Unnormalized Measures in Information Theory
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Information theory is built on probability measures and by definition a probability measure has total mass 1. Probability measures are used to model uncertainty, and one may ask how important it is that the total mass is one. We claim that the main reason to normalize measures is that probability measures are related to codes via Kraft's inequality. Using a minimum description length approach to statistics we will demonstrate with that measures that are not normalized require a new interpretation that we will call the Poisson interpretation. With the Poisson interpretation many problems can be simplified. The focus will shift from from probabilities to mean values. We give examples of improvements of test procedures, improved inequalities, simplified algorithms, new projection results, and improvements in our description of quantum systems.
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