Bayesian Testing and Decision Theory

Let \(X_1,X_2,\ldots ,X_n\) be a random sample from a \(N(\theta ,\sigma ^2)\) distribution (\(\sigma ^2\) known), and let the prior distribution of \(\theta \) be \(N(\mu _0,\sigma _0^2)\), where \(\mu _0\) and \(\sigma _0^2\) are known.

(a) Find the posterior distribution of \(\theta \).

(b) Find the Bayes estimator under squared error loss.

Let \(X_1,\ldots ,X_n\) be a random sample from a \(N(\theta ,1)\) distribution, and let the prior distribution of \(\theta \) be \(N(\mu _0,1)\), where \(\mu _0\) is known.

(a) Find the posterior distribution of \(\theta \).

(b) Find the Bayes estimator of \(\theta \) under squared error loss, and show that it is a weighted average of the prior mean \(\mu _0\) and the sample mean \(\bar X\).

Let \(X\) have the binomial distribution with parameters \(n=10\) and \(p=\theta \) unknown. Suppose we impose a uniform-\((0,1)\) prior on \(\theta \). Compute the posterior mean of \(\theta \). Is it consistent?

Suppose \(X_1,\ldots ,X_n\) is an i.i.d. sample from a normal distribution with mean \(\mu \) and variance 1.

(a) Find the UMVU estimator for \(\mu \). (Prove the optimality criterion.)

(b) Put a \(N(\theta ,\tau ^2)\) prior on \(\mu \) and find the minimax estimator of \(\mu \).

Suppose \(X\sim \operatorname {POISS}(\theta )\). Put a \(\operatorname {GAMMA}(\alpha ,\beta )\) prior on \(\theta \) and suppose we have the following loss function: \[ \ell (\theta ,a)=\frac {(a-\theta )^2}{\theta }. \] Find the Bayes’ estimator of \(\theta \).

Suppose \(\theta \in \{\theta _0,\theta _1\}\) is an unknown (\(\theta _0\) and \(\theta _1\) are, however, known.) Put some prior, \(\pi \), on \(\theta \). We are to test \(H_0:\theta =\theta _0\) versus \(H_1:\theta =\theta _1\). Our actions are 0 (accept) and 1 (reject). Find the Bayes’ procedure for doing this test, if the loss function, \(\ell (\theta ,a)\), is the 0–1 loss given by: \[ \ell (0,1)=0,\qquad \ell (0,0)=1, \] \[ \ell (1,0)=0,\qquad \ell (1,1)=1. \] Is this procedure minimax?

Suppose \(X\sim \operatorname {Poisson}(\theta )\). Put a \(\operatorname {Gamma}(\alpha ,\beta )\) prior on \(\theta \) and suppose we have the following loss function: \(\ell (\theta ,a)=(a-\theta )^2/\theta \). Find the Bayes’ estimator of \(\theta \).

Suppose \(\theta \in \{\theta _0,\theta _1\}\) is an unknown (\(\theta _0\) and \(\theta _1\) are, however, known.) Put some prior, \(\pi \), on \(\theta \). We are to test \(H_0:\theta =\theta _0\) versus \(H_1:\theta =\theta _1\). Our actions are 0 (accept) and 1 (reject). Find the Bayes’ procedure for doing this test, if the loss function, \(\ell (\theta ,a)\), is the 0-1 loss given by: \[ \begin {alignedat}{2} \ell (0,1)&=0,\qquad & \ell (0,0)&=1,\\ \ell (1,0)&=0,\qquad & \ell (1,1)&=1. \end {alignedat} \] Is this procedure minimax?