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Bayes' theorem and reasoning with probability

Thomas Bayes Richard Price Pierre-Simon Laplace

This page delivers no verdict; it presents claims and evidence with their sources. No theory in science is absolute. Method

For everyone

Thomas Bayes's essay, published after his death in 1763 by Richard Price, offered a method showing how confidence in a hypothesis should be updated when new evidence arrives. Laplace developed and generalised the idea independently. Today Bayes' theorem is used widely, from medical diagnosis and artificial intelligence to debates on scientific method and the weighing of evidence in courts. What probability means, however, is debated.

For specialists

The theorem reads P(H|E) = P(E|H)·P(H) / P(E): the prior probability P(H) is multiplied by the likelihood of the evidence P(E|H) to give the posterior. In the Bayesian approach probability is a degree of belief, and the choice of priors draws the criticism of subjectivity. In the frequentist approach probability is the long-run frequency in repeated trials, and no probability is assigned to hypotheses. The spread of Markov chain Monte Carlo methods in the 1990s made Bayesian computation feasible for complex models.

Claims

Confidence in a hypothesis should be updated by a definite rule, according to how expected the new evidence is under that hypothesis.^[1][2]

Positions

Bayesians: Probability is a degree of belief; taking prior knowledge explicitly into account makes scientific inference more coherent and transparent.^[2]
Frequentists: Probability is only the long-run frequency of repeatable events; subjective priors bring arbitrariness into scientific conclusions.^[2]

Connections

Parent

Mathematics

The p-value and the statistical significance debate Philosophy of science

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Sources

ArticleBayes, T.; Price, R. (1763). An essay towards solving a problem in the doctrine of chances. Philosophical Transactions of the Royal Society of London 53, 370–418DOI 10.1098/rstl.1763.0053 archive
EncyclopediaLin, H.Bayesian Epistemology. Stanford Encyclopedia of Philosophyplato.stanford.edu archive

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Cite this page

APA

Athena contributors. (2026). Bayes' theorem and reasoning with probability. Athena: The tree of human knowledge. Retrieved November 1, 2026, from https://latifkedi.github.io/athena/en/d/bayes/

BibTeX

@misc{athena_bayes,
  title        = {Bayes' theorem and reasoning with probability},
  author       = {{Athena contributors}},
  year         = {2026},
  howpublished = {\url{https://latifkedi.github.io/athena/en/d/bayes/}},
  note         = {Athena: The tree of human knowledge. Accessed: 2026-11-01}
}

Added: September 28, 2026 · Last changed: September 28, 2026 · Page history

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