\, \pi (\theta )=1, \forall \theta >0 \, and conditional on \, \theta \,, \, X_1,\ldots ,X_ n\stackrel{i.i.d.}{\sim } \mathcal U([0,\theta ]) \,.

Question

What is the Bayesian estimator \hat{\theta }^{\text {Bayes}}?

(If applicable, enter barX_n for \bar{X_ n},\, max_n for \displaystyle \text {max}_{i=1\ldots n} X_ i. )

\hat{\theta }^{\text {Bayes}}=\quad

Answer 1

The prior distribution given is uniform, which means that all values of theta are equally likely. Therefore, the prior distribution does not provide any additional information about the value of theta.

In this case, the Bayesian Estimator is given by the Maximum Likelihood Estimator, which is the maximum value of the observed data. Therefore,

\(\hat{\theta }^{\text {Bayes}}=\displaystyle \text {max}_{i=1\ldots n} X_ i\)