FACIAL SOFT BIOMETRICS - Library of Ph.D. Theses | EURASIP

110 B. APPENDIX TO SECTION 4we conclude thatlog− limN→∞ n/ρ logP(α 1|α 0 ) = ∑ fα 0,f I f ( α 1,fα 0,f).(B.7)Finally given that P(α,τ) = P(α 0 )P(α 1 |α 0 ), we conclude that −lim N→∞logn/ρ logP(α,τ) =D(α 0 ||ρp)+ ∑ f α 0,fI f ( α 1,fα 0,f).□Proof of Theorem 2: The proof is direct from the method of types (cf. [CT06]), which appliesafter noting that |V(τ)| ≤ n 2ρ ˙≤ e nδ ∀δ > 0, and that sup α∈V(τ) P(α) ≤ P(τ) ≤|V(τ)|sup α∈V(τ) P(α).□Proof of Theorem 3: The proof is direct by noting that for anyδ > 0, then for τ ≥ τ 0 we have− limN→∞logn/ρ P(|S| > (τ +δ)n ) >− limρ N→∞logn/ρ P(|S| > τn ρ ),(B.8)and similarly for τ < τ 0 we have− limN→∞logn/ρ P(|S| < (τ −δ)n ρ ) > − limN→∞logn/ρ P(|S| < τn ρ ).(B.9)□B.2 Confusion matrices and population characteristics for proposedsystemsLight eyesDark eyesLight eyes 0.9266 0.0734Dark eyes 0.0759 0.9238p C 0.3762 0.6238Table B.1: Confusion matrix related to ‘2e’: 2 eye color categoriesNo moustacheMoustacheNo moustache 0.8730 0.1270Moustache 0.2720 0.7280p C 0.7340 0.1623Table B.2: Confusion matrix related to ‘m’: moustache detectionNo glassesGlassesNo glasses 0.9283 0.0717Glasses 0.0566 0.9434p C 0.0849 0.0188Table B.3: Confusion matrix related to ‘g’: glasses detection