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The nature of dependence between random variables has always been the subject of many statistical problems for over a century. Yet today, there is a great deal of research on this topic, especially focusing on the analysis of nonlinearity. Shannon mutual information has been considered to be the most comprehensive measure of dependence for evaluating total dependence, and several methods have been suggested for discerning the linear and nonlinear components of dependence between two variables. We, in this study, propose employing the Rényi and Tsallis mutual information measures for measuring total dependence because of their parametric nature. We first use a residual analysis in order to remove linear dependence between the variables, and then we compare the Rényi and Tsallis mutual information measures of the original data with that the lacking linear component to determine the degree of nonlinearity. A comparison against the values of the Shannon mutual information measure is also provided. Finally, we apply our method to the environmental Kuznets curve (EKC) and demonstrate the validity of the EKC hypothesis for Eastern Asian and Asia-Pacific countries. [ABSTRACT FROM AUTHOR]

We consider a novel application of inverse reinforcement learning with behavioral economics constraints to model, learn and predict the commenting behavior of YouTube viewers. Each group of users is modeled as a rationally inattentive Bayesian agent which solves a contextual bandit problem. Our methodology integrates three key components. First, to identify distinct commenting patterns, we use deep embedded clustering to estimate framing information (essential extrinsic features) that clusters users into distinct groups. Second, we present an inverse reinforcement learning algorithm that uses Bayesian revealed preferences to test for rationality: does there exist a utility function that rationalizes the given data, and if yes, can it be used to predict commenting behavior? Finally, we impose behavioral economics constraints stemming from rational inattention to characterize the attention span of groups of users. The test imposes a Rényi mutual information cost constraint which impacts how the agent can select attention strategies to maximize their expected utility. After a careful analysis of a massive YouTube dataset, our surprising result is that in most YouTube user groups, the commenting behavior is consistent with optimizing a Bayesian utility with rationally inattentive constraints. The paper also highlights how the rational inattention model can accurately predict commenting behavior. The massive YouTube dataset and analysis used in this paper are available on GitHub and completely reproducible. [ABSTRACT FROM AUTHOR]

The analogues of Arimoto’s definition of conditional Rényi entropy and Rényi mutual information are explored for abstract alphabets. These quantities, although dependent on the reference measure, have some useful properties similar to those known in the discrete setting. In addition to laying out some such basic properties and the relations to Rényi divergences, the relationships between the families of mutual informations defined by Sibson, Augustin-Csiszár, and Lapidoth-Pfister, as well as the corresponding capacities, are explored.

Motivated by a horse betting problem, a new conditional Rényi divergence is introduced. It is compared with the conditional Rényi divergences that appear in the definitions of the dependence measures by Csiszár and Sibson, and the properties of all three are studied with emphasis on their behavior under data processing. In the same way that Csiszár’s and Sibson’s conditional divergence lead to the respective dependence measures, so does the new conditional divergence lead to the Lapidoth−Pfister mutual information. Moreover, the new conditional divergence is also related to the Arimoto−Rényi conditional entropy and to Arimoto’s measure of dependence. In the second part of the paper, the horse betting problem is analyzed where, instead of Kelly’s expected log-wealth criterion, a more general family of power-mean utility functions is considered. The key role in the analysis is played by the Rényi divergence, and in the setting where the gambler has access to side information, the new conditional Rényi divergence is key. The setting with side information also provides another operational meaning to the Lapidoth−Pfister mutual information. Finally, a universal strategy for independent and identically distributed races is presented that—without knowing the winning probabilities or the parameter of the utility function—asymptotically maximizes the gambler’s utility function.

The analogues of Arimoto's definition of conditional Rényi entropy and Rényi mutual information are explored for abstract alphabets. These quantities, although dependent on the reference measure, have some useful properties similar to those known in the discrete setting. In addition to laying out some such basic properties and the relations to Rényi divergences, the relationships between the families of mutual informations defined by Sibson, Augustin-Csiszár, and Lapidoth-Pfister, as well as the corresponding capacities, are explored. [ABSTRACT FROM AUTHOR]

Motivated by a horse betting problem, a new conditional Rényi divergence is introduced. It is compared with the conditional Rényi divergences that appear in the definitions of the dependence measures by Csiszár and Sibson, and the properties of all three are studied with emphasis on their behavior under data processing. In the same way that Csiszár's and Sibson's conditional divergence lead to the respective dependence measures, so does the new conditional divergence lead to the Lapidoth–Pfister mutual information. Moreover, the new conditional divergence is also related to the Arimoto–Rényi conditional entropy and to Arimoto's measure of dependence. In the second part of the paper, the horse betting problem is analyzed where, instead of Kelly's expected log-wealth criterion, a more general family of power-mean utility functions is considered. The key role in the analysis is played by the Rényi divergence, and in the setting where the gambler has access to side information, the new conditional Rényi divergence is key. The setting with side information also provides another operational meaning to the Lapidoth–Pfister mutual information. Finally, a universal strategy for independent and identically distributed races is presented that—without knowing the winning probabilities or the parameter of the utility function—asymptotically maximizes the gambler's utility function. [ABSTRACT FROM AUTHOR]