Supplemental Material for the AISTATS 2014 paper "Decontamination of Mutually Contaminated Models"

gC(μ) := { F0(C) Fμ(C) if Fμ(C) > 0 ; +∞ if Fμ(C) = 0. Then fC is an upper semicontinuous function: if μ ∈ SM is such that Fμ(C) > 0, then fC is continuous at point μ. Otherwise, fC(μ) = ∞ and fC is trivially upper semicontinuous at point μ. Clearly, one has G(μ) = infC∈C fC(μ) ; as an infimum of upper semicontinuous functions, it is itself upper semicontinuous, and therefore attains its maximum on the compact set SM .