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Register a new userEquational Axioms for Expected Value Operators
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Jan A. Bergstra
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An equational axiomatisation of probability functions for one-dimensional event spaces in the language of signed meadows is expanded with conditional values. Conditional values constitute a so-called signed vector meadow. In the presence of a probability function, equational axioms are provided for expected value, variance, covariance, and correlation squared, each defined for conditional values. Finite support summation is introduced as a binding operator on meadows which simplifies formulating requirements on probability mass functions with finite support. Conditional values are related to probability mass functions and to random variables. The definitions are reconsidered in a finite dimensional setting.
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A wide range of intuitionistic type theories may be presented as equational theories within a logical framework. This method was formulated by Per Martin-L{o}f in the mid-1980s and further developed by Uemura, who used it to prove an initiality result for a class of models. Herein is presented a logical framework for type theories that includes an extensional equality type so that a type theory may be given by a signature of constants. The framework is illustrated by a number of examples of type-theoretic concepts, including identity and equality types, and a hierarchy of universes.
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