A Family of Computationally Efficient and Simple Estimators for Unnormalized Statistical Models

Miika Pihlaja, Michael Gutmann, Aapo Hyvärinen
Proceedings of the 26th Conference on Uncertainty in Artificial Intelligence, PMLR R8:441-448, 2010.

Abstract

We introduce a new family of estimators for unnormalized statistical models. Our fam- ily of estimators is parameterized by two nonlinear functions and uses a single sam- ple from an auxiliary distribution, general- izing Maximum Likelihood Monte Carlo esti- mation of Geyer and Thompson (1992). The family is such that we can estimate the parti- tion function like any other parameter in the model. The estimation is done by optimiz- ing an algebraically simple, well defined ob- jective function, which allows for the use of dedicated optimization methods. We estab- lish consistency of the estimator family and give an expression for the asymptotic covari- ance matrix, which enables us to further an- alyze the influence of the nonlinearities and the auxiliary density on estimation perfor- mance. Some estimators in our family are particularly stable for a wide range of auxil- iary densities. Interestingly, a specific choice of the nonlinearity establishes a connection between density estimation and classification by nonlinear logistic regression. Finally, the optimal amount of auxiliary samples relative to the given amount of the data is consid- ered from the perspective of computational efficiency.

Cite this Paper

BibTeX
@InProceedings{pmlr-vR8-pihlaja10a, title = {A Family of Computationally Efficient and Simple Estimators for Unnormalized Statistical Models}, author = {Pihlaja, Miika and Gutmann, Michael and Hyv{\"a}rinen, Aapo}, booktitle = {Proceedings of the 26th Conference on Uncertainty in Artificial Intelligence}, pages = {441--448}, year = {2010}, editor = {Grünwald, Peter and Spirtes, Peter}, volume = {R8}, series = {Proceedings of Machine Learning Research}, month = {08--11 Jul}, publisher = {PMLR}, pdf = {https://raw.githubusercontent.com/mlresearch/r8/main/assets/pihlaja10a/pihlaja10a.pdf}, url = {https://proceedings.mlr.press/r8/pihlaja10a.html}, abstract = {We introduce a new family of estimators for unnormalized statistical models. Our fam- ily of estimators is parameterized by two nonlinear functions and uses a single sam- ple from an auxiliary distribution, general- izing Maximum Likelihood Monte Carlo esti- mation of Geyer and Thompson (1992). The family is such that we can estimate the parti- tion function like any other parameter in the model. The estimation is done by optimiz- ing an algebraically simple, well defined ob- jective function, which allows for the use of dedicated optimization methods. We estab- lish consistency of the estimator family and give an expression for the asymptotic covari- ance matrix, which enables us to further an- alyze the influence of the nonlinearities and the auxiliary density on estimation perfor- mance. Some estimators in our family are particularly stable for a wide range of auxil- iary densities. Interestingly, a specific choice of the nonlinearity establishes a connection between density estimation and classification by nonlinear logistic regression. Finally, the optimal amount of auxiliary samples relative to the given amount of the data is consid- ered from the perspective of computational efficiency.}, note = {Reissued by PMLR on 04 October 2026.} }
Endnote
%0 Conference Paper %T A Family of Computationally Efficient and Simple Estimators for Unnormalized Statistical Models %A Miika Pihlaja %A Michael Gutmann %A Aapo Hyvärinen %B Proceedings of the 26th Conference on Uncertainty in Artificial Intelligence %C Proceedings of Machine Learning Research %D 2010 %E Peter Grünwald %E Peter Spirtes %F pmlr-vR8-pihlaja10a %I PMLR %P 441--448 %U https://proceedings.mlr.press/r8/pihlaja10a.html %V R8 %X We introduce a new family of estimators for unnormalized statistical models. Our fam- ily of estimators is parameterized by two nonlinear functions and uses a single sam- ple from an auxiliary distribution, general- izing Maximum Likelihood Monte Carlo esti- mation of Geyer and Thompson (1992). The family is such that we can estimate the parti- tion function like any other parameter in the model. The estimation is done by optimiz- ing an algebraically simple, well defined ob- jective function, which allows for the use of dedicated optimization methods. We estab- lish consistency of the estimator family and give an expression for the asymptotic covari- ance matrix, which enables us to further an- alyze the influence of the nonlinearities and the auxiliary density on estimation perfor- mance. Some estimators in our family are particularly stable for a wide range of auxil- iary densities. Interestingly, a specific choice of the nonlinearity establishes a connection between density estimation and classification by nonlinear logistic regression. Finally, the optimal amount of auxiliary samples relative to the given amount of the data is consid- ered from the perspective of computational efficiency. %Z Reissued by PMLR on 04 October 2026.
APA
Pihlaja, M., Gutmann, M. & Hyvärinen, A.. (2010). A Family of Computationally Efficient and Simple Estimators for Unnormalized Statistical Models. Proceedings of the 26th Conference on Uncertainty in Artificial Intelligence, in Proceedings of Machine Learning Research R8:441-448 Available from https://proceedings.mlr.press/r8/pihlaja10a.html. Reissued by PMLR on 04 October 2026.

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