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Semirings Modeling Confidence and Uncertainty in Speech Recognition

In automatic speech recognition, the confidence in a recognition result, i.e., the "degree of belief that a user uttered a specific phrase", can be computed as Bayesian posterior probability of the specific phrase, given a language model defining the Bayesian prior, and the observed utterance. 
 
This paper studies the mathematical structures of this confidence measure and connects it to a reasonable measure for uncertainty. A thorough analysis of logical operations leads to semiring structures for these measures. The two arising semirings turn out to be isomorphic, and also isomorphic to the well-known max-plus algebra. It is proved that, for any two of these semirings, the set of semiring isomorphisms between them is a one-parameter family.