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Bayesian Statistics

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Bayesian Probability

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Interprets probability as a measure of believability or confidence that an individual can place on a statement.

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Bayes' Theorem

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A formula that describes how to update the probabilities of hypotheses when given evidence.

P(AB)=P(BA)P(A)P(B) P(A|B) = \frac{P(B|A)P(A)}{P(B)}

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Prior Probability

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The probability of an event before new data is considered, often denoted as P(A)P(A).

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Posterior Probability

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Updated probability of a hypothesis after considering new evidence, often denoted as P(AB)P(A|B).

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Likelihood

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A function of the parameters of a statistical model given data. Often expressed as P(BA)P(B|A), it measures the plausibility of the data under different hypotheses.

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Bayesian Inference

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The process of deducing properties about a population or probability distribution from data using Bayes' theorem.

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Markov Chain Monte Carlo (MCMC)

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A class of algorithms for sampling from a probability distribution by constructing a Markov chain that has the desired distribution as its equilibrium distribution.

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Conjugate Priors

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A prior distribution that, when combined with a likelihood function from a certain family, results in a posterior distribution that is in the same family.

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Jeffreys Prior

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A non-informative prior that is invariant under reparameterization. Typically used when prior knowledge is minimal.

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Credible Interval

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An interval estimate of a parameter, constructed so that it encloses the parameter with a specified probability, analogous to a confidence interval in frequentist statistics.

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Bayesian Network

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A probabilistic graphical model that represents a set of variables and their conditional dependencies via a directed acyclic graph (DAG).

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Maximum A Posteriori (MAP) Estimation

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The mode of the posterior distribution, which can be used as a point estimate of an unknown parameter.

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Predictive Distribution

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The distribution of a new data point, inferred from the model and past observations, without requiring this point's actual observation.

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Exchangeability

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The property of a sequence of random variables where any finite permutation of the sequence has the same joint probability distribution.

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Bayesian Hierarchical Modeling

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A type of model that includes parameters at multiple hierarchical levels, allowing for structured, hierarchical data analysis.

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Bayesian Linear Regression

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A regression analysis approach where the statistical analysis is undertaken within the context of Bayesian inference.

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Monte Carlo Integration

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A technique for numerically estimating the value of an integral, which is especially useful when dealing with high-dimensional integrals often encountered in Bayesian analysis.

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Propensity Score

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The conditional probability of assignment to a particular treatment given a set of observed covariates, often used in observational studies within a Bayesian framework.

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Decision Theory

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A framework for making decisions based on the principles of maximizing utility and often incorporates Bayesian probabilities.

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Bayes Factor

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A Bayesian alternative to classical hypothesis testing, the Bayes factor is a ratio of the posterior odds to the prior odds for two competing hypotheses.

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