
Gibbs Mo dels 527
where
β
1
and
β
2
are non-negative constants.
The problem here is that the conditional intensity cannot be chosen at will; the values
λ
(u | x)
for different lo cations u and different patterns x must be logically consistent.
The con ditional intensity
λ
(u | x) of a point pr ocess is related to its probability density f (x )
through equ a tion (13.4). Rearranging this equation gives f (x ∪{u}) = f(x)
λ
(u | x). If we know
the probability density for x and the conditional intensity at a new point u given x, we can calculate
the probability density for the augmented pattern x ∪{u}. Applying this idea repeatedly, we can
determine the p robability ...