Free energy methods for Bayesian inference: efficient exploration of univariate Gaussian mixture posteriors
Abstract Because of their multimodality, mixture posterior distributions are difficult to sample with standard Markov chain Monte Carlo (MCMC) methods. We propose a strategy to enhance the sampling of MCMC in this context, using a biasing procedure which originates from computational Statistical Phy...
Ausführliche Beschreibung
Gespeichert in:
| Autor*in: |
Chopin, Nicolas [verfasserIn] |
|---|
| Format: |
Artikel |
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| Sprache: |
Englisch |
| Erschienen: |
2011 |
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| Schlagwörter: |
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| Anmerkung: |
© Springer Science+Business Media, LLC 2011 |
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| Übergeordnetes Werk: |
Enthalten in: Statistics and computing - Springer US, 1991, 22(2011), 4 vom: 23. Juni, Seite 897-916 |
|---|---|
| Übergeordnetes Werk: |
volume:22 ; year:2011 ; number:4 ; day:23 ; month:06 ; pages:897-916 |
| Links: |
|---|
| DOI / URN: |
10.1007/s11222-011-9257-9 |
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OLC2033745031 |
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| 520 | |a Abstract Because of their multimodality, mixture posterior distributions are difficult to sample with standard Markov chain Monte Carlo (MCMC) methods. We propose a strategy to enhance the sampling of MCMC in this context, using a biasing procedure which originates from computational Statistical Physics. The principle is first to choose a “reaction coordinate”, that is, a “direction” in which the target distribution is multimodal. In a second step, the marginal log-density of the reaction coordinate with respect to the posterior distribution is estimated; minus this quantity is called “free energy” in the computational Statistical Physics literature. To this end, we use adaptive biasing Markov chain algorithms which adapt their targeted invariant distribution on the fly, in order to overcome sampling barriers along the chosen reaction coordinate. Finally, we perform an importance sampling step in order to remove the bias and recover the true posterior. The efficiency factor of the importance sampling step can easily be estimated a priori once the bias is known, and appears to be rather large for the test cases we considered. A crucial point is the choice of the reaction coordinate. One standard choice (used for example in the classical Wang-Landau algorithm) is minus the log-posterior density. We discuss other choices. We show in particular that the hyper-parameter that determines the order of magnitude of the variance of each component is both a convenient and an efficient reaction coordinate. We also show how to adapt the method to compute the evidence (marginal likelihood) of a mixture model. We illustrate our approach by analyzing two real data sets. | ||
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10.1007/s11222-011-9257-9 doi (DE-627)OLC2033745031 (DE-He213)s11222-011-9257-9-p DE-627 ger DE-627 rakwb eng 004 620 VZ Chopin, Nicolas verfasserin aut Free energy methods for Bayesian inference: efficient exploration of univariate Gaussian mixture posteriors 2011 Text txt rdacontent ohne Hilfsmittel zu benutzen n rdamedia Band nc rdacarrier © Springer Science+Business Media, LLC 2011 Abstract Because of their multimodality, mixture posterior distributions are difficult to sample with standard Markov chain Monte Carlo (MCMC) methods. We propose a strategy to enhance the sampling of MCMC in this context, using a biasing procedure which originates from computational Statistical Physics. The principle is first to choose a “reaction coordinate”, that is, a “direction” in which the target distribution is multimodal. In a second step, the marginal log-density of the reaction coordinate with respect to the posterior distribution is estimated; minus this quantity is called “free energy” in the computational Statistical Physics literature. To this end, we use adaptive biasing Markov chain algorithms which adapt their targeted invariant distribution on the fly, in order to overcome sampling barriers along the chosen reaction coordinate. Finally, we perform an importance sampling step in order to remove the bias and recover the true posterior. The efficiency factor of the importance sampling step can easily be estimated a priori once the bias is known, and appears to be rather large for the test cases we considered. A crucial point is the choice of the reaction coordinate. One standard choice (used for example in the classical Wang-Landau algorithm) is minus the log-posterior density. We discuss other choices. We show in particular that the hyper-parameter that determines the order of magnitude of the variance of each component is both a convenient and an efficient