View - Statistics - University of Washington

84⇒ (L ˆX (Y |K)) exp(−(D K − D KT ) log(N)/2)(L ˆX (Y |K T ))< 1 (5.34)⇒ (L ˆX(Y |K))(L ˆX (Y |K T )) < exp((D K − D KT ) log(N)/2) (5.35)⇒∏ ∑ 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 )(5.36)⇒ 1 N⎛∑log ⎝i∑ 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 T), ˆφ⎠K T )∑ KT< ((D K − D KT ) log(N)/2)Ni(5.37)Define h i as shown in equation 5.38.⎛h i = log ⎝∑ 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 T), ˆφ⎠ (5.38)K T )∑ KTLet ZY 2 be the subset of Z 2 on which Y is defined, and define a process H =h i , i ∈ ZY 2 . Consider a translation in Z 2 denoted by τ, such that the distributionof H does not change under τ. The terms in h i are Gaussian densities and localcharacteristics of a Markov random field process; we are using the Potts modelof equation 5.2 to model the local characteristics of this process. The Gaussiandensities model the spatially independent noise at each pixel, while the Markovrandom field terms capture the spatial dependence of the image. Now, theselocal characteristics do not change for any interior pixel; they differ only at theboundaries, which (as noted previously) have been excluded from this analysis. Inexcluding the boundaries (as well as in letting the image size increase to infinity ini