
Gibbs Mo dels 557
That is, the indicator s I
i
satisfy a conditional autologistic regression. The model can be fitted by
maximising the logistic composite log-likelihoo d
LRL(θ) =
∑
x
i
∈x
log
λ
θ
(x
i
| x)
δ
(x
i
) +
λ
θ
(x
i
| x)
+
∑
d
j
∈d
log
δ
(d
j
)
δ
(d
j
) +
λ
θ
(d
j
| x)
(13.84)
which is Besag’s discrete log-pseudolikelihood for the indicators (I
i
) given the locations y. This can
be maximised using standard software for logistic regression. The loglikeliho od is a concave func-
tion of θ, and conditions for existence and uniqueness of the maximum are well known [616]. The
pseudosco re
∂
/(
∂
θ)LRL(θ) is an unbiased estimating function, so that the estimates are con sistent
and asymptotically ...