Bayesian statistics expresses uncertainty about an unknown statistical parameter through a probability distribution, then uses Bayes' theorem to update it after observing data. The prior distribution describes uncertainty before the current observations; the likelihood function describes how the statistical model generates those observations at each parameter value; the posterior distribution describes uncertainty conditional on the observations and that model. A physical parameter can have one fixed value while its value remains uncertain to us. Assigning it a probability distribution represents that uncertainty under the chosen model, rather than requiring the physical parameter to change randomly between measurements.
For example, suppose ten Bernoulli trials are independent conditional on an unknown heads probability , with a beta distribution prior . Eight heads and two tails give the posterior distribution by Beta-binomial conjugacy. Its posterior mean is , between the prior mean and observed binomial proportion . The Bayesian posterior below computes the whole update, and the credible interval below summarizes its remaining uncertainty. The conjugate update is explained in MIT 18.05, Reading 15: beta and normal conjugate priors.
The useful feature is that one posterior distribution supports several questions: a posterior mean estimates a value, a credible interval reports a range, and a posterior predictive distribution predicts a future observation while averaging over parameter uncertainty. With conditionally independent new data, today's posterior distribution can become tomorrow's prior distribution. These conclusions remain conditional on the statistical model and prior distribution; a precise update does not establish that either assumption fits the experiment.
A historical starting point is Thomas Bayes's essay in the 1763 volume of the Royal Society's Philosophical Transactions, communicated after his death by Richard Price. Its opening problem asks for uncertainty about an unknown event probability given observed successes and failures: the same direction of reasoning illustrated by the coin example. See Bayes and Price, An Essay towards solving a Problem in the Doctrine of Chances, opening letter and problem.
Bayesian statistics is a branch of statistics that incorporates prior knowledge or beliefs into the analysis of data. It is based on Bayes' theorem, which describes how to update the probability of a hypothesis as more evidence or information becomes available. The core components of Bayesian statistics include: 1. **Prior Distribution**: This represents the initial beliefs or knowledge about a parameter before observing any data.
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