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79denoted by µ K and σK; 2 equations 5.18 and 5.19 give formulas for µ K and σK 2 interms of the θ K T parameters. Let P1 be the proportion of pixels for which the true(unobservable) state X i is 1, and similarly let P 2 be the proportion of pixels instate 2.µ K = P 1 µ 1 + P 2 µ 2 (5.18)σ 2 K = P 1 (σ 2 1 + µ 2 1 − 2P 1 µ 2 1 − 2P 2 µ 1 µ 2 + P 2 1 µ 2 1 + P 2 2 µ 2 2 + 2P 1 P 2 µ 1 µ 2 )+P 2 (σ 2 2 + µ 2 2 − 2P 2 µ 2 2 − 2P 1 µ 1 µ 2 + P 2 1 µ 2 1 + P 2 2 µ 2 2 + 2P 1 P 2 µ 1 µ 2 )(5.19)Suppose, without loss of generality, that σ 1 > σ 2 . Condition A is given byequation 5.20.log(σ K ) − log(σ 1 ) − 8φ > 0 (5.20)Note from equation 5.57 that σ K becomes larger as the two true mixture componentsbecome more separated; that is, σ K can be made arbitrarily large bymoving µ 1 and µ 2 farther apart. Thus, condition A can be thought of as a regularitycondition which requires a certain amount of separability between the twotrue segments.When φ = 0 (the spatial independence case), condition A reduces to log(σ K ) −log(σ 1 ) > 0, assuming σ 1 > σ 2 . If, in addition, we have σ 1 = σ 2 , then condition Ais guaranteed to hold as long as µ 1 ≠ µ 2 .Lemma 1: IntegrabilitySuppose we define g i , a function of Y i , as shown in equation 5.21.