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86the object of the expectation is constant.⎛ ⎛log ⎝E KT⎝∑ Kj=1f(Y i |X i = j, ˆθ K )p(X i = j|N( ˆX i K ), ˆφ⎞⎞K )j=1 f(Y i |X i = j, ˆθ K T )p(Xi = j|N( ˆX K Ti ), ˆφ⎠⎠ (5.41)K T )∑ KTIf the inequality in equation 5.42 holds, then consistency is implied.⎛E KT⎝∑ Kj=1f(Y i |X i = j, ˆθ K )p(X i = j|N( ˆX i K ), ˆφ⎞K )j=1 f(Y i |X i = j, ˆθ K T )p(Xi = j|N( ˆX K Ti ), ˆφ⎠K T )( )((DK − D KT ) log(N)/2)≤ expN∑ KT(5.42)When K T = 1, there is only one possible configuration for X; in other words,X is constant. Equation 5.43 follows.f(Y ) = f(Y |X) = ∏ if(Y i |X i ) = ∏ if(Y i ) (5.43)Note that f(Y i |ˆθ K T , Xi ) is equal to f(Y i |ˆθ K T ) when KT = 1, and this in turn isasymptotically equal to f(Y i |θ K T ), since in this case ˆθ is just the usual maximumlikelihood estimate with independent data. The expected value in equation 5.42can be rewritten as equation 5.44.⎛∑ Kj=1f(YE KT⎝i |X i = j, ˆθ K )p(X i = j|N( ˆX i K ), ˆφ⎞K )⎠ (5.44)f(Y i |ˆθ K T )I now show that equation 5.44 simplifies to equation 5.45. Consider the value ofˆφ K when K T = 1 and K > K T . The estimate ˆφ K is a maximum pseudolikelihoodestimate. Maximum pseudolikelihood estimates were shown to be consistent byGeman and Graffigne (1986), so we know that ˆφ K → φ K Twhen K T = 1 all points are independent, so φ K Tas N → ∞. However,= 0. When φ = 0, the probabilityof any X i is (1/K), regardless of its neighbors. It follows that the expectation inequation 5.44 is equal to the expectation in equation 5.45.