reaction coordinate. We also show how to adapt the method to compute the evidence (marginal likelihood) of a mixture model. We illustrate our approach by analyzing two real data sets. Adaptive biasing force Adaptive biasing potential Adaptive Markov chain Monte Carlo Importance sampling Mixture models Lelièvre, Tony aut Stoltz, Gabriel aut Enthalten in Statistics and computing Springer US, 1991 22(2011), 4 vom: 23. Juni, Seite 897-916 (DE-627)131007963 (DE-600)1087487-2 (DE-576)052732894 0960-3174 nnns volume:22 year:2011 number:4 day:23 month:06 pages:897-916 https://doi.org/10.1007/s11222-011-9257-9 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_OLC SSG-OLC-TEC SSG-OLC-MAT GBV_ILN_70 GBV_ILN_2012 GBV_ILN_4126 AR 22 2011 4 23 06 897-916 |
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10.1007/s11222-011-9257-9 doi (DE-627)OLC2033745031 (DE-He213)s11222-011-9257-9-p DE-627 ger DE-627 rakwb eng 004 620 VZ Chopin, Nicolas verfasserin aut Free energy methods for Bayesian inference: efficient exploration of univariate Gaussian mixture posteriors 2011 Text txt rdacontent ohne Hilfsmittel zu benutzen n rdamedia Band nc rdacarrier © Springer Science+Business Media, LLC 2011 Abstract Because of their multimodality, mixture posterior distributions are difficult to sample with standard Markov chain Monte Carlo (MCMC) methods. We propose a strategy to enhance the sampling of MCMC in this context, using a biasing procedure which originates from computational Statistical Physics. The principle is first to choose a “reaction coordinate”, that is, a “direction” in which the target distribution is multimodal. In a second step, the marginal log-density of the reaction coordinate with respect to the posterior distribution is estimated; minus this quantity is called “free energy” in the computational Statistical Physics literature. To this end, we use adaptive biasing Markov chain algorithms which adapt their targeted invariant distribution on the fly, in order to overcome sampling barriers along the chosen reaction coordinate. Finally, we perform an importance sampling step in order to remove the bias and recover the true posterior. The efficiency factor of the importance sampling step can easily be estimated a priori once the bias is known, and appears to be rather large for the test cases we considered. A crucial point is the choice of the reaction coordinate. One standard choice (used for example in the classical Wang-Landau algorithm) is minus the log-posterior density. We discuss other choices. We show in particular that the hyper-parameter that determines the order of magnitude of the variance of each component is both a convenient and an efficient reaction coordinate. We also show how to adapt the method to compute the evidence (marginal likelihood) of a mixture model. We illustrate our approach by analyzing two real data sets. Adaptive biasing force Adaptive biasing potential Adaptive Markov chain Monte Carlo Importance sampling Mixture models Lelièvre, Tony aut Stoltz, Gabriel aut Enthalten in Statistics and computing Springer US, 1991 22(2011), 4 vom: 23. Juni, Seite 897-916 (DE-627)131007963 (DE-600)1087487-2 (DE-576)052732894 0960-3174 nnns volume:22 year:2011 number:4 day:23 month:06 pages:897-916 https://doi.org/10.1007/s11222-011-9257-9 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_OLC SSG-OLC-TEC SSG-OLC-MAT GBV_ILN_70 GBV_ILN_2012 GBV_ILN_4126 AR 22 2011 4 23 06 897-916 |
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10.1007/s11222-011-9257-9 doi (DE-627)OLC2033745031 (DE-He213)s11222-011-9257-9-p DE-627 ger DE-627 rakwb eng 004 620 VZ Chopin, Nicolas verfasserin aut Free energy methods for Bayesian inference: efficient exploration of univariate Gaussian mixture posteriors 2011 Text txt rdacontent ohne Hilfsmittel zu benutzen n rdamedia Band nc rdacarrier © Springer Science+Business Media, LLC 2011 Abstract Because of their multimodality, mixture posterior distributions are difficult to sample with standard Markov chain Monte Carlo (MCMC) methods. We propose a strategy to enhance the sampling of MCMC in this context, using a biasing procedure which originates from computational Statistical Physics. The principle is first to choose a “reaction coordinate”, that is, a “direction” in which the target distribution is multimodal. In a second step, the marginal log-density of the reaction coordinate with respect to the posterior distribution is estimated; minus this quantity is called “free energy” in the computational Statistical Physics literature. To this end, we use adaptive biasing Markov chain algorithms which adapt their targeted invariant distribution on the fly, in order to overcome sampling barriers along the chosen reaction coordinate. Finally, we perform an importance sampling step in order to remove the bias and recover the true posterior. The efficiency factor of the importance sampling step can easily