Knot a Bad Idea: Testing BLISS Mapping for Spitzer Space Telescope Eclipse Observations

[Abridged] Much of transiting exoplanet science relies on high-precision photometry. The current generation of instruments exhibit sensitivity variations greater than the astrophysical signals. For the InfraRed Array Camera (IRAC) on the Spitzer Space Telescope, a popular way to handle this is BiLinearly-Interpolated Subpixel Sensitivity mapping (BLISS). We use examples of posterior probability functions to show that this scheme can misfit or bias astrophysical parameters, and a toy model to show that underestimated uncertainties may even happen in very simple cases. BLISS maps of detector sensitivity can also be unreliable if the noise in the data is low. To know the astrophysical and detector models a priori, we construct a model of \emph{Spitzer} light curves with $\sim10^{3}$ data. We compare standard BLISS to a variant in which the knot values are full-fledged parameters in the MCMC, and to a standard polynomial model. Both types of BLISS fit the eclipse depth similarly, and the standard BLISS knots vary about as much as the fitted knots, meaning the standard knots behave like real parameters. The polynomial model is almost always more precise, and usually more accurate, than BLISS. When applied to real \emph{Spitzer} data, however, BLISS yields better fits and more precise astrophysical parameters than polynomial models. When we increase the number of BLISS knots, add Brownian noise to the photometry, or make the sensitivity variations more pixelated, then BLISS fits the eclipse depth about as accurately and precisely as---but not more than---the polynomial model. In these and other cases, the detector and astrophysical signals are generally distinct, implying there can be many ways to model sensitivity variations well. Thus, we deem that BLISS could be an acceptable shortcut when used on \emph{Spitzer} IRAC eclipse photometry.