  The problem of bias, meaning over- or underestimation, of the component
perpendicular to the line-of-sight, Bperp, in vector magnetic field maps is
discussed. Previous works on this topic have illustrated that the problem
exists; here we perform novel investigations to quantify the bias, fully
understand its source(s), and provide mitigation strategies. First, we develop
quantitative metrics to measure the Bperp bias and quantify the effect in both
local (physical) and native image-plane components. Second we test and evaluate
different inversion options and data sources, to systematically characterize
the impacts of choices, including explicitly accounting for the magnetic fill
fraction ff. Third we deploy a simple model to test how noise and different
models of the bias may manifest. From these three investigations we find that
while the bias is dominantly present in under-resolved structures, it is also
present in strong-field pixel-filling structures. Noise in the magnetograms can
exacerbate the problem, but it is not the primary cause. We show that fitting
ff explicitly provides significant mitigation, but that other considerations
such as choice of chi^2 weights and optimization algorithms can impact the
results as well. Finally, we demonstrate a straightforward "quick fix" that can
be applied post-facto but prior to solving the 180deg ambiguity in Bperp, and
which may be useful when global-scale structures are, e.g., used for model
boundary input. The conclusions of this work support the deployment of
inversion codes that explicitly fit ff or, as with the new SyntHIA neural-net,
that are trained on data that did so.
