Explain what is meant by a prior distribution, a posterior distribution, and a Bayes estimator. Relate the Bayes estimator to the posterior distribution for both quadratic and absolute error loss functions.

Suppose $X_{1}, \ldots, X_{n}$ are independent identically distributed random variables from a distribution uniform on $(\theta-1, \theta+1)$, and that the prior for $\theta$ is uniform on $(20,50)$.

Calculate the posterior distribution for $\theta$, given $\mathbf{x}=\left(x_{1}, \ldots, x_{n}\right)$, and find the point estimate for $\theta$ under both quadratic and absolute error loss function.