Frequentist probability theory

It is well established that probabilistic values can be characterized by two general classes of interpretation relative frequencies of an observed outcome and Bayesian probabilities (or so-called subjective probabilities). The class of subjective probabilities allows for a broader context of probability theory. This interpretation proposes that probability can be justified not necessarily by the objective or frequentist basis (a frequency of occurrence among trials ) but to single occurrence events in the form of a measure of one s uncertainty about a particular event (Dubois and Prade 1988 Vick 2002). From this, Bayes theorem serves as a mathematical basis for manipulating relationships between prior and new probabilistic information. As such, the axioms of probability theory serve as a foundation for expressing uncertainty in multiple contexts. 

As statistics is based upon probability theory, it is helpfiil to cite again two possible means of defining probabilities. One way to think about probability - the frequentist... 

The above approach, which was attacked as being too vague to be the starting point of any theory of probability, led eventually to the frequentist approach, where probability was defined in a manner that assigns a numerical value, albeit a value that cannot ever be measured, since it requires an inhnite number of trials... 

The frequentist approach to this sort of problem is closely related to the following philosophical position. If I have a theory which states all swans are white , then it doesn t matter how many white swans are observed, I cannot prove that it is true. However, a single swan of a different colour will succeed in proving that it is false. Now, if we return to the problem of the CD player, we can see that if the first track played is not W then the theory that I pressed play is wrong, whatever the prior probability of its being true. (It is worth pointing out that the Bayesian will also reach this conclusion.)... 

Au SK (2011) Fast Bayesian FFT method for ambient modal identification with separated modes. J Eng Mech-ASCE 137(3) 214-226 Au SK (2012a) Fast Bayesian ambient modal identification in the frequency domain, Part I posterior most probable value. Mech Syst Signal Process 26(l) 60-75 Au SK (2012b) Connecting Bayesian and frequentist quantification of parameter uncertainty in system Identification. Mech Syst Signal Process 29 328-342 Au SK (2014a) Uncertainty law in ambient modal identification. Part I theory. Mech Syst Signal Process 48(1-2) 15-33... 

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