A Sufficient Statistics Construction of Exponential Family Lévy Measure Densities for Nonparametric Conjugate Models

Robert Finn, Brian Kulis
Proceedings of the Eighteenth International Conference on Artificial Intelligence and Statistics, PMLR 38:250-258, 2015.

Abstract

Conjugate pairs of distributions over infinite dimensional spaces are prominent in machine learning, particularly due to the widespread adoption of Bayesian nonparametric methodologies for a host of models and applications. Much of the existing literature in the learning community focuses on processes possessing some form of computationally tractable conjugacy as is the case for the beta process and the gamma process (and, via normalization, the Dirichlet process). For these processes, conjugacy is proved via statistical machinery tailored to the particular model. We seek to address the problem of obtaining a general construction of prior distributions over infinite dimensional spaces possessing distributional properties amenable to conjugacy. Our result is achieved by generalizing Hjort’s construction of the beta process via appropriate utilization of sufficient statistics for exponential families.

Cite this Paper

BibTeX
@InProceedings{pmlr-v38-finn15, title = {A Sufficient Statistics Construction of Exponential Family {L}\'evy Measure Densities for Nonparametric Conjugate Models}, author = {Finn, Robert and Kulis, Brian}, booktitle = {Proceedings of the Eighteenth International Conference on Artificial Intelligence and Statistics}, pages = {250--258}, year = {2015}, editor = {Lebanon, Guy and Vishwanathan, S. V. N.}, volume = {38}, series = {Proceedings of Machine Learning Research}, address = {San Diego, California, USA}, month = {09--12 May}, publisher = {PMLR}, pdf = {http://proceedings.mlr.press/v38/finn15.pdf}, url = {https://proceedings.mlr.press/v38/finn15.html}, abstract = {Conjugate pairs of distributions over infinite dimensional spaces are prominent in machine learning, particularly due to the widespread adoption of Bayesian nonparametric methodologies for a host of models and applications. Much of the existing literature in the learning community focuses on processes possessing some form of computationally tractable conjugacy as is the case for the beta process and the gamma process (and, via normalization, the Dirichlet process). For these processes, conjugacy is proved via statistical machinery tailored to the particular model. We seek to address the problem of obtaining a general construction of prior distributions over infinite dimensional spaces possessing distributional properties amenable to conjugacy. Our result is achieved by generalizing Hjort’s construction of the beta process via appropriate utilization of sufficient statistics for exponential families.} }
Endnote
%0 Conference Paper %T A Sufficient Statistics Construction of Exponential Family Lévy Measure Densities for Nonparametric Conjugate Models %A Robert Finn %A Brian Kulis %B Proceedings of the Eighteenth International Conference on Artificial Intelligence and Statistics %C Proceedings of Machine Learning Research %D 2015 %E Guy Lebanon %E S. V. N. Vishwanathan %F pmlr-v38-finn15 %I PMLR %P 250--258 %U https://proceedings.mlr.press/v38/finn15.html %V 38 %X Conjugate pairs of distributions over infinite dimensional spaces are prominent in machine learning, particularly due to the widespread adoption of Bayesian nonparametric methodologies for a host of models and applications. Much of the existing literature in the learning community focuses on processes possessing some form of computationally tractable conjugacy as is the case for the beta process and the gamma process (and, via normalization, the Dirichlet process). For these processes, conjugacy is proved via statistical machinery tailored to the particular model. We seek to address the problem of obtaining a general construction of prior distributions over infinite dimensional spaces possessing distributional properties amenable to conjugacy. Our result is achieved by generalizing Hjort’s construction of the beta process via appropriate utilization of sufficient statistics for exponential families.
RIS
TY - CPAPER TI - A Sufficient Statistics Construction of Exponential Family Lévy Measure Densities for Nonparametric Conjugate Models AU - Robert Finn AU - Brian Kulis BT - Proceedings of the Eighteenth International Conference on Artificial Intelligence and Statistics DA - 2015/02/21 ED - Guy Lebanon ED - S. V. N. Vishwanathan ID - pmlr-v38-finn15 PB - PMLR DP - Proceedings of Machine Learning Research VL - 38 SP - 250 EP - 258 L1 - http://proceedings.mlr.press/v38/finn15.pdf UR - https://proceedings.mlr.press/v38/finn15.html AB - Conjugate pairs of distributions over infinite dimensional spaces are prominent in machine learning, particularly due to the widespread adoption of Bayesian nonparametric methodologies for a host of models and applications. Much of the existing literature in the learning community focuses on processes possessing some form of computationally tractable conjugacy as is the case for the beta process and the gamma process (and, via normalization, the Dirichlet process). For these processes, conjugacy is proved via statistical machinery tailored to the particular model. We seek to address the problem of obtaining a general construction of prior distributions over infinite dimensional spaces possessing distributional properties amenable to conjugacy. Our result is achieved by generalizing Hjort’s construction of the beta process via appropriate utilization of sufficient statistics for exponential families. ER -
APA
Finn, R. & Kulis, B.. (2015). A Sufficient Statistics Construction of Exponential Family Lévy Measure Densities for Nonparametric Conjugate Models. Proceedings of the Eighteenth International Conference on Artificial Intelligence and Statistics, in Proceedings of Machine Learning Research 38:250-258 Available from https://proceedings.mlr.press/v38/finn15.html.

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