178.

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Page 258

(6.39)

where

(6.40)

The function ψ(a, b) is Boolean and is defined as ψ(a, b)=−1 if a=b, ψ(a, b)=1 otherwise. For example, in Figure 6.15b, the function ψ(w_r0, w_Nr)=[2, −2, 0, 0], while in Figure 6.15c, ψ(w_r0, w_Nr) is [0, 0, −2, 2].

The conditional probability P(w_r0 |w_Nr) is related to the energy function U(w_ro, w_Nr) by (see Equation (6.5)):

(6.41)

The conditional probability can also be expressed in terms of joint probability as:

(6.42)

Comparing Equations (6.41) and (6.42), one obtains:

(6.43)

Note that the right-hand side of Equation (6.43) is independent of w_r0. One can extend the idea to show that the left-hand side is also independent of w_r0. Thus, if one defines w′_r0 as another label on site r₀, according to Equation (6.43), the following relation can be created:

(6.44)

By using the natural logarithm operator, Equation (6.44) is reduced to:

 

or, equivalently:

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