Geometric convergence of elliptical slice sampling

Viacheslav Natarovskii, Daniel Rudolf, Björn Sprungk
Proceedings of the 38th International Conference on Machine Learning, PMLR 139:7969-7978, 2021.

Abstract

For Bayesian learning, given likelihood function and Gaussian prior, the elliptical slice sampler, introduced by Murray, Adams and MacKay 2010, provides a tool for the construction of a Markov chain for approximate sampling of the underlying posterior distribution. Besides of its wide applicability and simplicity its main feature is that no tuning is necessary. Under weak regularity assumptions on the posterior density we show that the corresponding Markov chain is geometrically ergodic and therefore yield qualitative convergence guarantees. We illustrate our result for Gaussian posteriors as they appear in Gaussian process regression in a fully Gaussian scenario, which for example is exhibited in Gaussian process regression, as well as in a setting of a multi-modal distribution. Remarkably, our numerical experiments indicate a dimension-independent performance of elliptical slice sampling even in situations where our ergodicity result does not apply.

Cite this Paper

BibTeX
@InProceedings{pmlr-v139-natarovskii21a, title = {Geometric convergence of elliptical slice sampling}, author = {Natarovskii, Viacheslav and Rudolf, Daniel and Sprungk, Bj{\"o}rn}, booktitle = {Proceedings of the 38th International Conference on Machine Learning}, pages = {7969--7978}, year = {2021}, editor = {Meila, Marina and Zhang, Tong}, volume = {139}, series = {Proceedings of Machine Learning Research}, month = {18--24 Jul}, publisher = {PMLR}, pdf = {http://proceedings.mlr.press/v139/natarovskii21a/natarovskii21a.pdf}, url = {https://proceedings.mlr.press/v139/natarovskii21a.html}, abstract = {For Bayesian learning, given likelihood function and Gaussian prior, the elliptical slice sampler, introduced by Murray, Adams and MacKay 2010, provides a tool for the construction of a Markov chain for approximate sampling of the underlying posterior distribution. Besides of its wide applicability and simplicity its main feature is that no tuning is necessary. Under weak regularity assumptions on the posterior density we show that the corresponding Markov chain is geometrically ergodic and therefore yield qualitative convergence guarantees. We illustrate our result for Gaussian posteriors as they appear in Gaussian process regression in a fully Gaussian scenario, which for example is exhibited in Gaussian process regression, as well as in a setting of a multi-modal distribution. Remarkably, our numerical experiments indicate a dimension-independent performance of elliptical slice sampling even in situations where our ergodicity result does not apply.} }
Endnote
%0 Conference Paper %T Geometric convergence of elliptical slice sampling %A Viacheslav Natarovskii %A Daniel Rudolf %A Björn Sprungk %B Proceedings of the 38th International Conference on Machine Learning %C Proceedings of Machine Learning Research %D 2021 %E Marina Meila %E Tong Zhang %F pmlr-v139-natarovskii21a %I PMLR %P 7969--7978 %U https://proceedings.mlr.press/v139/natarovskii21a.html %V 139 %X For Bayesian learning, given likelihood function and Gaussian prior, the elliptical slice sampler, introduced by Murray, Adams and MacKay 2010, provides a tool for the construction of a Markov chain for approximate sampling of the underlying posterior distribution. Besides of its wide applicability and simplicity its main feature is that no tuning is necessary. Under weak regularity assumptions on the posterior density we show that the corresponding Markov chain is geometrically ergodic and therefore yield qualitative convergence guarantees. We illustrate our result for Gaussian posteriors as they appear in Gaussian process regression in a fully Gaussian scenario, which for example is exhibited in Gaussian process regression, as well as in a setting of a multi-modal distribution. Remarkably, our numerical experiments indicate a dimension-independent performance of elliptical slice sampling even in situations where our ergodicity result does not apply.
APA
Natarovskii, V., Rudolf, D. & Sprungk, B.. (2021). Geometric convergence of elliptical slice sampling. Proceedings of the 38th International Conference on Machine Learning, in Proceedings of Machine Learning Research 139:7969-7978 Available from https://proceedings.mlr.press/v139/natarovskii21a.html.

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