross-validation and Bayes

There is a relation between cross-validation and Bayes evidence, a.k.a. marginal likelihood - see (Claeskens and Hjort 2008; Fong and Holmes 2019).

Evidence/marginal likelihood/type II maximum likelihood

See model selection by model evidence maximisation.

Incoming

John Mount on applied variable selection

We have also always felt a bit exposed in this, as feature selection seems unjustified in standard explanations of regression. One feels that if a coefficient were meant to be zero, the fitting procedure would have set it to zero. Under this misapprehension, stepping in and removing some variables feels unjustified.
Regardless of intuition or feelings, it is a fair question: is variable selection a natural justifiable part of modeling? Or is it something that is already done (therefore redundant). Or is it something that is not done for important reasons (such as avoiding damaging bias)?
In this note we will show that feature selection is in fact an obvious justified step when using a sufficiently sophisticated model of regression. This note is long, as it defines so many tiny elementary steps. However this note ends with a big point: variable selection is justified. It naturally appears in the right variation of Bayesian Regression. You should select variables, using your preferred methodology. And you shouldn’t feel bad about selecting variables.

References

Bhadra, Anindya, Jyotishka Datta, Nicholas G. Polson, and Brandon Willard. 2016. “Default Bayesian Analysis with Global-Local Shrinkage Priors.” Biometrika 103 (4): 955–69.

Bondell, Howard D., and Brian J. Reich. 2012. “Consistent High-Dimensional Bayesian Variable Selection via Penalized Credible Regions.” Journal of the American Statistical Association 107 (500): 1610–24.

Bürkner, Paul-Christian, Jonah Gabry, and Aki Vehtari. 2020. “Approximate Leave-Future-Out Cross-Validation for Bayesian Time Series Models.” Journal of Statistical Computation and Simulation 90 (14): 2499–2523.

Carvalho, Carlos M., Nicholas G. Polson, and James G. Scott. 2010. “The Horseshoe Estimator for Sparse Signals.” Biometrika 97 (2): 465–80.

Castillo, Ismaël, Johannes Schmidt-Hieber, and Aad van der Vaart. 2015. “Bayesian Linear Regression with Sparse Priors.” The Annals of Statistics 43 (5).

Chipman, Hugh, Edward I. George, Robert E. McCulloch, and P Lahiri. 2001. “The Practical Implementation of Bayesian Model Selection.” In Model Selection. Vol. 38. IMS Lecture Notes - Monograph Series. Beachwood, OH: Institute of Mathematical Statistics.

Claeskens, Gerda, and Nils Lid Hjort. 2008. Model Selection and Model Averaging. Cambridge Series in Statistical and Probabilistic Mathematics. Cambridge ; New York: Cambridge University Press.

Efron, Bradley. 2012. “Bayesian Inference and the Parametric Bootstrap.” The Annals of Applied Statistics 6 (4): 1971–97.

Filippone, Maurizio, and Raphael Engler. 2015. “Enabling Scalable Stochastic Gradient-Based Inference for Gaussian Processes by Employing the Unbiased LInear System SolvEr (ULISSE).” In Proceedings of the 32nd International Conference on Machine Learning, 1015–24. PMLR.

Fong, Edwin, and Chris Holmes. 2019. “On the Marginal Likelihood and Cross-Validation.” arXiv:1905.08737 [Stat], May.

Gelman, Andrew, and Donald B. Rubin. 1995. “Avoiding Model Selection in Bayesian Social Research.” Sociological Methodology 25: 165–73.

George, Edward I., and Robert McCulloch. 1997. “Approaches for bayesian variable selection.” Statistica Sinica 7 (2): 339–73.

Hirsh, Seth M., David A. Barajas-Solano, and J. Nathan Kutz. 2022. “Sparsifying Priors for Bayesian Uncertainty Quantification in Model Discovery.” Royal Society Open Science 9 (2): 211823.

Ishwaran, Hemant, and J. Sunil Rao. 2005. “Spike and Slab Variable Selection: Frequentist and Bayesian Strategies.” The Annals of Statistics 33 (2): 730–73.

Kadane, Joseph B., and Nicole A. Lazar. 2004. “Methods and Criteria for Model Selection.” Journal of the American Statistical Association 99 (465): 279–90.

