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88Thus, equation 5.42 holds, so BIC P L (K) is consistent for K in case 1. Notethat in the limit as N → ∞, the inequalities in equations 5.50 and 5.51 becomeequalities; equation 5.42 still holds since its inequality is not strict.A comment about the proof for case 1 is needed. In equation 5.45, the numeratorcorresponds to a certain mixture density with K components, while thedenominator has K T = 1 component. I emphasize that this consistency result doesnot hold for the general Gaussian mixture model in which mixture proportions areestimated along with θ, since the mixture implied by the numerator of equation5.45 has the mixture proportions held constant at 1/K.Case 2: K T = 2, K = 1, and condition ABegin as in case 1, up to equation 5.36 which is rewritten here as equation 5.52.∏ ∑ Kj=1i f(Y i |X i = j, ˆθ K )p(X i = j|N( ˆX i K ), ˆφ K )∏ ∑ KTi j=1 f(Y i |X i = j, ˆθ K T )p(Xi = j|N( ˆX K Ti ), ˆφ< exp((D K − D K KT ) log(N)/2)T )(5.52)Inverting the fraction we obtain equation 5.53∏ ∑ KTi j=1 f(Y i |X i = j, ˆθ K T )p(Xi = j|N( ˆX K Ti ), ˆφ K T )∏ ∑ Kj=1i f(Y i |X i = j, ˆθ K )p(X i = j|N( ˆX i K ), ˆφ K )> exp((D KT − D K ) log(N)/2)(5.53)Since K = 1, this simplifies to equation 5.54.∏ ∑ KTi j=1 f(Y i |X i = j, ˆθ K T )p(Xi = j|N( ˆX K Ti ), ˆφ K T )∏i f(Y i |ˆθ K )> exp((D KT − D K ) log(N)/2)(5.54)The inequality in equation 5.54 is equivalent to the inequality in equation 5.55.