In the field of statistics, the Neyman–Pearson lemma states that the most powerful way to distinguish between the two distributions P and Q based on an observation Y (drawn from one of them) is through the log of the ratio of their likelihoods: log ⁡ ⁡ P ( Y ) − − log ⁡ ⁡ Q ( Y ) . The KL divergence is the expected value of this statistic if Y is actually drawn from P. Kullback motivated the statistic as an expected log likelihood ratio.