An expression is provided for the self-lensing optical depth of the thin LMC disk surrounded by a shroud of stars at larger scale heights. The formula is written in terms of the vertical velocity dispersion of the thin disk population. If tidal forcing causes 1-5 % of the disk mass to have a height larger than 6 kpc and 10-15 % to have a height above 3 kpc, then the self-lensing optical depth of the LMC is $0.7 - 1.9 \times 10^{-7}$, which is within the observational uncertainties. The shroud may be composed of bright stars provided they are not in stellar hydrodynamical equilibrium. Alternatively, the shroud may be built from low mass stars or compact objects, though then the self-lensing optical depths are overestimates of the true optical depth by a factor of roughly 3. The distributions of timescales of the events and their spatial variation across the face of the LMC disk offer possibilities of identifying the dominant lens population. In propitious circumstances, an experiment lifetime of less than 5 years is sufficient to decide between the competing claims of Milky Way halos and LMC lenses. However, LMC disks can sometimes mimic the microlensing properties of Galactic halos for many years and then decades of survey work are needed. In this case observations of parallax or binary caustic events offer the best hope for current experiments to deduce the lens population. The difficult models to distinguish are Milky Way halos in which the lens fraction is low (< 10 %) and fattened LMC disks composed of lenses with a typical mass of low luminosity stars or greater. A next-generation wide-area microlensing survey, such as the proposed ``SuperMACHO'' experiment, will be able to distinguish even these difficult models with just a year or two of data.

25 pages, 4 figures, The Astrophysical Journal (in press)