Above are papers related to the following paper: [Paper_ID] Paper #1110628. 	[Title] Mutual Information as a Bayesian Measure of Independence [Abstract] 0.0 Abstract. The problem of hypothesis testing is examined from both the historical and the Bayesian points of view in the case that sampling is from an underlying joint probability distribution and the hypotheses tested for are those of independence and dependence of the underlying distribution. Exact results for the Bayesian method are provided. Asymptotic Bayesian results and historical method quantities are compared, and historical method quantities are interpreted in terms of clearly defined Bayesian quantities. The asymptotic Bayesian test relies upon a statistic that is predominantly mutual information. Problems of hypothesis testing arise ubiquitously in situations where observed data is produced by an unknown process and the question is asked From what process did this observed data arise? Historically, the hypothesis testing problem is approached from the point of view of sampling, whereby several fixed hypotheses to be tested for are given, and all measures of the test and its quality are found directly from the likelihood, i.e. by what amounts to sampling the likelihood [2] [3]. (To be specific, a hypothesis is a set of possible parameter vectors, each parameter vector completely specifying a sampling distribution. A simple hypothesis is a hypothesis set that contains one parameter vector. A composite hypothesis occurs when the (nonempty) hypothesis set is not a single parameter vector.) Generally, the test procedure chooses as true the hypothesis that gives the largest test value, although the notion of procedure is not specific and may refer to any method for choosing the hypothesis given the test values. Since it is of interest to quantify the quality of the test, a level of significance is generated, this level being the probability that, under the chosen hypothesis and test procedure, an incorrect hypothesis choice is made. The significance is generated using the sampling distribution, or likelihood. For simple hypotheses the level of significance is found using the single parameter value of the hypothesis. When a test is applied in the case of a composite hypothesis, a size for the test is found that is given by the supremum probability (ranging over the parameter vectors in the hypothesis set) that under the chosen Please summarise all the given information.

Above are papers related to the following paper: [Paper_ID] Paper #180301. 	[Title] ESTIMATING FUNCTIONS OF PROBABILITY DISTRIBUTIONS FROM A FINITE SET OF SAMPLES Part II: Bayes Estimators [Abstract] This paper is the second in a series of two on the problem of estimating a function of a probability distribution from a finite set of samples of that distribution. In the first paper 1 , the Bayes estimator for a function of a probability distribution was introduced, the optimal properties of the Bayes estimator were discussed, and the Bayes and frequency-counts estimators for the Sh-annon entropy were derived and graphically contrasted. In the current paper the analysis of the first paper is extended by the derivation of Bayes estimators for several other functions of interest in statistics and information theory. These functions are (powers of) the mutual information, chi-squared for tests of independence, variance, covariance, and average. Finding Bayes estimators for several of these functions requires extensions to the analytical techniques developed in the first paper, and these extensions form the main body of this paper. This paper extends the analysis in other ways as well, for example by enlarging the class of potential priors beyond the uniform prior assumed in the first paper. In particular, the use of the entropic and Dirichlet priors is considered. Please summarise all the given information.

Paper #1110628 has the title `Mutual Information as a Bayesian Measure of Independence  
` and the abstract `0.0 Abstract. The problem of hypothesis testing is examined from both the historical and the Bayesian points of view in the case that sampling is from an underlying joint probability distribution and the hypotheses tested for are those of independence and dependence of the underlying distribution. Exact results for the Bayesian method are provided. Asymptotic Bayesian results and historical method quantities are compared, and historical method quantities are interpreted in terms of clearly defined Bayesian quantities. The asymptotic Bayesian test relies upon a statistic that is predominantly mutual information. Problems of hypothesis testing arise ubiquitously in situations where observed data is produced by an unknown process and the question is asked From what process did this observed data arise? Historically, the hypothesis testing problem is approached from the point of view of sampling, whereby several fixed hypotheses to be tested for are given, and all measures of the test and its quality are found directly from the likelihood, i.e. by what amounts to sampling the likelihood [2] [3]. (To be specific, a hypothesis is a set of possible parameter vectors, each parameter vector completely specifying a sampling distribution. A simple hypothesis is a hypothesis set that contains one parameter vector. A composite hypothesis occurs when the (nonempty) hypothesis set is not a single parameter vector.) Generally, the test procedure chooses as true the hypothesis that gives the largest test value, although the notion of procedure is not specific and may refer to any method for choosing the hypothesis given the test values. Since it is of interest to quantify the quality of the test, a level of significance is generated, this level being the probability that, under the chosen hypothesis and test procedure, an incorrect hypothesis choice is made. The significance is generated using the sampling distribution, or likelihood. For simple hypotheses the level of significance is found using the single parameter value of the hypothesis. When a test is applied in the case of a composite hypothesis, a size for the test is found that is given by the supremum probability (ranging over the parameter vectors in the hypothesis set) that under the chosen 
`.
Question: Which category should Paper #1110628 be classified as ?
You can select one from ['Neural_Networks', 'Case_Based', 'Theory', 'Reinforcement_Learning', 'Probabilistic_Methods', 'Rule_Learning', 'Genetic_Algorithms'].


Paper #1110628 has the title `Mutual Information as a Bayesian Measure of Independence  
` and the abstract `0.0 Abstract. The problem of hypothesis testing is examined from both the historical and the Bayesian points of view in the case that sampling is from an underlying joint probability distribution and the hypotheses tested for are those of independence and dependence of the underlying distribution. Exact results for the Bayesian method are provided. Asymptotic Bayesian results and historical method quantities are compared, and historical method quantities are interpreted in terms of clearly defined Bayesian quantities. The asymptotic Bayesian test relies upon a statistic that is predominantly mutual information. Problems of hypothesis testing arise ubiquitously in situations where observed data is produced by an unknown process and the question is asked From what process did this observed data arise? Historically, the hypothesis testing problem is approached from the point of view of sampling, whereby several fixed hypotheses to be tested for are given, and all measures of the test and its quality are found directly from the likelihood, i.e. by what amounts to sampling the likelihood [2] [3]. (To be specific, a hypothesis is a set of possible parameter vectors, each parameter vector completely specifying a sampling distribution. A simple hypothesis is a hypothesis set that contains one parameter vector. A composite hypothesis occurs when the (nonempty) hypothesis set is not a single parameter vector.) Generally, the test procedure chooses as true the hypothesis that gives the largest test value, although the notion of procedure is not specific and may refer to any method for choosing the hypothesis given the test values. Since it is of interest to quantify the quality of the test, a level of significance is generated, this level being the probability that, under the chosen hypothesis and test procedure, an incorrect hypothesis choice is made. The significance is generated using the sampling distribution, or likelihood. For simple hypotheses the level of significance is found using the single parameter value of the hypothesis. When a test is applied in the case of a composite hypothesis, a size for the test is found that is given by the supremum probability (ranging over the parameter vectors in the hypothesis set) that under the chosen 
`.
Question: Which category should Paper #1110628 be classified as ?
You can select one from ['Neural_Networks', 'Case_Based', 'Theory', 'Reinforcement_Learning', 'Probabilistic_Methods', 'Rule_Learning', 'Genetic_Algorithms'].

This paper should be classified into: Probabilistic_Methods