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第291行： − + − + − 如何进行互信息的贝叶斯估计是一个众所周知的问题+ − − of a joint distribution based on samples of that distribution. The − − of a joint distribution based on samples of that distribution. The − − 联合分布的基础上，该分布的样本。这个 − − first work to do this, which also showed how to do Bayesian estimation of many − − first work to do this, which also showed how to do Bayesian estimation of many − − 第一个工作这样做，这也显示了如何做贝叶斯估计的许多 − − other information-theoretic properties besides mutual information, was <ref>{{cite journal | last1 = Wolpert | first1 = D.H. | last2 = Wolf | first2 = D.R. | year = 1995 | title = Estimating functions of probability distributions from a finite set of samples | journal = Physical Review E | volume = 52 | issue = 6 | pages = 6841–6854 | doi = 10.1103/PhysRevE.52.6841 | pmid = 9964199 | citeseerx = 10.1.1.55.7122 | bibcode = 1995PhRvE..52.6841W }}</ref>. Subsequent researchers have rederived <ref>{{cite journal | last1 = Hutter | first1 = M. | year = 2001 | title = Distribution of Mutual Information | journal = Advances in Neural Information Processing Systems 2001 }}</ref> − − other information-theoretic properties besides mutual information, was . Subsequent researchers have rederived − − 除了互信息之外，还有其他信息论性质。后来的研究人员重新推导出 − − and extended <ref>{{cite journal | last1 = Archer | first1 = E. | last2 = Park | first2 = I.M. | last3 = Pillow | first3 = J. | year = 2013 | title = Bayesian and Quasi-Bayesian Estimators for Mutual Information from Discrete Data | journal = Entropy| volume = 15 | issue = 12 | pages = 1738–1755 | doi = 10.3390/e15051738 | citeseerx = 10.1.1.294.4690 | bibcode = 2013Entrp..15.1738A }}</ref> − − and extended − − 还有延伸 − − this analysis. See <ref>{{cite journal | last1 = Wolpert | first1 = D.H | last2 = DeDeo | first2 = S. | year = 2013 | title = Estimating Functions of Distributions Defined over Spaces of Unknown Size | journal = Entropy | volume = 15 | issue = 12 | pages = 4668–4699 | doi = 10.3390/e15114668 | arxiv = 1311.4548 | bibcode = 2013Entrp..15.4668W }}</ref> − − this analysis. See − − 这个分析。看 − − for a recent paper based on a prior specifically tailored to estimation of mutual − − for a recent paper based on a prior specifically tailored to estimation of mutual − − 在最近的一篇论文中，基于之前专门针对相互性的评估 − − information per se. Besides, recently an estimation method accounting for continuous and multivariate outputs, <math>Y</math>, was proposed in − − information per se. Besides, recently an estimation method accounting for continuous and multivariate outputs, <math>Y</math>, was proposed in − − 信息本身。此外，最近提出了一种考虑连续和多变量输出的估计方法，即数学 y / math − − <ref>{{citation| journal = [[PLOS Computational Biology]]|volume = 15|issue = 7|pages = e1007132|doi = 10.1371/journal.pcbi.1007132|pmid = 31299056|pmc = 6655862|title=Information-theoretic analysis of multivariate single-cell signaling responses|author1= Tomasz Jetka|author2= Karol Nienaltowski|author3= Tomasz Winarski| author4=Slawomir Blonski| author5= Michal Komorowski|year=2019|bibcode = 2019PLSCB..15E7132J|arxiv = 1808.05581}}</ref>. − − . − − . + + + +
→互信息的贝叶斯估计 Bayesian estimation of mutual information
=== 互信息的贝叶斯估计 Bayesian estimation of mutual information ===
=== 互信息的贝叶斯估计 Bayesian estimation of mutual information ===
It is well-understood how to do Bayesian estimation of the mutual information
It is well-understood how to do Bayesian estimation of the mutual information of a joint distribution based on samples of that distribution.
It is well-understood how to do Bayesian estimation of the mutual information
It is well-understood how to do Bayesian estimation of the mutual information of a joint distribution based on samples of that distribution.
如何根据联合分布的样本对联合分布的互信息进行贝叶斯估计，是一个众所周知的问题
The first work to do this, which also showed how to do Bayesian estimation of many other information-theoretic properties besides mutual information, was <ref>{{cite journal | last1 = Wolpert | first1 = D.H. | last2 = Wolf | first2 = D.R. | year = 1995 | title = Estimating functions of probability distributions from a finite set of samples | journal = Physical Review E | volume = 52 | issue = 6 | pages = 6841–6854 | doi = 10.1103/PhysRevE.52.6841 | pmid = 9964199 | citeseerx = 10.1.1.55.7122 | bibcode = 1995PhRvE..52.6841W }}</ref>. Subsequent researchers have rederived <ref>{{cite journal | last1 = Hutter | first1 = M. | year = 2001 | title = Distribution of Mutual Information | journal = Advances in Neural Information Processing Systems 2001 }}</ref>
and extended <ref>{{cite journal | last1 = Archer | first1 = E. | last2 = Park | first2 = I.M. | last3 = Pillow | first3 = J. | year = 2013 | title = Bayesian and Quasi-Bayesian Estimators for Mutual Information from Discrete Data | journal = Entropy| volume = 15 | issue = 12 | pages = 1738–1755 | doi = 10.3390/e15051738 | citeseerx = 10.1.1.294.4690 | bibcode = 2013Entrp..15.1738A }}</ref>this analysis.
See <ref>{{cite journal | last1 = Wolpert | first1 = D.H | last2 = DeDeo | first2 = S. | year = 2013 | title = Estimating Functions of Distributions Defined over Spaces of Unknown Size | journal = Entropy | volume = 15 | issue = 12 | pages = 4668–4699 | doi = 10.3390/e15114668 | arxiv = 1311.4548 | bibcode = 2013Entrp..15.4668W }}</ref>for a recent paper based on a prior specifically tailored to estimation of mutual information per se.
Besides, recently an estimation method accounting for continuous and multivariate outputs, <math>Y</math>, was proposed in <ref>{{citation| journal = [[PLOS Computational Biology]]|volume = 15|issue = 7|pages = e1007132|doi = 10.1371/journal.pcbi.1007132|pmid = 31299056|pmc = 6655862|title=Information-theoretic analysis of multivariate single-cell signaling responses|author1= Tomasz Jetka|author2= Karol Nienaltowski|author3= Tomasz Winarski| author4=Slawomir Blonski| author5= Michal Komorowski|year=2019|bibcode = 2019PLSCB..15E7132J|arxiv = 1808.05581}}</ref>.
=== 独立性假设 Independence assumptions ===
=== 独立性假设 Independence assumptions ===