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Bayesian estimate

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Suppose theta is a parameter that takes values on the real line. Consider the loss function

l(theta, theta_hat) = 0 if |q-theta_hat|<=c, 1 otherwise

Show that the Bayes estimate of theta is the midpoint of the interval I of length 2c that maximizes P(theta in I|x).

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Solution Summary

This solutions shows that the Bayes estimate of theta is the midpoint of the interval I of length 2c that maximizes P(theta in I|x).

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Bayes estimate; median of the posterior distribution

Suppose that the loss function is l(theta, theta_hat) = |theta - theta_hat|. Show that the Bayes estimate of theta is any median of the posterior distribution (any number m(x) such that P(theta <= m(x)|x)>=.5 and P(theta>=m(x)|x)>=.5).

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