Bayesian statistics is a branch of statistics that deals with the probability of events based on prior knowledge or beliefs. It's named after the mathematician Thomas Bayes, who introduced the principle of Bayesian probability.

In Bayesian statistics, probability is viewed as a measure of belief or confidence that an event will occur. These beliefs may be updated as new evidence or data becomes available. This is different from frequentist statistics, where probability is seen as the long-run frequency of an event, based on repeated sampling.

The fundamental theorem in Bayesian statistics is Bayes' Theorem, which is a formula that describes how to update the probabilities of hypotheses when given evidence. It is written as:

P(H|E) = [P(E|H) * P(H)] / P(E)

Where:

P(H|E) is the posterior probability, or the updated probability of the hypothesis (H) given the observed evidence (E).

P(E|H) is the likelihood, or the probability of the evidence given that the hypothesis is true.

P(H) is the prior probability, or the original probability of the hypothesis before seeing the evidence.

P(E) is the total probability of the evidence.

A significant part of the Bayesian approach is determining the prior probability, which can be subjective and based on personal belief or external information. As new evidence is observed, the prior is updated using Bayes' theorem to form the posterior probability, which is the updated belief.

One of the advantages of Bayesian statistics is that it allows for the incorporation of prior knowledge or belief into the analysis, which can be particularly useful when dealing with complex models or small datasets. However, the choice of the prior can also be a subject of controversy, as it might introduce bias if not chosen carefully.

In practical applications, Bayesian methods have been used in a wide variety of fields, from data science and machine learning to genetics and astrophysics. In recent years, with the advent of modern computational methods, the application of Bayesian statistics has become easier and more widespread.

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