A Bayesian estimator is an estimator of an unknown parameter θ that minimizes the expected loss for all observations x of X.

In other words, it’s a term that estimates your unknown parameter in a way that you lose the least amount of accuracy (as compared with having used the true value of that parameter).

## Bayes Estimators & The Loss Function

A bayesian estimator is a function of observable random variables, variables you observed in the process of your research. Call your observable variables X_{1} X_{2}…X_{n}

Your data may be able to be represented by the function f(x|θ), where θ is a prior distribution. However, you don’t know the actual value of θ, so you have to estimate it. An estimator of θ is a real valued function δ(X_{1}… X_{n}), not to be confused with Δx in calculus, which means a small change.

The** loss function** L(θ, *a*) where a ε R, is also a real valued function of θ. Our estimate here is *a*, and L(θ, *a*) tells us how much we lose by using *a* as an estimate when the true, real value of a parameter is θ.

There are **different possible loss functions**. For instance, the *squared error loss function* is given by L(θ *a*) = (θ –*a*)^{2}. The *absolute error loss function* would be L(θ *a*) = |θ –*a*|.

In the first definition of Bayesian estimator at the beginning of this page, we said it was an estimator that minimized expected loss. That loss is represented by a loss function like one of those we’ve just described.

We can find the **minimum expected loss** by integrating. For a given X = x, the expected loss (E) is:

In this formula the Ω is the range over which θ is defined. p(θ | x) is the likelihood function; the prior distribution for the parameter θ over observations **x**. Call a*(x) the point where we reach the minimum expected loss. Then, for a*(x) = δ*(x), δ*(x) is the Bayesian estimate of θ.

## Bayesian Estimator: References

Brynjarsdottir, Jenny. STAT 611 Lecture Notes: Lecture 12, Estimation.

Retrieved from https://www2.stat.duke.edu/courses/Fall12/sta611/Lecture12.pdf on March 9, 2018

Shiryaev, A. N. Bayesian estimator, Encyclopedia of Mathematics. Retrieved from : http://www.encyclopediaofmath.org/index.php?title=Bayesian_estimator&oldid=19043 on March 4, 2018.

Zhu, Wei. Bayesian Inference for the Normal Decision. Retrieved from http://www.ams.sunysb.edu/~zhu/ams571/Bayesian_Normal_wide.pdf on March 4, 2018

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