Lecture

Bayesian Estimation: Overview and Examples

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Description

This lecture covers Bayesian estimation, starting with an overview of classical statistical inference versus Bayesian inference. It then delves into topics such as conjugate priors, Markov Chain Monte Carlo (MCMC) methods, and examples of Bayesian estimation for binary logit and logit mixture models. The instructor explains the importance of priors, likelihood, and posteriors in Bayesian inference, showcasing a temperature example to illustrate the concepts. The lecture also explores the use of Gibbs sampling and Metropolis-Hastings algorithms in MCMC, along with the assessment of convergence using Gelman and Rubin diagnostics. A case study on the choice of grapes demonstrates the practical application of Bayesian estimation in predicting individual behavior and market demand.

Official source

Ontological neighbourhood

Statistics

Statistical inference: Mathematical statistics, Bayesian statistics

Related concepts (32)

Posterior predictive distribution

In Bayesian statistics, the posterior predictive distribution is the distribution of possible unobserved values conditional on the observed values. Given a set of N i.i.d. observations , a new value will be drawn from a distribution that depends on a parameter , where is the parameter space. It may seem tempting to plug in a single best estimate for , but this ignores uncertainty about , and because a source of uncertainty is ignored, the predictive distribution will be too narrow.

Beta distribution

In probability theory and statistics, the beta distribution is a family of continuous probability distributions defined on the interval [0, 1] or (0, 1) in terms of two positive parameters, denoted by alpha (α) and beta (β), that appear as exponents of the variable and its complement to 1, respectively, and control the shape of the distribution. The beta distribution has been applied to model the behavior of random variables limited to intervals of finite length in a wide variety of disciplines.

Conjugate prior

In Bayesian probability theory, if the posterior distribution is in the same probability distribution family as the prior probability distribution , the prior and posterior are then called conjugate distributions, and the prior is called a conjugate prior for the likelihood function . A conjugate prior is an algebraic convenience, giving a closed-form expression for the posterior; otherwise, numerical integration may be necessary. Further, conjugate priors may give intuition by more transparently showing how a likelihood function updates a prior distribution.

Bayesian inference

Bayesian inference (ˈbeɪziən or ˈbeɪʒən ) is a method of statistical inference in which Bayes' theorem is used to update the probability for a hypothesis as more evidence or information becomes available. Bayesian inference is an important technique in statistics, and especially in mathematical statistics. Bayesian updating is particularly important in the dynamic analysis of a sequence of data. Bayesian inference has found application in a wide range of activities, including science, engineering, philosophy, medicine, sport, and law.

Prior probability

A prior probability distribution of an uncertain quantity, often simply called the prior, is its assumed probability distribution before some evidence is taken into account. For example, the prior could be the probability distribution representing the relative proportions of voters who will vote for a particular politician in a future election. The unknown quantity may be a parameter of the model or a latent variable rather than an observable variable.

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