Ordinal Graphical Models: A Tale of Two Approaches

Arun Sai Suggala, Eunho Yang, Pradeep Ravikumar
Proceedings of the 34th International Conference on Machine Learning, PMLR 70:3260-3269, 2017.

Abstract

Undirected graphical models or Markov random fields (MRFs) are widely used for modeling multivariate probability distributions. Much of the work on MRFs has focused on continuous variables, and nominal variables (that is, unordered categorical variables). However, data from many real world applications involve ordered categorical variables also known as ordinal variables, e.g., movie ratings on Netflix which can be ordered from 1 to 5 stars. With respect to univariate ordinal distributions, as we detail in the paper, there are two main categories of distributions; while there have been efforts to extend these to multivariate ordinal distributions, the resulting distributions are typically very complex, with either a large number of parameters, or with non-convex likelihoods. While there have been some work on tractable approximations, these do not come with strong statistical guarantees, and moreover are relatively computationally expensive. In this paper, we theoretically investigate two classes of graphical models for ordinal data, corresponding to the two main categories of univariate ordinal distributions. In contrast to previous work, our theoretical developments allow us to provide correspondingly two classes of estimators that are not only computationally efficient but also have strong statistical guarantees.

Cite this Paper


BibTeX
@InProceedings{pmlr-v70-suggala17a, title = {Ordinal Graphical Models: A Tale of Two Approaches}, author = {Arun Sai Suggala and Eunho Yang and Pradeep Ravikumar}, booktitle = {Proceedings of the 34th International Conference on Machine Learning}, pages = {3260--3269}, year = {2017}, editor = {Precup, Doina and Teh, Yee Whye}, volume = {70}, series = {Proceedings of Machine Learning Research}, month = {06--11 Aug}, publisher = {PMLR}, pdf = {http://proceedings.mlr.press/v70/suggala17a/suggala17a.pdf}, url = { http://proceedings.mlr.press/v70/suggala17a.html }, abstract = {Undirected graphical models or Markov random fields (MRFs) are widely used for modeling multivariate probability distributions. Much of the work on MRFs has focused on continuous variables, and nominal variables (that is, unordered categorical variables). However, data from many real world applications involve ordered categorical variables also known as ordinal variables, e.g., movie ratings on Netflix which can be ordered from 1 to 5 stars. With respect to univariate ordinal distributions, as we detail in the paper, there are two main categories of distributions; while there have been efforts to extend these to multivariate ordinal distributions, the resulting distributions are typically very complex, with either a large number of parameters, or with non-convex likelihoods. While there have been some work on tractable approximations, these do not come with strong statistical guarantees, and moreover are relatively computationally expensive. In this paper, we theoretically investigate two classes of graphical models for ordinal data, corresponding to the two main categories of univariate ordinal distributions. In contrast to previous work, our theoretical developments allow us to provide correspondingly two classes of estimators that are not only computationally efficient but also have strong statistical guarantees.} }
Endnote
%0 Conference Paper %T Ordinal Graphical Models: A Tale of Two Approaches %A Arun Sai Suggala %A Eunho Yang %A Pradeep Ravikumar %B Proceedings of the 34th International Conference on Machine Learning %C Proceedings of Machine Learning Research %D 2017 %E Doina Precup %E Yee Whye Teh %F pmlr-v70-suggala17a %I PMLR %P 3260--3269 %U http://proceedings.mlr.press/v70/suggala17a.html %V 70 %X Undirected graphical models or Markov random fields (MRFs) are widely used for modeling multivariate probability distributions. Much of the work on MRFs has focused on continuous variables, and nominal variables (that is, unordered categorical variables). However, data from many real world applications involve ordered categorical variables also known as ordinal variables, e.g., movie ratings on Netflix which can be ordered from 1 to 5 stars. With respect to univariate ordinal distributions, as we detail in the paper, there are two main categories of distributions; while there have been efforts to extend these to multivariate ordinal distributions, the resulting distributions are typically very complex, with either a large number of parameters, or with non-convex likelihoods. While there have been some work on tractable approximations, these do not come with strong statistical guarantees, and moreover are relatively computationally expensive. In this paper, we theoretically investigate two classes of graphical models for ordinal data, corresponding to the two main categories of univariate ordinal distributions. In contrast to previous work, our theoretical developments allow us to provide correspondingly two classes of estimators that are not only computationally efficient but also have strong statistical guarantees.
APA
Suggala, A.S., Yang, E. & Ravikumar, P.. (2017). Ordinal Graphical Models: A Tale of Two Approaches. Proceedings of the 34th International Conference on Machine Learning, in Proceedings of Machine Learning Research 70:3260-3269 Available from http://proceedings.mlr.press/v70/suggala17a.html .

Related Material