# Bayes estimator

**BayesianBayesian decision theoryBayesian estimatorBayesian estimationBayes riskasymptoticly efficientBayesBayes actionBayes ResponseBayes rule**

In estimation theory and decision theory, a Bayes estimator or a Bayes action is an estimator or decision rule that minimizes the posterior expected value of a loss function (i.e., the posterior expected loss).wikipedia

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### Estimation theory

**parameter estimationestimationestimated**

In estimation theory and decision theory, a Bayes estimator or a Bayes action is an estimator or decision rule that minimizes the posterior expected value of a loss function (i.e., the posterior expected loss).

### Conjugate prior

**conjugateconjugate distributionconjugate prior distribution**

If there is no inherent reason to prefer one prior probability distribution over another, a conjugate prior is sometimes chosen for simplicity.

The concept, as well as the term "conjugate prior", were introduced by Howard Raiffa and Robert Schlaifer in their work on Bayesian decision theory.

### Decision rule

**rule for making a decision**

In estimation theory and decision theory, a Bayes estimator or a Bayes action is an estimator or decision rule that minimizes the posterior expected value of a loss function (i.e., the posterior expected loss).

### Maximum a posteriori estimation

**maximum a posterioriMAPposterior mode**

An alternative way of formulating an estimator within Bayesian statistics is maximum a posteriori estimation.

as c goes to 0, the Bayes estimator approaches the MAP estimator, provided that the distribution of \theta is quasi-concave.

### Admissible decision rule

**admissibleadmissibilityinadmissible**

that minimizes is called a Bayes rule with respect to.

### Maximum likelihood estimation

**maximum likelihoodmaximum likelihood estimatormaximum likelihood estimate**

and variance of the marginal distribution of using the maximum likelihood approach:

A maximum likelihood estimator coincides with the most probable Bayesian estimator given a uniform prior distribution on the parameters.

### Empirical Bayes method

**empirical BayesEmpirical Bayes methodsEmpirical Bayesian**

A Bayes estimator derived through the empirical Bayes method is called an empirical Bayes estimator.

### Decision theory

**decision sciencestatistical decision theorydecision sciences**

In estimation theory and decision theory, a Bayes estimator or a Bayes action is an estimator or decision rule that minimizes the posterior expected value of a loss function (i.e., the posterior expected loss).

### Expected value

**expectationexpectedmean**

### Loss function

**objective functioncost functionrisk function**

### Utility

**utility functionutility theoryutilities**

Equivalently, it maximizes the posterior expectation of a utility function.

### Bayesian statistics

**BayesianBayesian methodsBayesian analysis**

An alternative way of formulating an estimator within Bayesian statistics is maximum a posteriori estimation.

### Posterior probability

**posterior distributionposteriorposterior probability distribution**

### Parametric family

**parameterized familyfamilyparametrized family**

A conjugate prior is defined as a prior distribution belonging to some parametric family, for which the resulting posterior distribution also belongs to the same family.

### Independent and identically distributed random variables

**independent and identically distributedi.i.d.iid**

Let θ be an unknown random variable, and suppose that are iid samples with density.

### Poisson distribution

**PoissonPoisson-distributedPoissonian**

### Gamma distribution

**gammagamma distributedGamma variate**

### Uniform distribution (continuous)

**uniform distributionuniformuniformly distributed**

### Pareto distribution

**Paretogeneralized Pareto distributionParetian tail**

### Mean squared error

**mean square errorsquared error lossMSE**

The most common risk function used for Bayesian estimation is the mean square error (MSE), also called squared error risk.

### Robust statistics

**robustbreakdown pointrobustness**

Other loss functions are used in statistics, particularly in robust statistics.

### Prior probability

**prior distributionpriorprior probabilities**

Suppose an unknown parameter \theta is known to have a prior distribution \pi. :Such measures p(\theta), which are not probability distributions, are referred to as improper priors.

### Measure (mathematics)

**measuremeasure theorymeasurable**

:Such measures p(\theta), which are not probability distributions, are referred to as improper priors.

### Bayes' theorem

**Bayes' ruleBayes theoremBayes's theorem**

:This is a definition, and not an application of Bayes' theorem, since Bayes' theorem can only be applied when all distributions are proper.