Laud, Purushottam W., and Joseph G. Ibrahim. 1995. “Predictive Model Selection.” Journal of the Royal Statistical Society. Series B (Methodological) 57 (1): 247–62.

Li, Meng, and David B. Dunson. 2016. “A Framework for Probabilistic Inferences from Imperfect Models.” arXiv:1611.01241 [Stat], November.

Linden, Sander van der, and Breanne Chryst. 2017. “No Need for Bayes Factors: A Fully Bayesian Evidence Synthesis.” Frontiers in Applied Mathematics and Statistics 3.

Lorch, Lars, Jonas Rothfuss, Bernhard Schölkopf, and Andreas Krause. 2021. “DiBS: Differentiable Bayesian Structure Learning.” In.

Mackay, David J. C. 1995. “Probable Networks and Plausible Predictions — a Review of Practical Bayesian Methods for Supervised Neural Networks.” Network: Computation in Neural Systems 6 (3): 469–505.

MacKay, David JC. 1999. “Comparison of Approximate Methods for Handling Hyperparameters.” Neural Computation 11 (5): 1035–68.

Madigan, David, and Adrian E. Raftery. 1994. “Model Selection and Accounting for Model Uncertainty in Graphical Models Using Occam’s Window.” Journal of the American Statistical Association 89 (428): 1535–46.

Navarro, Danielle J. 2019. “Between the Devil and the Deep Blue Sea: Tensions Between Scientific Judgement and Statistical Model Selection.” Computational Brain & Behavior 2 (1): 28–34.

Ohn, Ilsang, and Yongdai Kim. 2021. “Posterior Consistency of Factor Dimensionality in High-Dimensional Sparse Factor Models.” Bayesian Analysis, January.

Ohn, Ilsang, and Lizhen Lin. 2021. “Adaptive Variational Bayes: Optimality, Computation and Applications.” arXiv:2109.03204 [Math, Stat], September.

Ormerod, John T., Michael Stewart, Weichang Yu, and Sarah E. Romanes. 2017. “Bayesian Hypothesis Tests with Diffuse Priors: Can We Have Our Cake and Eat It Too?” arXiv:1710.09146 [Math, Stat], October.

Piironen, Juho, and Aki Vehtari. 2017. “Comparison of Bayesian Predictive Methods for Model Selection.” Statistics and Computing 27 (3): 711–35.

Polson, Nicholas G., and James G. Scott. 2012. “Local Shrinkage Rules, Lévy Processes and Regularized Regression.” Journal of the Royal Statistical Society: Series B (Statistical Methodology) 74 (2): 287–311.

Raftery, Adrian E. 1995. “Bayesian Model Selection in Social Research.” Sociological Methodology 25: 111–63.

Ročková, Veronika, and Edward I. George. 2018. “The Spike-and-Slab LASSO.” Journal of the American Statistical Association 113 (521): 431–44.

Schmidt, Daniel F., and Enes Makalic. 2020. “Log-Scale Shrinkage Priors and Adaptive Bayesian Global-Local Shrinkage Estimation.” arXiv.

Simchoni, Giora, and Saharon Rosset. 2023. “Integrating Random Effects in Deep Neural Networks.” arXiv.

Stein, Michael L. 2008. “A Modeling Approach for Large Spatial Datasets.” Journal of the Korean Statistical Society 37 (1): 3–10.

Tang, Xueying, Xiaofan Xu, Malay Ghosh, and Prasenjit Ghosh. 2016. “Bayesian Variable Selection and Estimation Based on Global-Local Shrinkage Priors.” arXiv.

Vehtari, Aki, and Janne Ojanen. 2012. “A Survey of Bayesian Predictive Methods for Model Assessment, Selection and Comparison.” Statistics Surveys 6: 142–228.

Wieringen, Wessel N. van. 2021. “Lecture Notes on Ridge Regression.” arXiv:1509.09169 [Stat], May.

Xu, Zemei, Daniel F. Schmidt, Enes Makalic, Guoqi Qian, and John L. Hopper. 2017. “Bayesian Sparse Global-Local Shrinkage Regression for Selection of Grouped Variables.” arXiv.

Zanella, Giacomo, and Gareth Roberts. 2019. “Scalable Importance Tempering and Bayesian Variable Selection.” Journal of the Royal Statistical Society Series B: Statistical Methodology 81 (3): 489–517.