Batch normalization variance calculation
Well, the only reason, I would say here, is the numerical stability. For instance, if the Batch size $m =1$ during the training, then you may end up with $NaN$ variance!

Update: I doubled check the BN paper again, they clearly mentioned the condition that $m&gt;1$. However, I found an interesting post discussing the cases when biased estimator preferable than the unbiased one. This might explain why the authors used the biased one during the training and the unbiased one inference.