Bayesianism

Bayesianism is a philosophical and statistical approach that emphasizes the use of probability theory and Bayesian inference to reason and make decisions under uncertainty. Bayesianism is named after the Reverend Thomas Bayes, an 18th-century British mathematician and theologian, who developed a mathematical formula for updating beliefs based on new evidence.

In Bayesianism, probabilities are treated as degrees of belief or degrees of uncertainty, rather than as objective frequencies. Bayesian inference involves updating beliefs based on new evidence, using Bayes' theorem to calculate the probability of a hypothesis given the available data. This approach is often contrasted with frequentist statistics, which emphasizes the objective measurement of frequencies and probabilities in repeated experiments.

Bayesianism has been applied to a wide range of fields, including philosophy, psychology, economics, artificial intelligence, and machine learning. In philosophy, Bayesianism has been used to analyze the nature of knowledge and justification, to argue for a subjective or relativistic theory of probability, and to provide a framework for decision-making under uncertainty.

Beliefs
Bayesian probability is a type of probability that is used to measure the degree of confidence one has in a belief or hypothesis, rather than the frequency of an event. This approach allows for the application of probability to a wide range of propositions, beyond those that are associated with a reference class. The term "Bayesian" has been used to describe this type of probability since the 1950s, and recent advancements in computing technology have enabled its use in various scientific disciplines.

It's worth noting that while the concept of Bayesian probability is often associated with Thomas Bayes, he may not have fully embraced the modern interpretation. Pierre-Simon Laplace is credited with pioneering and popularizing the broader interpretation of Bayesian probability.

Bayes' original definition of probability focused on the ratio between the value of an expectation that depends on an event and the value of the thing expected upon the event's occurrence. This definition is subjective and does not require repeated events, but it does require the event to be observable. Modern Bayesian statisticians have expanded on Bayes' original definition in various ways, but some argue that Bayes intended his results to be applied in a more limited fashion.

The philosophy of Bayesian statistics is foundational to many modern estimation techniques, such as probabilistic machine learning, risk assessment, and simultaneous localization and mapping. While probability theory as a whole was developed much later, Bayesianism has been an influential and widely used approach in various fields.

Criticism
Critics of Bayesianism argue that it is limited by its reliance on subjective probabilities, and that it fails to account for the objective reality of the world. Additionally, Bayesianism has been criticized for its potential to lead to circular reasoning and for its inability to handle certain types of uncertainty, such as Knightian uncertainty, which involves situations where the probability of an event cannot be assigned at all.

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 * Bayesian Statistics