6 Coupling from the Past for a Conditional Boolean M od el
6.2 T h e Underlying Process
We start by constructing an underlying process U from which we can derive a Markov chain which converges to the target conditional Boolean model. Following Lantu£joul [79], we use a spatial birth-and-death process which converges to the unconditional Boolean model and prohibit transformations which do not respect the covering condition. As we will show in Theorem 6.3 the resultant conditional spatial birth-and-death process then converges to the conditional Boolean model.
Consider first an unconditional Boolean model with germ intensity measure n and grain distribution H on the space o f primary grains Q. We derive a Boolean model evolving in time as follows. Consider a space-time
Poisson process o f mean measure /¿(-) x Leb(-) on W x R. We assume that
¡1 is positive on W . Each point o f this process is marked independently with a (random) grain drawn from H and an Exponential variable o f unit mean which represents the life time o f the grain. Thus we obtain a marked space-time Poisson process Q on W x R x £/ x (0, oo) based on space-time points
(.x ,s ,G ,l ) € W x R x Q x (0,oo).
Here s represents the birth time and l represents the life time o f germ-grain pair (x ,G ).
We now derive a time-evolving marked point process Z = { Z t,t € R}, where Zt is the list o f the germ-grain pairs o f Q alive at time t:
Zt = {(x ,G ) : ( x ,s , G ,l ) e Q , s < t < s + l}.
Notice that Z evolves as a birth-death process o f objects, each object being a germ-grain pair. Also note that { Z t,t e R} parametrizes a space-time Boolean model € R }, where:
H, = U ( x ® G ) .
(x.G)eZt
The random set the induced set o f Zt, is distributed according to the unconditional Boolean model if Z is started in equilibrium. Figure 17 illustrates the relation between Q and E.
Figure 17: The marked space-time Poisson process Q parametrizes the cylin der model shown in this figure. The process E describes the time-slices through the cylinder process.
We now alter Z allowing only transformations which respect the covering condition, that is we do not permit any transitions which uncover a point in C. Let X = { X t : t 6 R} denote this conditional spatial birth-and-death process. Note that we say a germ-grain pair (x, G) covers a conditioning point
z € C if z is an element o f the induced set x(BG. Clearly births never violate
the covering condition, whereas deaths might uncover a conditioning point. Consequently, whenever a germ-grain pair is born in Z it is also born in X . If a germ-grain pair dies in Z the germ-grain pair becomes a candidate for death in X but in contrast to Z does not necessarily die. A death is only permitted if the removal o f the germ-grain pair does not uncover a conditioning point, for an illustration see Figure 18. If a germ-grain pair survives a death time, we assign a new independent unit rate Exponential life time to it. At the end
Figure 18: Perpetuation: the figure shows the conditioning points as crosses. The lighter shaded grains would be allowed to die in this configuration, the darker shaded grains would be perpetuated.
o f this new life time we again check whether the germ-grain pair is allowed to die subject to the covering condition. We call a germ-grain pair which does not die at its earliest possible death time a perpetuated germ-grain pair and the new life tim e(s) assigned to it perpetuation time(s).
The space-time marked Poisson process Z has a birth rate given by the measure p(-) and a unit death rate per grain, and so X has also birth rate p( ) but the death rate o f X varies according to coverage. Like Z the process
X parametrizes a space-time random set { © t,t € R}:
0f
=
(J
(x 0 G ).(x , G ) e X i
space-time random set {©t, t € R} and the process Z with { S t,< € R}. Let (E , £ ) denote the state space o f X . We now show that X converges to the target conditional Boolean model. To prove the result, we need the following definition of a petite set, see also Meyn and Tweedie [91].
D e fin itio n 6.1 (P e tite S et) Let $ be a continuous-time Markov process
with transition kernel P l(x ,A ) = P($t € = x). For some probability
measure a on R+ define K a = / P l a(dt). The non-empty Borel set C is va-small, if va is a non-trivial measure with
K a(x, •) > »'„( ) fo r all x e C.
If there is any such a and va, we call C petite.
L em m a 6 . 2 Suppose Cm¡„ is the set o f germ-grain lists satisfying the fol
lowing conditions:
1. each germ-grain pair covers at least one conditioning point;
2. if we remove one germ-grain pair from the list, then a conditioning point becomes uncovered.
Then C mi„ is petite.
