Auto-logistic model

Consider a two dimensional lattice, each node {(i,j)} of which has a random variable {X_{i,j}} associated with it. One definition of nearest-neighbor models which might be applied to describe the interactions between the variables {X_{i,j}} in this situation is through conditional probabilities of the form

\displaystyle P\{x_{i,j}| \text{all other values} \} \equiv P\{x_{i,j}|x_{i-1, j}, x_{i+1, j}, x_{i, j-1}, x_{i, j+1}\} \ \ \ \ \ (1)

Although (1) provides a intuitive representation, it has some drawbacks. For instance, there is no direct method of evaluating the joint probability distribution on the lattice, and the functional form of the conditional probability on the right-hand side of (1) is subject to severe consistency conditions.

Assuming that the model is spatially homogeneous, and that we have binary data Besag (1972) have shown that we may write

\displaystyle p(x_{i,j}|x_{i-1, j}, x_{i+1, j}, x_{i, j-1}, x_{i, j+1}) = \frac {\exp\{x[\alpha + \beta_1(x_{i-1, j} + x_{i+1, j}) + \beta_2(x_{i, j-1} + x_{i, j+1})]\}} {1 + \exp\{\alpha + \beta_1(x_{i-1, j} + x_{i+1, j}) + \beta_2(x_{i, j-1} + x_{i, j+1})\}}

where {\alpha}, {\beta_1} and {\beta_2} are arbitrary real numbers. The similarity of the above model with the logistic regression model led it to be called an auto-logistic model.

If a boundary of zeros {\bold{x}_\bold{B} = 0} surrounds the inner array of the lattice {\bold{x}_{\bold{I}}}, then

\displaystyle P\{\bold{x}_\bold{I}|\bold{x}_\bold{B} = 0\} = \frac{\exp\{\sum (\alpha + \beta_1 x_{i-1, j} + \beta_2x_{i, j-1})x_{i,j}\}}{C(\alpha, \beta_1, \beta_2)}

where the summation extends over all {(i,j) \in I} and {C(\alpha, \beta_1, \beta_2)} is a normalizing function, dependent also upon the dimensions of the array.

The above results are valid for any shape of closed boundary. Notice that with this result we get a valid functional form for the conditional probability in Eq. (1), but we still don’t have a direct method of evaluating the joint probability distribution on the lattice.

Reference:

Besag, J.E., 1972. Nearest-neighbor systems and the auto-logistic model for binary data. Journal of the Royal Statistical Society. Series B (Methodological), 75-83

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