
502 Spatial Point Patterns: Methodology and Applications with R
For a finite Gibbs process X with probability density f(x), if we condition on the presence
of a point at lo cation u, then X is no longer a Gibbs process, because it contains a fixed point.
However, the rest of the process, say Y = X \u, is a finite Gibbs pr ocess with probability density
h(y) = c f (y ∪{u}) = c f (y)
λ
(u | y) wher e c is a normalising constant. In technical terms, the
reduced Palm distribution of a Gibbs model is a Gibbs model.
For example, in the Gibbs hard core process with parameters
β
and h in a window W , if we
condition on the pr esence of a point at location u , then ...