In fact we only need to know the m numbers zi 1 yi i

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Unformatted text preview: e parameter θ and function c. (a) When c(x) = x and D = Rn , what is the associated family of densities? What is the set of + valid values of θ? (b) Consider the case with D = {0, 1}, with c(0) = 0, c(1) = 1. What is the associated exponential family of distributions? What are the valid values of the parameter θ ∈ R? (c) Explain how to represent the normal family N (µ, Σ) as an exponential family. Hint. Use parameter (z, Y ) = (Σ−1 µ, Σ−1 ). With this parameter, θT c(x) has the form z T c1 (x)+ tr Y C2 (x), where C2 (x) ∈ Sn . (d) Log-likelihood function. Show that for any x ∈ D, the log-likelihood function log pθ (x) is concave in θ. This means that maximum-likelihood estimation for an exponential family leads to a convex optimization problem. You don’t have to give a formal proof of concavity of log pθ (x) in the general case: You can just consider the case when D is finite, and state that the other cases (discrete but infinite D, continuous D) can be handled by taking limits of finite sums. (...
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This note was uploaded on 09/10/2013 for the course C 231 taught by Professor F.borrelli during the Fall '13 term at Berkeley.

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