Proof:
Consider the following germ-grain listwhere {zi, i = 1 ,... k } are the conditioning points and D is a disc o f radius 2. Each o f the germ-grain pairs in L max is given an independent unit mean Exponential life time. Assume we start the conditional birth-and-death pro cess $ in Lmax and evolve it similarly to X , that is germ-grain pairs are born at rate p() and have unit mean Exponential life times. At the end of a life time, a germ-grain pair dies or is perpetuated subject to the cover ing condition. Except for the starting configuration Lmax we assume that any germ-grain pair which is born in $ has a grain o f distribution H . Let
Q*{x, A ) = P ($t € A|$o = x ). Fix positive T , 0 < Sx < S2 < T, and set Ai = { point Zi is covered at time T — <5i by a germ-grain pair which
was born in [T - S2, T - ¿1] and whose grain has distribution i f } .
The FKG inequality [51, Theorem (2.4)] applies to the events A i, . . . A t , since adding further germ-grain pairs increases the family o f these events which are satisfied by a realization. Consequently we have
* ( n * ) £
t i
On the other hand we can find p > 0 such that P(Aj) > p for all i. It follows that p ( f l j is positive. Now suppose that at time T — Si the conditioning points are all covered by l < k newly born germ-grain pairs, so that 0* M holds. By symmetry considerations and the memoryless property o f the Exponential distribution we deduce that the probability o f each o f the t new
err * ay > °
It follows that there is a positive probability that $7- is in a configuration which is E\ thus the restriction o f QT{Lmix, ■) to (E ,£ ) is a non-trivial mea sure on (E ,£ ).
Recall that the grains o f distribution H are almost surely contained in a disc o f unit radius. As the coverage o f any germ-grain pair in Lmax is greater than that o f any germ-grain pair from a configuration in C m it follows that the probability that a germ-grain pair in Lmax is allowed to die is smaller than that for a germ-grain pair from a configuration in Cmjn. Thus if P l(x ,A ) = P (X t 6 A\X0 = x ) and a the probability distribution which gives unit mass to T then it follows that for all A € £ and x 6 Cm¡„
K a(x ,A ) = P T(x, A) > QT{Lmax,A ).
Thus Cmin is petite. □
We now show the following convergence result.
T h e o r e m 6.3 The process X is an ergodic Markov chain whose invariant
distribution is the distribution o f the target conditional Boolean model.
P r o o f: We first show that X satisfies detailed balance with respect to the conditional Boolean model. Thus the target conditional Boolean model spec
ifies the invariant distribution o f X . It is sufficient to show that the germ- process o f X satisfies detailed balance with respect to the germ distribution of the conditional Boolean model. Let <pdenote a point configuration. Recall from Section 4.3.4 (27) the detailed balance condition for spatial birth-and- death processes:
&(£, <P) = d(C> V) 9(<fi U { £ }) for ? e W,
where g is the density o f the equilibrium process with respect to a stationary unit rate Poisson process on the bounded window W . In our setting 6(£, y?) = p(£). Let N c{ip, Gv) denote the number o f conditioning points which are covered by the induced set U^iev, (ys* © Gi). Then the death rate o f X is given by
d(t,<p) = l { N c(<p,Gv ) = N c( < p \ { t } , G l M ( ] ) ) ,
where / denotes the indicator function. Now, the germ-process o f the target conditional Boolean model has the density
/(¥>) °c I I l { N e{<p,Gv ) = k)
V>i€v>
and so X satisfies detailed balance with respect to the target conditional Boolean model.
We proceed to show that X is positive Harris recurrent. According to [91, Theorem 4.3] the following condition is sufficient for positive Harris recur rence. Let C = Cmin as defined in Lemma 6.2 and tc = inf{t > 0 : X t € C ),
th en we require that
1P(tc < oo | Xo = x) = 1 for all x
e
E. (46)Let
7\(x) = min{t > 0 : X t covers all conditioning points | X0 = x } . For each conditioning point z € C, each time a germ-grain pair is born, there is a positive probability t that this germ-grain pair covers z. As soon as z is covered, it cannot be uncovered again. Thus T i(x) is almost surely finite for any x € E. Now let
T2 = min{t > T i(x) : Z ( = 0 },
then T2 is almost surely finite. As r c < T2 it follows that (46) is satisfied and thus that X is positive Harris recurrent.
To deduce ergodicity from positive Harris recurrence, we need to show that some skeleton chain X o f X is ^-irreducible, see [91, Theorem 6.1]. Let
X „ be the discrete-time, skeleton Markov chain with transition kernel P T
where P T(x ,A ) = P(Xt € A\X0 = x). Let <t>(A) = QT(L mtx,A ) as defined in the previous lemma and let n(x) be large enough such that
P(-^Tn(i) covers all conditioning points | Xo = x) = S > 0. Then if 0(A ) = QT(Lmxx, A) > 0 and n = n(x)
> 6 P ( Z nT = 0 ) Q T ( L m A) > 0.
