Abstract
Classical statistics and Bayesian statistics refer to the frequentist and subjective theories of probability respectively. Von Mises and De Finetti, who authored those conceptualizations, provide interpretations of the probability that appear incompatible. This discrepancy raises ample debates and the foundations of the probability calculus emerge as a tricky, open issue so far. Instead of developing philosophical discussion, this research resorts to analytical and mathematical methods. We present two theorems that sustain the validity of both the frequentist and the subjective views on the probability. Secondly we show how the double facets of the probability turn out to be consistent within the present logical frame.
Highlights
When Hilbert prepared the list of the most significant mathematical issues to tackle in the future, he included the probability foundations in the group of 23 famous problems
The Kolmogorovian axioms of non-negativity, normalization and finite additivity provide the rigorous base for calculations; they solve the issue in the terms posed by Hilbert but do not entirely unravel the fundamentals of the probability calculus
De Finetti and other subjectivists calculate the probability of the single occurrence A1 in contrast with Von Mises who coined the term “collective” to emphasize the large number of occurrences pertaining to An
Summary
When Hilbert prepared the list of the most significant mathematical issues to tackle in the future, he included the probability foundations in the group of 23 famous problems. The relative frequency in a large sample of trials offered the indisputable evidence for the probability calculated on the paper but after the First World War a significant turning point occurred. Many problems that arose in this environment dealt with making decisions under uncertainty and the method based on the frequency was not applicable in these cases as it is usually impossible to accumulate wide experience to assess the probability of an economical event. The pragmatic adoption of different statistical methods appears inconsistent due to the conceptual divergence emerging between the probability seen as a degree of belief and the probability as long-term relative frequency. Popper sums up this situation by the ensuing statement:. We mean to prove that the dualist view on probability turns out to be consistent using two theorems
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