at the

American Institute of Mathematics, San Jose, California

organized by

Russell Steele, Bernd Sturmfels, and Sumio Watanabe

Singular statistical learning is an approach for statistical learning and model selection that can be applied to singular parameter spaces, i.e. can be used for non-regular statistical models. The methodology uses the method of resolution of singularities to generalize the criteria for regular statistical models to non-regular models. Examples of non-regular statistical models that have been studied as part of singular learning theory include hidden Markov models, finite mixture models, and multi-layer neural network models. Although there exists a large body of recent published work in this area, it is has not yet been integrated or even well-cited by the larger statistical community.

The workshop has three primary goals:

- To introduce statisticians and computer scientists working the area of model selection to the topic of singular learning theory, in particular the application of the method of resolution of singularities to model selection for non-regular statistical models.
- To generate a list of open problems in algebraic geometry motivated by complex statistical models that cannot be covered by current results.
- To collaboratively develop a set of core materials that will define the area of singular statistical learning that will be accessible to geometers and statisticians.

- Exploring connections of Widely Applicable Information Criteria (WAIC) from singular learning theory to other model selection criteria, including the Deviance Information Criterion (DIC), regular statistical versions of the AIC and BIC, and other criteria specific to particular non-regular statistical models (for example, the scan statistic from spatial statistics).
- Identifying fundamental problems in algebraic geometry are related to generalizing these information criteria to model selection problems for Generalized Estimating Equations (GEE), which use ideas from semi-parametric inference to obtain estimates of parameters without assuming a parametric form for the likelihood of the observed data.
- Generalizing the singular learning theory information criteria be to statistical problems where some observations contain missing information and/or measurement error.
- Establishing the finite sample properties of WAIC, in particular for problems where one can incorporate prior knowledge in a fully Bayesian modelling approach.

The workshop schedule.

A report on the workshop activities.

Papers arising from the workshop:

Geometry of Higher-Order Markov Chains

by Bernd Sturmfels, *J. Algebr. Stat. 3 (2012), no. 1, 1-10 * MR3016418

A widely applicable Bayesian information criterion

by Sumio Watanabe, *J. Mach. Learn. Res. 14 (2013), 867-897 * MR3049492