\, \pi (\theta )=1, \forall \theta >0 \, and conditional on \, \theta \,, \, X_1,\ldots ,X_ n\stackrel{i.i.d.}{\sim } \mathcal U([0,\theta ]) \,.
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
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,
\(\hat{\theta }^{\text {Bayes}}=\displaystyle \text {max}_{i=1\ldots n} X_ i\)
To find the Bayesian estimator, we need to calculate the posterior distribution and then derive the estimator based on that distribution.
The prior distribution in this case is given by:
$$\pi(\theta) = 1, \forall \theta > 0$$
The likelihood function, assuming iid uniform distribution, can be written as:
$$L(\theta|X_1,\ldots,X_n) = \prod_{i=1}^n \frac{1}{\theta} \mathbf{1}_{[0,\theta]}(X_i)$$
The posterior distribution is obtained by multiplying the prior and likelihood and normalizing:
$$\pi(\theta|X_1,\ldots,X_n) \propto \pi(\theta) L(\theta|X_1,\ldots,X_n) = \frac{1}{\theta^n} \mathbf{1}_{[0,\theta]}(X_{(n)})$$
where $X_{(n)} = \max_{i=1\ldots n} X_i$ is the maximum observed value.
To compute the normalization constant, we integrate the joint distribution over all possible values of $\theta$:
$$\int_0^\infty \frac{1}{\theta^n} \mathbf{1}_{[0,\theta]}(X_{(n)}) d\theta$$
The posterior distribution can now be obtained by dividing the unnormalized joint distribution by the normalization constant:
$$\pi(\theta|X_1,\ldots,X_n) = \frac{\frac{1}{\theta^n} \mathbf{1}_{[0,\theta]}(X_{(n)})}{\int_0^\infty \frac{1}{\theta^n} \mathbf{1}_{[0,\theta]}(X_{(n)}) d\theta}$$
To find the Bayesian estimator, we can take the expected value of the posterior distribution:
$$\hat{\theta}^{\text{Bayes}} = E[\theta|X_1,\ldots,X_n] = \int_0^\infty \theta \pi(\theta|X_1,\ldots,X_n) d\theta$$
However, because the posterior distribution depends on the maximum observed value $X_{(n)}$, it is not possible to find a closed-form expression for the expected value. The Bayesian estimator would involve calculating the integral numerically or using approximation methods like numerical integration or Monte Carlo simulation.
Therefore, the Bayesian estimator $\hat{\theta}^{\text{Bayes}}$ does not have a simple analytical expression in this case.