Thus X „ is (^-irreducible and hence X is an ergodic Markov chain. q
The CFTP algorithm for the conditional Boolean model will use the pro cess Z together with the set of perpetuation times as an underlying process
U. The process X can be realized as an adapted functional o f U using Z -t
as the starting pattern at time — T. The process Z satisfies detailed balance with respect to the Boolean model \t, thus if we start Z in a realisation of
then Z is time-homogeneous.
We can only use U within a CFTP algorithm if we are able to extend
U backwards in time. As Z is time-reversible, we can easily extend it back
wards in time, but how about the perpetuation times? The time instances in an interval [—T, 0] at which we assign perpetuation times and the total number o f perpetuation times assigned during the interval crucially depend on how far in the past the process U is started. Thus extending the collec tion o f perpetuation times backwards in time is not at all straight-forward. However, recall that we only have perpetuation if otherwise a conditioning point becomes uncovered. This link between perpetuation and conditioning points is the basis o f the following procedure which solves the problem of backwards extension. To each point in C we assign a Poisson process P* o f unit mean, supplying potential death times which will be used as follows. If a germ-grain pair o f X is not allowed to die then the incident times o f the
Poisson process assigned to the conditioning point covered by the germ-grain pair provide us with new potential death times. A t the next o f these death times the germ-grain pair becomes again a candidate for death. The proper ties o f a unit rate Poisson process imply that the time till the next potential death time is Exponentially distributed with unit mean. In case o f ambigu ity, when the germ-grain pair covers more than one conditioning point, we need to decide which o f the Poisson processes assigned to these conditioning points should provide the potential death times for the germ-grain pair. We only consider those Poisson processes which do not yet provide death times for a perpetuated germ-grain pair. It is arbitrary which o f these processes we choose. There will always be at least one Poisson process to choose from because a germ-grain pair will only be perpetuated if it covers at least one conditioning point which is not covered by any other germ-grain pair. As we can simulate Poisson processes backwards in time, the underlying process U consisting o f Z and the k Poisson processes Pi, i = 1 , . . . , k can be extended backwards in time.
The following pseudo-code M C M C -C on d ition a lB oolea n M od el describes how we can produce a realization o f
X
over the time-interval [—T, 0] from a realization o f U over the same time interval. Thus it describes explicitly howX
can be realized as an adapted functional o f U.we could start X in any configuration whose non-perpetuated germ-grain pairs coincide with Z - T and whose perpetuated germ-grain pairs are in C min. To each o f the perpetuated germ-grain pairs we would assign a Poisson pro cess according to the above rules.
The following definitions will be useful for the description of the algo rithm. If the nth unit-rate Poisson process Pn, n = 1 supplies the potential death times o f the perpetuated germ-grain pair (x, G), then the germ-grain pair is assigned the index n. This assignment is carried out by the algorithm in d e x * (•). The algorithm BooleanModel produces a realiza tion o f the process Z, the algorithm P oisson samples a unit mean Poisson process. The set I in the algorithm keeps track o f the selection o f Poisson processes which currently supply death times for perpetuated germ-grain pairs. Finally let N c( X t) denote the number o f conditioning points which are covered by the induced set o f X t.
MCMC-ConditionalBooleanModel(/r, H,C = { z i , . . . , z*}, W x Q x [—T, 0 ]): Z ([ —T, 0]) <— BooleanModel (¿1, Leb(-), H, W x Q x [—T, 0])
f o r 1 = 1 to k
P j ( [ - T ,0 ] ) < - P o is s o n ([—T ,0])
{< i,. . . ,tm(_ r)} <— ord ered l i s t o f in c id e n t times o f Z([—T, 0]) and P j( [ - T , 0]) f o r a l l j = l , . . . , k X <- Z -t 1 <- 0 f o r i = l t o m (—T ) i f —fj b ir th time o f (z, G) then X <- X u ( ( i , G ) }
i f — ti death time o f (x ,G ) then i f (7Vc( X \ {(a ;,G )}) = N C( X ) ) then
X < - X \ { ( z , G ) } e ls e
I « - I U {in d e x x ( ( z ,G ) ) }
i f — ti in cid en t tim e o f Pj and j € / then
(x , G) <— grain w ith index j
i f (7Vc( X \ { ( z , G ) } ) = N C( X ) ) then X < - X \ { ( z , G ) }
i <- A O )
re tu rn (X )