Towards a Frequentist Interpretation of Sets of Measures

We explore an objective, frequentist-related interpretation for a set of measures M such as would determine upper and lower envelopes; M also specifies the classical frequentist concept of a compound hypothesis. However, in contrast to the compound hypothesis case, in which there is a true measure µθ0 ∈M that is assumed either unknown or random selected, we do not believe that any single measure is the true description for the random phenomena in question. Rather, it is the whole set M, itself, that is the appropriate imprecise probabilistic description. Envelope models have hitherto been used almost exclusively in subjective settings to model the uncertainty or strength of belief of individuals or groups. Our interest in these imprecise probability representations is as mathematical models for those objective frequentist phenomena of engineering and scientific significance where what is known may be substantial, but relative frequencies, nonetheless, lack (statistical) stability. A full probabilistic methodology needs not only an appropriate mathematical probability concept, enriched by such notions as expectation and conditioning, but also an interpretive component to identify data that is typical of the model and an estimation component to enable inference to the model from data and background knowledge. Our starting point is this first task of determining typicality. Kolmogorov complexity is used as the key non-probabilistic idea to enable us to create simulation data from an envelope model in an attempt to identify “typical” sequences. First steps in finite sequence frequentist modeling will also be taken towards inference of the set M from finite frequentist data and then applied to data on vowel production from an Internet message source.