Bayes’ Theorem (SEP)
Bayes’ Theorem is a simple mathematical formula used for calculating conditional probabilities. It figures prominently in subjectivist or Bayesian approaches to epistemology, statistics, and inductive logic. Subjectivists, who maintain that rational belief is governed by the laws of probability, lean heavily on conditional probabilities in their theories of evidence and their models of empirical learning. Bayes’ Theorem is central to these enterprises both because it simplifies the calculation of conditional probabilities and because it clarifies significant features of subjectivist position. Indeed, the Theorem’s central insight — that a hypothesis is confirmed by any body of data that its truth renders probable — is the cornerstone of all subjectivist methodology.
Read the rest here: https://plato.stanford.edu/entries/bayes-theorem/
Chatbot
Bayes’ Theorem is a formal rule for updating probabilities when new evidence appears. It tells you how rational belief should shift in light of data.
The formula is:
P(H | E) = [P(E | H) × P(H)] / P(E)
H is a hypothesis.
E is evidence.
P(H) is the prior probability of the hypothesis (your initial degree of belief).
P(E | H) is the likelihood (how probable the evidence would be if the hypothesis were true).
P(E) is the overall probability of the evidence.
P(H | E) is the posterior probability (your updated belief after seeing the evidence).
Conceptually, Bayes’ theorem says:
Posterior = Likelihood × Prior / Evidence
The structure matters more than the algebra. You start with a prior belief. You ask: if this hypothesis were true, how likely would this evidence be? If the evidence strongly favors the hypothesis over its alternatives, the posterior probability increases. If not, it decreases.
A simple example. Suppose 1% of a population has a disease. A test detects it 99% of the time when present, but has a 5% false positive rate.
Let:
P(Disease) = 0.01
P(Positive | Disease) = 0.99
P(Positive | No Disease) = 0.05
If someone tests positive, the intuitive mistake is to think there is a 99% chance they are sick. Bayes shows that is wrong.
You compute:
P(Positive) = (0.99 × 0.01) + (0.05 × 0.99)
The second term dominates because most people are healthy. The result is that the probability of actually having the disease given a positive test is roughly 16–17%, not 99%. The low base rate matters.
Philosophically, Bayes’ theorem is central to epistemology. It formalizes rational belief revision. Evidence does not stand alone; it interacts with prior probability. This is relevant to many of the themes you often circle around: belief formation, skepticism, and why people can rationally disagree. Two people with different priors can see the same evidence and still end up with different posteriors.
It also clarifies debates about extraordinary claims. If a hypothesis has a very low prior probability, it needs extremely strong evidence (a high likelihood ratio) to become plausible. That principle explains why alien salvation stories or miracle claims require more than anecdote to be rationally persuasive.
Wikipedia
Bayes’ theorem (alternatively Bayes’ law or Bayes’ rule), named after Thomas Bayes, gives a mathematical rule for inverting conditional probabilities, allowing the probability of a cause to be found given its effect. For example, with Bayes’ theorem, the probability that a patient has a disease given that they tested positive for that disease can be found using the probability that the test yields a positive result when the disease is present. The theorem was developed in the 18th century by Bayes and independently by Pierre-Simon Laplace.
One of Bayes’ theorem’s many applications is Bayesian inference, an approach to statistical inference, where it is used to invert the probability of observations given a model configuration (i.e., the likelihood function) to obtain the probability of the model configuration given the observations (i.e., the posterior probability).
Read the rest here: https://en.wikipedia.org/wiki/Bayes%27_theorem
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