  In various data settings, it is necessary to compare observations from
disparate data sources. We assume the data is in the dissimilarity
representation and investigate a joint embedding method that results in a
commensurate representation of disparate dissimilarities. We further assume
that there are "matched" observations from different conditions which can be
considered to be highly similar, for the sake of inference. The joint embedding
results in the joint optimization of fidelity (preservation of within-condition
dissimilarities) and commensurability (preservation of between-condition
dissimilarities between matched observations). We show that the tradeoff
between these two criteria can be made explicit using weighted raw stress as
the objective function for multidimensional scaling. In our investigations, we
use a weight parameter, $w$, to control the tradeoff, and choose match
detection as the inference task. Our results show weights that are optimal
(with respect to the inference task) are different than equal weights for
commensurability and fidelity and the proposed weighted embedding scheme
provides significant improvements in statistical power.
