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Fisher's Information

Fisher information measures the amount of information that an observable random variable X {\displaystyle X} carries about an unknown parameter p {\displaystyle p} upon which the probability of X {\displaystyle X} depends. For the Bernoulli distribution, the Fisher information with respect to the parameter p {\displaystyle p} is given by:

Proof:

This represents the probability of observing X {\displaystyle X} given the parameter p {\displaystyle p}.

It is maximized when p = 0.5 {\displaystyle p=0.5}, reflecting maximum uncertainty and thus maximum information about the parameter p {\displaystyle p}.