Existence of an Unbiased Entropy Estimator for the Special Bernoulli Measure
Let \(\Omega = A^{N}\) be a space of right-sided infinite sequences drawn from a finite alphabet \(A = \{0,1\}\), \(N = \{1,2,\dots \}\), \[\label{rho} \rho(\boldsymbol{x},\boldsymbol{y}) = \sum_{k=1}^{\infty}|x_{k} - y_{k}|2^{-k} \] a metric on \(\Omega = A^{N}\), and \(\mu\) is a probability m...
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| Format: | Article |
| Language: | English |
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Yaroslavl State University
2017-10-01
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| Series: | Моделирование и анализ информационных систем |
| Subjects: | |
| Online Access: | https://www.mais-journal.ru/jour/article/view/578 |
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| Summary: | Let \(\Omega = A^{N}\) be a space of right-sided infinite sequences drawn from a finite alphabet \(A = \{0,1\}\), \(N = \{1,2,\dots \}\), \[\label{rho} \rho(\boldsymbol{x},\boldsymbol{y}) = \sum_{k=1}^{\infty}|x_{k} - y_{k}|2^{-k} \] a metric on \(\Omega = A^{N}\), and \(\mu\) is a probability measure on \(\Omega\). Let \(\boldsymbol{\xi_0}, \boldsymbol{\xi_1}, \dots, \boldsymbol{\xi_n}\) be independent identically distributed points on \(\Omega\). We study the estimator \(\eta_n^{(k)}(\gamma)\) of the reciprocal of the entropy \(1/h\) that are defined as\[ \label{etan} \eta_n^{(k)}(\gamma) = k \left(r_{n}^{(k)}(\gamma) - r_{n}^{(k+1)}(\gamma)\right),\] where \[\label{def_r} r_n^{(k)}(\gamma) = \frac{1}{n+1}\sum_{j=0}^{n} \gamma\left(\min_{i:i \neq j} {^{(k)}} \rho(\boldsymbol{\xi_{i}}, \boldsymbol{\xi_{j}})\right), \] \(\min ^{(k)}\{X_1,\dots,X_N\}= X_k\), if \(X_1\leq X_2\leq \dots\leq X_N\). The number \(k\) and the function \(\gamma(t)\) are auxiliary parameters.The main result of this paper isTheorem. Let \(\mu\) be the Bernoulli measure with probabilities \(p_0,p_1>0\), \(p_0+p_1=1\), \(p_0=p_1^2\). There exists a function \(\gamma(t)\) such that \[E\eta_n^{(k)}(\gamma) = \frac1h.\] |
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| ISSN: | 1818-1015 2313-5417 |