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I was wondering if the the following is an example of the law of iterated expectations.

Say we observe the entire population of the random variable $X$. Call the mean of the population $\mu$ . Now, let us arbitrarily resample a fraction of this population n times. For each of these samples, obtain an unbiased estimate of the mean as $\hat{\mu_{i}}$ where i denotes the sample. Now, we can obtain the average of these sample averages as:$$\tilde{\mu}=\frac{1}{N}\sum_{i}\hat{\mu_{i}}$$ Given that our estimator is unbiased, it should be that $\tilde{\mu}$ is extremely close to $\mu$ . Now, can this also be thought of as the Law of Iterated Expectations where $$E[X]=E[E[X|i]]$$

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This appears to be a trivial application of the Law of Iterated Expectations, because $X$ is mean-independent of the choice of the sample that contains realizations of it (i.e. the true mean of $X$ does not change as we change the sample). So

$$E[E(X|i)] = E[E(X)] = E(X)$$

The expression is not wrong, but it does not appear to me to be a useful example for the application of the tower property, since the result comes about due to independence, rather than from "averaging".

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