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(Solved): Assume x1,x2,,xn are normally distributed with mean and variance 2. Hold 2 const ...
Assume are normally distributed with mean and variance . Hold constant. (a) Show that this distribution is in the exponential family. (b) What is the conjugate prior? Express in canonical form. (c) What is the posterior distribution using the conjugate prior in (b)? Express in canonical form. Challenge question: Repeat (a)-(c) without holding constant.
Expert Answer
(a) To show that the distribution of , which are normally distributed with mean and variance , is in the exponential family, we need to express it in the general form of the exponential family: ,where is the natural parameter, is the log partition function, and is the carrier measure.For the normal distribution, we have: ,where is the number of samples.Let's rewrite this expression in the form of the exponential family: .Comparing this with the general form, we have: , , .Therefore, the distribution is in the exponential family.