There has been endless debate over the source and status of probability

Subjectivist vs Objectivist

Frequentist (Objectivist)

Frequentists are typically objectivists, however:The frequentist position is that the numbers can come only from experiments: if we test 100 people and find that 10 of them have a cavity, then we can say that the probability of a cavity is approximately 0.1. In this view, the assertion “the probability of a cavity is 0.1” means that 0.1 is the fraction that would be observed in the limit of infinitely many samples. From any finite sample, we can estimate the true fraction and also calculate how accurate our estimate is likely to be.

The objectivist view is that probabilities are real aspects of the universe— propensities of objects to behave in certain ways—rather than being just descriptions of an observer’s degree of belief. For example, the fact that a fair coin comes up heads with a probability 0.5 is a propensity of the coin itself. In this view, frequentist measurements are an observation of these propensities. Most physicists agree that quantum phenomena are objectively probabilistic, but uncertainty at the macroscopic scale—e.g., in coin tossing—usually arises from ignorance of initial conditions and does not seem consistent with the propensity view.

In the end, even a strict frequentist position involves subjective analysis because of the reference class problem.

Bayesian (Subjectivist)

Bayesians are typically subjectivists

The subjectivist view describes probabilities as a way of characterizing an agent’s beliefs, rather than as having any external physical significance. The subjective Bayesian view allows any self-consistent ascription of prior probabilities to propositions but then insists on proper Bayesian updating as evidence arrives.

Bayesian vs Frequentist

• Frequentist - a probability 𝑝 represents a long-run frequency, and never changes
• Bayesian - a probability 𝑝 simply reflects our tentative state of knowledge. Given new information, we may update 𝑝

more on Bayesian Inference vs Frequentist Inference

Closing StatementsThe principle of indifference attributed to Laplace (1816) states that propositions that are syntactically “symmetric” with respect to the evidence should be accorded equal probability. Various refinements have been proposed, culminating in the attempt by Carnap and others to develop a rigorous inductive logic, capable of computing the correct probability for any proposition from any collection of observations. Currently, it is believed that no unique inductive logic exists; rather, any such logic rests on a subjective prior probability distribution whose effect is diminished as more observations are collected.