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A Quick Tour of Maximum-Likelihood Estimation

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A tool for understanding where our observations may have come from

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“Given the data, where did they come from?”

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S  is a set of parameters that describe the parent distribution from which  D  is most likely to have come.

The mean      Some measure of spread, say      The kurtosis (skew)

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…where do we set the       so that the likelihood                  is maximized?

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Analytically: Usually not so easy (solve y = f(x) for x) Computationally: OK most of the time

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Brute-force: Work within the hypercube of volume

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L := + inf S_l  :=  0   For  s  in S1 x S2 x S3  do:      l := Likelihood(S, D);       if (l < L) then:            L = l;             s _l  = s; Return L,  s_l         # ML and location

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There may be more than one choice of S that maximizes L     | S_i | may be large    Heuristics such as assuming the parent distribution is of a certain type may quickly lead to the wrong answer   “The parent distribution is obviously binomial”  “That 5% event will  never  happen. As a result, we make a  Gaussianity  assumption.”