be estimated a priori once the bias is known, and appears to be rather large for the test cases we considered. A crucial point is the choice of the reaction coordinate. One standard choice (used for example in the classical Wang-Landau algorithm) is minus the log-posterior density. We discuss other choices. We show in particular that the hyper-parameter that determines the order of magnitude of the variance of each component is both a convenient and an efficient reaction coordinate. We also show how to adapt the method to compute the evidence (marginal likelihood) of a mixture model. We illustrate our approach by analyzing two real data sets. Adaptive biasing force Adaptive biasing potential Adaptive Markov chain Monte Carlo Importance sampling Mixture models Lelièvre, Tony aut Stoltz, Gabriel aut Enthalten in Statistics and computing Springer US, 1991 22(2011), 4 vom: 23. Juni, Seite 897-916 (DE-627)131007963 (DE-600)1087487-2 (DE-576)052732894 0960-3174 nnns volume:22 year:2011 number:4 day:23 month:06 pages:897-916 https://doi.org/10.1007/s11222-011-9257-9 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_OLC SSG-OLC-TEC SSG-OLC-MAT GBV_ILN_70 GBV_ILN_2012 GBV_ILN_4126 AR 22 2011 4 23 06 897-916 |
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10.1007/s11222-011-9257-9 doi (DE-627)OLC2033745031 (DE-He213)s11222-011-9257-9-p DE-627 ger DE-627 rakwb eng 004 620 VZ Chopin, Nicolas verfasserin aut Free energy methods for Bayesian inference: efficient exploration of univariate Gaussian mixture posteriors 2011 Text txt rdacontent ohne Hilfsmittel zu benutzen n rdamedia Band nc rdacarrier © Springer Science+Business Media, LLC 2011 Abstract Because of their multimodality, mixture posterior distributions are difficult to sample with standard Markov chain Monte Carlo (MCMC) methods. We propose a strategy to enhance the sampling of MCMC in this context, using a biasing procedure which originates from computational Statistical Physics. The principle is first to choose a “reaction coordinate”, that is, a “direction” in which the target distribution is multimodal. In a second step, the marginal log-density of the reaction coordinate with respect to the posterior distribution is estimated; minus this quantity is called “free energy” in the computational Statistical Physics literature. To this end, we use adaptive biasing Markov chain algorithms which adapt their targeted invariant distribution on the fly, in order to overcome sampling barriers along the chosen reaction coordinate. Finally, we perform an importance sampling step in order to remove the bias and recover the true posterior. The efficiency factor of the importance sampling step can easily be estimated a priori once the bias is known, and appears to be rather large for the test cases we considered. A crucial point is the choice of the reaction coordinate. One standard choice (used for example in the classical Wang-Landau algorithm) is minus the log-posterior density. We discuss other choices. We show in particular that the hyper-parameter that determines the order of magnitude of the variance of each component is both a convenient and an efficient reaction coordinate. We also show how to adapt the method to compute the evidence (marginal likelihood) of a mixture model. We illustrate our approach by analyzing two real data sets. Adaptive biasing force Adaptive biasing potential Adaptive Markov chain Monte Carlo Importance sampling Mixture models Lelièvre, Tony aut Stoltz, Gabriel aut Enthalten in Statistics and computing Springer US, 1991 22(2011), 4 vom: 23. Juni, Seite 897-916 (DE-627)131007963 (DE-600)1087487-2 (DE-576)052732894 0960-3174 nnns volume:22 year:2011 number:4 day:23 month:06 pages:897-916 https://doi.org/10.1007/s11222-011-9257-9 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_OLC SSG-OLC-TEC SSG-OLC-MAT GBV_ILN_70 GBV_ILN_2012 GBV_ILN_4126 AR 22 2011 4 23 06 897-916 |
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Chopin, Nicolas Lelièvre, Tony Stoltz, Gabriel |
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free energy methods for bayesian inference: efficient exploration of univariate gaussian mixture posteriors |
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Free energy methods for Bayesian inference: efficient exploration of univariate Gaussian mixture posteriors |
