Let be a sequence of independent identically distributed random variables with values in the state space S with probability measure P.
Definition
- The empirical measure is defined for measurable subsets of S and given by
- where is the indicator function and is the Dirac measure.
For a fixed measurable set A, nPn(A) is a binomial random variable with mean nP(A) and variance nP(A)(1 − P(A)). In particular, is an unbiased estimator of P(A).
Definition
- is the empirical measure indexed by, a collection of measurable subsets of S.
To generalize this notion further, observe that the empirical measure Pn maps measurable functions to their empirical mean,
In particular, the empirical measure of A is simply the empirical mean of the indicator function, .
For a fixed measurable function f, is a random variable with mean and variance .
By the strong law of large numbers, converges to P(A) almost surely for fixed A. Similarly converges to almost surely for a fixed measurable function f. The problem of uniform convergence of to P was open until Vapnik and Chervonenkis solved it in 1968.
If the class (or ) is Glivenko–Cantelli with respect to P then converges to P uniformly over (or ). In other words, with probability 1 we have
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