Frequentist vs. Bayesian

Probability, by definition, is a map which attribute each set a non-negative number between 0 and 1, satisifying the Kolmogorov's axioms.

However,there could be different interpretation behind this notion, especialy how it is related to the observations/ samples/ realizations

The frequentists claim that, as their name indicates, the probability is the limit of frequency. Therefove in their mind experiment, every events have happened for sufficiently

enough times to compute or at teast approximate this probability. Before all, such events is repeatable (for experiments). Hence, for fequentists, (almost) all events don't have a well-defined probability due to their irrepeatable nature. The repeatable events/experiments are often mechanical, i.e. regardless of time and space, relatively easy to control the variables and permitting to ignore the tiny fluctuations.

But we humans tend to apply the probability on everything, after all,there is no restrictions in axioms preventing us to do so. Why not consider the probability to an event which may only happen one? The frequentists shake their heads, saying we are lacking the data and the Bayesian come out saying then fill it with your personal beliefs: Observation by observation, you update your knowledge and belief.

The shift from frequentist to the Bayesian is a shift from objectiveness to subjectiveness. There is no more unique probability for a certain event for everyone (the observer) but infinite beliefs on the same event. Nevertheless, when observations tends to infinity, the beliefs tends to a same value, the probability, in the sense of

frequentist. However, from the Bayesian point of view, a consensus. In brief, the objectivity is made of collective subjectivity. In philosophy, we conclude as universality is contained in particularity.