Inference for High-dimensional Exponential Family Graphical Models

Jialei Wang, Mladen Kolar
Proceedings of the 19th International Conference on Artificial Intelligence and Statistics, PMLR 51:1042-1050, 2016.

Abstract

Probabilistic graphical models have been widely used to model complex systems and aid scientific discoveries. Most existing work on high-dimensional estimation of exponential family graphical models, including Gaussian graphical models and Ising models, is focused on consistent model selection. However, these results do not characterize uncertainty in the estimated structure and are of limited value to scientists who worry whether their findings will be reproducible and if the estimated edges are present in the model due to random chance. In this paper, we propose a novel estimator for edge parameters in an exponential family graphical models. We prove that the estimator is \sqrtn-consistent and asymptotically Normal. This result allows us to construct confidence intervals for edge parameters, as well as, hypothesis tests. We establish our results under conditions that are typically assumed in the literature for consistent estimation. However, we do not require that the estimator consistently recovers the graph structure. In particular, we prove that the asymptotic distribution of the estimator is robust to model selection mistakes and uniformly valid for a large number of data-generating processes. We illustrate validity of our estimator through extensive simulation studies.

Cite this Paper

BibTeX
@InProceedings{pmlr-v51-wang16g, title = {Inference for High-dimensional Exponential Family Graphical Models}, author = {Wang, Jialei and Kolar, Mladen}, booktitle = {Proceedings of the 19th International Conference on Artificial Intelligence and Statistics}, pages = {1042--1050}, year = {2016}, editor = {Gretton, Arthur and Robert, Christian C.}, volume = {51}, series = {Proceedings of Machine Learning Research}, address = {Cadiz, Spain}, month = {09--11 May}, publisher = {PMLR}, pdf = {http://proceedings.mlr.press/v51/wang16g.pdf}, url = {https://proceedings.mlr.press/v51/wang16g.html}, abstract = {Probabilistic graphical models have been widely used to model complex systems and aid scientific discoveries. Most existing work on high-dimensional estimation of exponential family graphical models, including Gaussian graphical models and Ising models, is focused on consistent model selection. However, these results do not characterize uncertainty in the estimated structure and are of limited value to scientists who worry whether their findings will be reproducible and if the estimated edges are present in the model due to random chance. In this paper, we propose a novel estimator for edge parameters in an exponential family graphical models. We prove that the estimator is \sqrtn-consistent and asymptotically Normal. This result allows us to construct confidence intervals for edge parameters, as well as, hypothesis tests. We establish our results under conditions that are typically assumed in the literature for consistent estimation. However, we do not require that the estimator consistently recovers the graph structure. In particular, we prove that the asymptotic distribution of the estimator is robust to model selection mistakes and uniformly valid for a large number of data-generating processes. We illustrate validity of our estimator through extensive simulation studies.} }
Endnote
%0 Conference Paper %T Inference for High-dimensional Exponential Family Graphical Models %A Jialei Wang %A Mladen Kolar %B Proceedings of the 19th International Conference on Artificial Intelligence and Statistics %C Proceedings of Machine Learning Research %D 2016 %E Arthur Gretton %E Christian C. Robert %F pmlr-v51-wang16g %I PMLR %P 1042--1050 %U https://proceedings.mlr.press/v51/wang16g.html %V 51 %X Probabilistic graphical models have been widely used to model complex systems and aid scientific discoveries. Most existing work on high-dimensional estimation of exponential family graphical models, including Gaussian graphical models and Ising models, is focused on consistent model selection. However, these results do not characterize uncertainty in the estimated structure and are of limited value to scientists who worry whether their findings will be reproducible and if the estimated edges are present in the model due to random chance. In this paper, we propose a novel estimator for edge parameters in an exponential family graphical models. We prove that the estimator is \sqrtn-consistent and asymptotically Normal. This result allows us to construct confidence intervals for edge parameters, as well as, hypothesis tests. We establish our results under conditions that are typically assumed in the literature for consistent estimation. However, we do not require that the estimator consistently recovers the graph structure. In particular, we prove that the asymptotic distribution of the estimator is robust to model selection mistakes and uniformly valid for a large number of data-generating processes. We illustrate validity of our estimator through extensive simulation studies.
RIS
TY - CPAPER TI - Inference for High-dimensional Exponential Family Graphical Models AU - Jialei Wang AU - Mladen Kolar BT - Proceedings of the 19th International Conference on Artificial Intelligence and Statistics DA - 2016/05/02 ED - Arthur Gretton ED - Christian C. Robert ID - pmlr-v51-wang16g PB - PMLR DP - Proceedings of Machine Learning Research VL - 51 SP - 1042 EP - 1050 L1 - http://proceedings.mlr.press/v51/wang16g.pdf UR - https://proceedings.mlr.press/v51/wang16g.html AB - Probabilistic graphical models have been widely used to model complex systems and aid scientific discoveries. Most existing work on high-dimensional estimation of exponential family graphical models, including Gaussian graphical models and Ising models, is focused on consistent model selection. However, these results do not characterize uncertainty in the estimated structure and are of limited value to scientists who worry whether their findings will be reproducible and if the estimated edges are present in the model due to random chance. In this paper, we propose a novel estimator for edge parameters in an exponential family graphical models. We prove that the estimator is \sqrtn-consistent and asymptotically Normal. This result allows us to construct confidence intervals for edge parameters, as well as, hypothesis tests. We establish our results under conditions that are typically assumed in the literature for consistent estimation. However, we do not require that the estimator consistently recovers the graph structure. In particular, we prove that the asymptotic distribution of the estimator is robust to model selection mistakes and uniformly valid for a large number of data-generating processes. We illustrate validity of our estimator through extensive simulation studies. ER -
APA
Wang, J. & Kolar, M.. (2016). Inference for High-dimensional Exponential Family Graphical Models. Proceedings of the 19th International Conference on Artificial Intelligence and Statistics, in Proceedings of Machine Learning Research 51:1042-1050 Available from https://proceedings.mlr.press/v51/wang16g.html.

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