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Abstract Because of their multimodality, mixture posterior distributions are difficult to sample with standard Markov chain Monte Carlo (MCMC) methods. We propose a strategy to enhance the sampling of MCMC in this context, using a biasing procedure which originates from computational Statistical Physics. The principle is first to choose a “reaction coordinate”, that is, a “direction” in which the target distribution is multimodal. In a second step, the marginal log-density of the reaction coordinate with respect to the posterior distribution is estimated; minus this quantity is called “free energy” in the computational Statistical Physics literature. To this end, we use adaptive biasing Markov chain algorithms which adapt their targeted invariant distribution on the fly, in order to overcome sampling barriers along the chosen reaction coordinate. Finally, we perform an importance sampling step in order to remove the bias and recover the true posterior. The efficiency factor of the importance sampling step can easily be estimated a priori once the bias is known, and appears to be rather large for the test cases we considered. A crucial point is the choice of the reaction coordinate. One standard choice (used for example in the classical Wang-Landau algorithm) is minus the log-posterior density. We discuss other choices. We show in particular that the hyper-parameter that determines the order of magnitude of the variance of each component is both a convenient and an efficient reaction coordinate. We also show how to adapt the method to compute the evidence (marginal likelihood) of a mixture model. We illustrate our approach by analyzing two real data sets. © Springer Science+Business Media, LLC 2011 |
| abstractGer |
Abstract Because of their multimodality, mixture posterior distributions are difficult to sample with standard Markov chain Monte Carlo (MCMC) methods. We propose a strategy to enhance the sampling of MCMC in this context, using a biasing procedure which originates from computational Statistical Physics. The principle is first to choose a “reaction coordinate”, that is, a “direction” in which the target distribution is multimodal. In a second step, the marginal log-density of the reaction coordinate with respect to the posterior distribution is estimated; minus this quantity is called “free energy” in the computational Statistical Physics literature. To this end, we use adaptive biasing Markov chain algorithms which adapt their targeted invariant distribution on the fly, in order to overcome sampling barriers along the chosen reaction coordinate. Finally, we perform an importance sampling step in order to remove the bias and recover the true posterior. The efficiency factor of the importance sampling step can easily be estimated a priori once the bias is known, and appears to be rather large for the test cases we considered. A crucial point is the choice of the reaction coordinate. One standard choice (used for example in the classical Wang-Landau algorithm) is minus the log-posterior density. We discuss other choices. We show in particular that the hyper-parameter that determines the order of magnitude of the variance of each component is both a convenient and an efficient reaction coordinate. We also show how to adapt the method to compute the evidence (marginal likelihood) of a mixture model. We illustrate our approach by analyzing two real data sets. © Springer Science+Business Media, LLC 2011 |
| abstract_unstemmed |
Abstract Because of their multimodality, mixture posterior distributions are difficult to sample with standard Markov chain Monte Carlo (MCMC) methods. We propose a strategy to enhance the sampling of MCMC in this context, using a biasing procedure which originates from computational Statistical Physics. The principle is first to choose a “reaction coordinate”, that is, a “direction” in which the target distribution is multimodal. In a second step, the marginal log-density of the reaction coordinate with respect to the posterior distribution is estimated; minus this quantity is called “free energy” in the computational Statistical Physics literature. To this end, we use adaptive biasing Markov chain algorithms which adapt their targeted invariant distribution on the fly, in order to overcome sampling barriers along the chosen reaction coordinate. Finally, we perform an importance sampling step in order to remove the bias and recover the true posterior. The efficiency factor of the importance sampling step can easily be estimated a priori once the bias is known, and appears to be rather large for the test cases we considered. A crucial point is the choice of the reaction coordinate. One standard choice (used for example in the classical Wang-Landau algorithm) is minus the log-posterior density. We discuss other choices. We show in particular that the hyper-parameter that determines the order of magnitude of the variance of each component is both a convenient and an efficient reaction coordinate. We also show how to adapt the method to compute the evidence (marginal likelihood) of a mixture model. We illustrate our approach by analyzing two real data sets. © Springer Science+Business Media, LLC 2011 |
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Free energy methods for Bayesian inference: efficient exploration of univariate Gaussian mixture posteriors |
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