Ideal hypothesis testing and algorithmic information transfer
Bruno Bauwens · Ghent University Academic Bibliography (Ghent University) · 2009
In this talk we study influence testing for two discrete time series of equal length, by defining two types of universal influence-free enumerable semimeasures.We prove a coding result that characterizes the semimeasure approximately by a variant of conditional prefix-free Kolmogorov complexity, and describe how influence tests in engineering literature can be considered as approximations of these ideal definitions.We show that these have some nice additive properties.The results shed light on the much more involved case of general ideal hypothesis testing, where many questions are left open.In statistics a simple hypothesis is defined as a set of logical statements that allow the inference of a unique semimeasure over the set of all a priori possible outcomes of an experiment.Assume that two simple hypothesis have semimeasures P 0 and P 1 .The statistical test d(x) = P 1 (x)/P 0 (x) has an optimal power for any significance level [5].Composite tests are tests that specify a set S of semimeasures.Multiplicative dominance defines a partial order on the semimeasures, and in some cases the enumerable semimeasures in S have a maximal element, which is called the the universal element of the hypothesis.The likelihood-ratio test can now define in the same way notions of significance and power, which under some assumptions, can have the same meaning as traditional significances and powers.The hypothesis of a time series x being influence-free from another time series y with equal length l(x) = l(y) = n, is a composite hypothesis.We show that with this hypothesis there corresponds a universal enumerable length conditional semimeasure, whose logarithm is approximately