Neyman-Pearson Lemma

This lemma gives an optimum decision procedure for hypothesis testing. Suppose you have 2 hypotheses, H₀ and H₁. You wish to know which, in fact, holds, so you perform an experiment and observe the results O (for our purposes, O can be the result of a series of experiments). The Nayman-Pearson lemma tells you that you should check if this holds (the parameter t will be explained later):

P(H₁ | O) / P(H₀ | O) > t

 

If it holds, you should decide the hypothesis H₁ holds; otherwise H₀ (sometimes called the null hypothesis) holds. Of course, larger values of t make it harder for you to decide the H₁ holds. The Nayman-Pearson lemma says that for every (positive) value of t, this procedure is optimal in the following sense:

• any other procedure with fewer false positives will have more false negatives, and every other procedure with fewer false negatives will have more false positives.

Thus, we see that the value t controls the ratio of false negatives to false positives. Different applications will require different ratios; we choose it to give the desired ratio.