Partition function approach to non-Gaussian likelihoods: macrocanonical partitions and replicating Markov-chains
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| Main Authors: | , , , , |
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| Format: | Preprint |
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2023
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| _version_ | 1866912094457167872 |
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| author | Herzog, Maximilian Philipp von Campe, Heinrich Kuntz, Rebecca Maria Röver, Lennart Schäfer, Björn Malte |
| author_facet | Herzog, Maximilian Philipp von Campe, Heinrich Kuntz, Rebecca Maria Röver, Lennart Schäfer, Björn Malte |
| contents | Monte-Carlo techniques are standard numerical tools for exploring non-Gaussian and multivariate likelihoods. Many variants of the original Metropolis-Hastings algorithm have been proposed to increase the sampling efficiency. Motivated by Ensemble Monte Carlo we allow the number of Markov chains to vary by exchanging particles with a reservoir, controlled by a parameter analogous to a chemical potential $μ$, which effectively establishes a random process that samples microstates from a macrocanonical instead of a canonical ensemble. In this paper, we develop the theory of macrocanonical sampling for statistical inference on the basis of Bayesian macrocanonical partition functions, thereby bringing to light the relations between information-theoretical quantities and thermodynamic properties. Furthermore, we propose an algorithm for macrocanonical sampling, $\texttt{Avalanche Sampling}$, and apply it to various toy problems as well as the likelihood on the cosmological parameters $Ω_m$ and $w$ on the basis of data from the supernova distance redshift relation. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2311_16218 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Partition function approach to non-Gaussian likelihoods: macrocanonical partitions and replicating Markov-chains Herzog, Maximilian Philipp von Campe, Heinrich Kuntz, Rebecca Maria Röver, Lennart Schäfer, Björn Malte Cosmology and Nongalactic Astrophysics Instrumentation and Methods for Astrophysics Data Analysis, Statistics and Probability Monte-Carlo techniques are standard numerical tools for exploring non-Gaussian and multivariate likelihoods. Many variants of the original Metropolis-Hastings algorithm have been proposed to increase the sampling efficiency. Motivated by Ensemble Monte Carlo we allow the number of Markov chains to vary by exchanging particles with a reservoir, controlled by a parameter analogous to a chemical potential $μ$, which effectively establishes a random process that samples microstates from a macrocanonical instead of a canonical ensemble. In this paper, we develop the theory of macrocanonical sampling for statistical inference on the basis of Bayesian macrocanonical partition functions, thereby bringing to light the relations between information-theoretical quantities and thermodynamic properties. Furthermore, we propose an algorithm for macrocanonical sampling, $\texttt{Avalanche Sampling}$, and apply it to various toy problems as well as the likelihood on the cosmological parameters $Ω_m$ and $w$ on the basis of data from the supernova distance redshift relation. |
| title | Partition function approach to non-Gaussian likelihoods: macrocanonical partitions and replicating Markov-chains |
| topic | Cosmology and Nongalactic Astrophysics Instrumentation and Methods for Astrophysics Data Analysis, Statistics and Probability |
| url | https://arxiv.org/abs/2311.16218 |