I wonder what is an intrinsic value of using harmonic mean (for instance to calculate F-measures), as opposed to weighted arithmetic mean in combining precision and recall? I am thinking that weighted arithmetic average could play the role of harmonic mean, or am I missing something?
Solved – Why we don’t use weighted arithmetic mean instead of harmonic mean
harmonic meanmachine learningmonte carloprecision-recall
Best Answer
In general, harmonic means are preferred when one is trying to average rates, instead of whole numbers. In the case of an F1-measure, a harmonic mean will penalize very small precisions or recalls whereas the unweighted arithmetic mean won't. Imagine averaging 100% and 0%: Arithmetic mean is 50% and Harmonic mean is 0%. The harmonic mean requires that both precision and recall be high.
In addition, when the precision and recall are close together, the harmonic mean will be close to the arithmetic mean. Example: the harmonic mean of 95% and 90% is 92.4% compared to the arithmetic mean of 92.5%.
Whether this is a desirable property is probably dependent on your use case, but typically it's considered good.
Finally, note that, as @whuber stated in the comments, the harmonic mean is indeed a weighted arithmetic mean.