Bayesianism is the broader philosophy inspired by Bayes’ theorem. The core claim behind all varieties of Bayesianism is that probabilities are subjective degrees of belief (or confidence) — often operationalized as willingness to bet.
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As a consequence of the subjective interpretation of probability theory, Bayesians are more inclined to apply Bayes’ Theorem in practical statistical inference. The primary example of this is statistical hypothesis testing. Frequentists take the application of Bayes’ Theorem to be inappropriate, because “the probability of a hypothesis” is meaningless: a hypothesis is either true or false; you cannot define a repeated experiment in which it is sometimes true and sometimes false, so you cannot assign it an intermediate probability. - LessWrong
Bayesianism models epistemic uncertainty - our imperfect knowledge about objective reality. This is not subjective idealism (reality depends on beliefs) but recognizing we need to represent uncertainty about the true state of the world somehow.
Practical advantages:
- Incorporates prior knowledge explicitly (useful with limited data)
- Natural framework for sequential updating as new data arrives
- Can assign probabilities to hypotheses, not just data
- Gives credible intervals with direct probability interpretation: “95% probability the parameter is in this range”
Applications are common in ML: Gaussian processes, Bayesian neural networks