Parameter estimation for Gibbs distributions
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arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2020
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| _version_ | 1866909562278248448 |
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| author | Harris, David G. Kolmogorov, Vladimir |
| author_facet | Harris, David G. Kolmogorov, Vladimir |
| contents | We consider Gibbs distributions, which are families of probability distributions over a discrete space $Ω$ with probability mass function of the form $μ^Ω_β(ω) \propto e^{βH(ω)}$ for $β$ in an interval $[β_{\min}, β_{\max}]$ and $H( ω) \in \{0 \} \cup [1, n]$. The partition function is the normalization factor $Z(β)=\sum_{ω\inΩ}e^{βH(ω)}$.
Two important parameters of these distributions are the log partition ratio $q = \log \tfrac{Z(β_{\max})}{Z(β_{\min})}$ and the counts $c_x = |H^{-1}(x)|$. These are correlated with system parameters in a number of physical applications and sampling algorithms.
Our first main result is to estimate the counts $c_x$ using roughly $\tilde O( \frac{q}{\varepsilon^2})$ samples for general Gibbs distributions and $\tilde O( \frac{n^2}{\varepsilon^2} )$ samples for integer-valued distributions (ignoring some second-order terms and parameters), and we show this is optimal up to logarithmic factors. We illustrate with improved algorithms for counting connected subgraphs, independent sets, and perfect matchings.
As a key subroutine, we also develop algorithms to compute the partition function $Z$ using $\tilde O(\frac{q}{\varepsilon^2})$ samples for general Gibbs distributions and using $\tilde O(\frac{n^2}{\varepsilon^2})$ samples for integer-valued distributions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2007_10824 |
| institution | arXiv |
| publishDate | 2020 |
| record_format | arxiv |
| spellingShingle | Parameter estimation for Gibbs distributions Harris, David G. Kolmogorov, Vladimir Data Structures and Algorithms Computational Complexity Discrete Mathematics Probability We consider Gibbs distributions, which are families of probability distributions over a discrete space $Ω$ with probability mass function of the form $μ^Ω_β(ω) \propto e^{βH(ω)}$ for $β$ in an interval $[β_{\min}, β_{\max}]$ and $H( ω) \in \{0 \} \cup [1, n]$. The partition function is the normalization factor $Z(β)=\sum_{ω\inΩ}e^{βH(ω)}$. Two important parameters of these distributions are the log partition ratio $q = \log \tfrac{Z(β_{\max})}{Z(β_{\min})}$ and the counts $c_x = |H^{-1}(x)|$. These are correlated with system parameters in a number of physical applications and sampling algorithms. Our first main result is to estimate the counts $c_x$ using roughly $\tilde O( \frac{q}{\varepsilon^2})$ samples for general Gibbs distributions and $\tilde O( \frac{n^2}{\varepsilon^2} )$ samples for integer-valued distributions (ignoring some second-order terms and parameters), and we show this is optimal up to logarithmic factors. We illustrate with improved algorithms for counting connected subgraphs, independent sets, and perfect matchings. As a key subroutine, we also develop algorithms to compute the partition function $Z$ using $\tilde O(\frac{q}{\varepsilon^2})$ samples for general Gibbs distributions and using $\tilde O(\frac{n^2}{\varepsilon^2})$ samples for integer-valued distributions. |
| title | Parameter estimation for Gibbs distributions |
| topic | Data Structures and Algorithms Computational Complexity Discrete Mathematics Probability |
| url | https://arxiv.org/abs/2007.10824 |