Theoretical formulation and simulation
Let \boldsymbol{Y} be a finite set of translations of source segment \boldsymbol{x} , and let \vec{p}_{x|y} and \vec{p}_{y} denote log probability vectors that include \text{accuracy}_{M} and \text{fluency}_{M} scores for all \boldsymbol{y}\in\boldsymbol{Y} . 2 2 2 There are infinitely many possible translations, but here we only consider a pool of human and machine generations. We use the Pearson correlation between the two vectors:

\rho =\mathrm{corr}(\vec{p}_{x|y},\vec{p}_{y}) (1)

to quantify the tradeoff between \text{accuracy}_{M} and \text{fluency}_{M} across translations of \boldsymbol{x} .
If \rho>0 there is no tradeoff, and the translations with higher \text{accuracy}_{M} also tend to have higher \text{fluency}_{M} . If \rho<0 the dimensions trade off, and improving a translation along one dimension tends to leave it worse along the other.

Suppose that a translator is choosing among several good translations \boldsymbol{y} that all have near-maximal values of p(\boldsymbol{y}|\boldsymbol{x}) . Because p(\boldsymbol{y}|\boldsymbol{x})\propto p(\boldsymbol{x}|\boldsymbol{y})p(\boldsymbol{y}) , it follows that \text{accuracy}_{M} and \text{fluency}_{M} trade off within the set of candidates.
To validate this informal argument, we ran simulations in which \boldsymbol{x} and \boldsymbol{y} are both numeric vectors drawn from a Gaussian joint distribution P(\boldsymbol{x},\boldsymbol{y}) . 3 3 3 All simulation details are provided in Appendix. For each source “segment” \boldsymbol{x} there are infinitely many translations \boldsymbol{y} , and we focus on the best, or those with highest p(\boldsymbol{y|x}) .

We initially assume that both \boldsymbol{x} and \boldsymbol{y} are one-dimensional vectors. Figure 2(a) shows the relationship between p(\boldsymbol{y}) and p(\boldsymbol{x|y}) for 3 samples \boldsymbol{x} . Each point in each panel corresponds to a candidate translation \boldsymbol{y} , and candidates with highest p(\boldsymbol{y|x}) are shown in yellow. The correlation above each panel results from applying Equation 1 to all translations with p(\boldsymbol{y|x}) above the 90th percentile (i.e. all points in the brightest part of each plot). The first source “segment” \boldsymbol{x} (leftmost panel) has relatively high probability p(\boldsymbol{x}) , and no tradeoff is observed in this case. The tradeoff emerges, however, and becomes increasingly strong as \boldsymbol{x} moves away from the mode of the distribution p(\boldsymbol{x}) .

Figure 2(b) shows that the tradeoff persists when the dimensionality of \boldsymbol{x} and \boldsymbol{y} is increased. The density plot for each dimensionality is based on a sample of 100 source “segments” (rather than the 3 in Figure 2(a) ), and at all dimensionalities the majority of source “segments” induce tradeoffs. The tradeoffs are stronger (i.e. correlations more negative) when the candidate translations consist of the \boldsymbol{y} with highest p(\boldsymbol{y|x}) , but for all dimensions except n=1 most source “segments” still induce a tradeoff even if all candidate translations are considered. These results provide theoretical grounds to expect that tradeoffs will also occur in real translations generated by humans and machines.