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Suppose you have a discrete random variable $X_1$ with known probability mass function. I guess that choosing a variable drawn from the same pmf would be the best way to guess $X_1$ assuming all experiments are independent. If this is true, what is the name of this theorem?

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"Best" in what sense? –  cardinal Nov 21 '11 at 23:44
    
the one that minimizes the probability of error (it's a repeated experiment) –  ACAC Nov 21 '11 at 23:49
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If you want to minimize the probability of error, then you'd just always select as an estimator any mode of the distribution. –  cardinal Nov 21 '11 at 23:50
    
if you have a discrete r.v. with two values and probabilities $p_1$ and $1-p_1$ the mode does not minimize the probability of error. If you draw a r.v. from the same pmf the probability of error decreases. –  ACAC Nov 22 '11 at 0:00
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Either I'm not understanding your question quite right or you need to double-check your claim. –  cardinal Nov 22 '11 at 0:09

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up vote 2 down vote accepted

By the most usual criteria of "best guess", that's not an optimal strategy. For example:

If you want to miminize average square error : pick E(X)

If you want to miminize average absolute error : pick median(X)

If you want to miminize probability of error: pick mode(X) (maximum value of pmf)

To use random strategies for guessing can be optimal is some scenarios where there is an opponent who can vary his strategy observing yours (eg. rock-paper-scissors), but that doesn't seem your scenario.

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