There are several estimators of conditional probability from observed frequencies of features. In this paper, we propose using the lower limit of confidence interval on posterior distribution determined by the observed frequencies to ascertain conditional probability. In our experiments, this method outperformed other popular estimators.
Church and Hanks @cite_8 , proposed the concept of mutual information to measure the association between words. This method solves low-frequency problem, because infrequent features cannot sum up to a large mutual information quantity. However, if the true strength of the relationship between two features can be regarded as conditional probability, it may not be an appropriate measure to use because mutual information is symmetric towards two features, whereas conditional probability is asymmetric.
@cite_8
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We demonstrate that it is possible to approximate the mutual information arbitrarily closely in probability by calculating the relative frequencies on appropriate partitions and achieving conditional independence on the rectangles of which the partitions are made. Empirical results, including a comparison with maximum-likelihood estimators, are presented. How should one quantify the strength of association between two random variables without bias for relationships of a specific form? Despite its conceptual simplicity, this notion of statistical “equitability” has yet to receive a definitive mathematical formalization. Here we argue that equitability is properly formalized by a self-consistency condition closely related to Data Processing Inequality. Mutual information, a fundamental quantity in information theory, is shown to satisfy this equitability criterion. These findings are at odds with the recent work of [Reshef DN, et al (2011) Science 334(6062):1518–1524], which proposed an alternative definition of equitability and introduced a new statistic, the “maximal information coefficient” (MIC), said to satisfy equitability in contradistinction to mutual information. These conclusions, however, were supported only with limited simulation evidence, not with mathematical arguments. Upon revisiting these claims, we prove that the mathematical definition of equitability proposed by cannot be satisfied by any (nontrivial) dependence measure. We also identify artifacts in the reported simulation evidence. When these artifacts are removed, estimates of mutual information are found to be more equitable than estimates of MIC. Mutual information is also observed to have consistently higher statistical power than MIC. We conclude that estimating mutual information provides a natural (and often practical) way to equitably quantify statistical associations in large datasets. Wyner defined the notion of common information of two discrete random variables as the minimum of I(W; X, Y) where W induces conditional independence between X and Y. Its generalization to multiple dependent random variables revealed a surprising monotone property in the number of variables. Motivated by this monotonicity property, this paper explores the application of Wyner's common information to inference problems and its connection with other performance metrics. A central question is that under what conditions Wyner's common information captures the entire information contained in the observations about the inference object under a simple Bayesian model. For infinitely exchangeable random variables, it is shown using the de Finetti-Hewitt-Savage theorem that the common information is asymptotically equal to the information of the inference object. For finite exchangeable random variables, such conclusion is no longer true even for infinitely extendable sequences. However, for some special cases, including both the binary and the Gaussian cases, concrete connection between common information and inference performance metrics can be established even for finite samples. The term word association is used in a very particular sense in the psycholinguistic literature. (Generally speaking, subjects respond quicker than normal to the word nurse if it follows a highly associated word such as doctor. ) We will extend the term to provide the basis for a statistical description of a variety of interesting linguistic phenomena, ranging from semantic relations of the doctor nurse type (content word content word) to lexico-syntactic co-occurrence constraints between verbs and prepositions (content word function word). This paper will propose an objective measure based on the information theoretic notion of mutual information, for estimating word association norms from computer readable corpora. (The standard method of obtaining word association norms, testing a few thousand subjects on a few hundred words, is both costly and unreliable.) The proposed measure, the association ratio, estimates word association norms directly from computer readable corpora, making it possible to estimate norms for tens of thousands of words.