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Question 1. Take a moment to convince yourself that the exponential and gamma distributions are exponential family models. Show that, if the data is exponentially distributed as above with a gamma prior q(0) Gamma(00,%) the posterior is again a gamma, and find the formula for the posterior parameters. (In other words, adapt the computation we performed in class for general exponential families to the specific case of the exponential/gamma model.) In detail:Ignore multiplicative constants and normalization terms, such as the evidence term in Bayes formula. Show that the posterior is proportional to a gamma distribution. . Deduce the parameters by comparing your result for the posterior to the definition of the gamma distribution. Machine learning problems are often online problems, where each data point has to be processed immediately when it is recorded (as opposed to batch problems, where the entire data set is recorded first and then processed as a whole). Conjugate priors are particularly useful for online problems, since, roughly speaking, the posterior given the first (n -1) observations can be used as a prior for processing the nth observation:

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