  In most machine learning tasks, we evaluate a model $M$ on a given data
population $S$ by measuring a population-level metric $F(S;M)$. Examples of
such evaluation metric $F$ include precision/recall for (binary) recognition,
the F1 score for multi-class classification, and the BLEU metric for language
generation. On the other hand, the model $M$ is trained by optimizing a
sample-level loss $G(S_t;M)$ at each learning step $t$, where $S_t$ is a subset
of $S$ (a.k.a. the mini-batch). Popular choices of $G$ include cross-entropy
loss, the Dice loss, and sentence-level BLEU scores. A fundamental assumption
behind this paradigm is that the mean value of the sample-level loss $G$, if
averaged over all possible samples, should effectively represent the
population-level metric $F$ of the task, such as, that $\mathbb{E}[ G(S_t;M) ]
\approx F(S;M)$.
  In this paper, we systematically investigate the above assumption in several
NLP tasks. We show, both theoretically and experimentally, that some popular
designs of the sample-level loss $G$ may be inconsistent with the true
population-level metric $F$ of the task, so that models trained to optimize the
former can be substantially sub-optimal to the latter, a phenomenon we call it,
Simpson's bias, due to its deep connections with the classic paradox known as
Simpson's reversal paradox in statistics and social sciences.
