Gibbs Sampling

In the previous section we saw how continuous-time spatial birth-death processes can be employed to sample conditional Boolean models. It turns out that one can also use a discrete-time Gibbs sam- pler for the same sampling problem. To the best of our knowledge a Gibbs sampler for conditional Boolean models has not yet been considered in the simulation literature. The exact coupling con- struction presented here bears a loose resemblance to the Häggströmet al. (1999) algorithm for the bivariate penetrable spheres model. Their method (Algorithm 4.5) is described in Section 4.4, and can be used to perfectly sample the area-interaction process. For point processes there is atypically no maximal state; however they introduce quasi-minimaland quasi-maximalelements in order to carry out perfect sampling via Monotone CFTP.

Recall thatC ={c1, . . . , ck}denotes the finite set of conditioning nodes and for each non-empty

A⊆ Cthe regionEA, defined by Eq.(3.4), is such that a germ inEAcovers only those conditioning

nodes inAand no others. The (conditional) distribution of the conditional Boolean model on some

EA, given the process in the other regions{EB; B 6=A}, is just Poissonian. Hence devising Gibbs

sampler for the conditional Boolean model is straightforward. Due to the sequential nature of a Gibbs updating scheme it is convenient to consider the set{A; A⊆ C, A6=∅}as an ordered set

{A1, . . . , AN}so that eachi∈ {1, . . . , N}corresponds uniquely to someA, whereN is the number

of non-emptyA⊆ C.

If X ∼ Poisson(λ), Xi its restriction to EAi and X˜ = ˜ X1, . . . ,X˜N a collection of Poisson processes with UX˜

⊃ C, then X˜ ∼ πC_λ. Consider a discrete-time component process Φ =˜

˜

Φ0, . . . ,Φ˜N

with equilibrium distribution πC_λ. Let the updating order of the components Φ˜i be

{1, . . . , N}and define ˜ Φ−i(n) = [ j<i ˜ Φj(n) [ j>i ˜ Φj(n−1) ; χi(n) = 1_{U(Φ˜_{−i(n))⊃Ai}}.

Denote the distribution of an unconditional Poisson(λ)process byπλ and that conditioned to have

at least one germ by π0_λ. The density of a Poisson(λ) process conditioned to cover C is given by f(x) ∝ λn(x)₁

{U(x)⊃C}. So the density of the process restricted to the ‘cell’ EAi is given by

fi(x) ∝ λn(x)1{U(x)⊃Ai}, since a germ inEAi only coversAi and the processes in distinct regions

EAi andEAj are independent. Therefore, given n

˜ Φ−i

o

, the conditional density ofΦ˜iis given by

fi · |Φ˜−i ∝    λn(·) ifχi =1_{U_(Φ˜_−i)⊃Ai} = 1; λn(·)1{n(·)>0} ifχi = 0. (3.9) Thus, givenΦ˜−i, the next update ofΦ˜i is (i)Xi ∼ πλ onEAi ifχi = 1; or (ii)X

0

i ∼ πλ0 ifχi = 0.

The Gibbs sampler for the n

˜ Φi

oN

i=1 with updating order

{1, . . . , N}is summarized as follows. Algorithm 3.5 (Gibbs: Conditional Boolean Model).

fori= 1, . . . , N: initializeΦ˜i(0) ={ξi}, whereξi ∈EAi. forn = 1,2, . . .: fori= 1, . . . , N: ifχi(n) = 1: drawXi ∼πλ onEAi; setΦ˜i(n) = Xi. else ifχi(n) = 0: drawXi0 ∼π 0 λ onEAi; setΦ˜i(n) = X 0 i. 74

3.5.1

Exact Gibbs Sampler

In this section a perfect variant of the Gibbs sampler (Algorithm 3.5) is presented. The construction here has been motivated by that of Häggströmet al.(1999) who present an exact 2-component Gibbs sampler for the bivariate penetrable spheres model. In order to devise a CFTP-based procedure one requires maximal and minimal elements of the state space to exist. For point processes a maximal element does not exist since the state space is uncountable. However Häggström et al. (1999) go round this problem by defining quasi-minimal and quasi-maximal elements in order to carry out perfect sampling via Monotone CFTP. In the spirit of this we define such quasi-minimal and -maximal elements for each of theN regions or cells EAi, and devise a CFTP-based construction that outputs samples with the required target distribution.

Consider anN-tuple of spatial point configurationsx= (x1, . . . , xN), wherexiis a configuration

on cellEAi. DefineIx = 1{x16=∅}, . . . ,1{xN6=∅}

and a relation₄byx ₄ yif Ix ≤ Iy, where the

inequality is interpreted component-wise. Let x = (∅, . . . ,∅) be an N-tuple of empty configura- tions, so that Ix = (0, . . . ,0); andx = ({ξ1}, . . . ,{ξN}), withξi ∈ EAi, so that Ix = (1, . . . ,1). Refer toxand xasquasi-minimalandquasi-maximalelements respectively. Denote byΦ˜T ,min ₌

˜ ΦT ,min₁ , . . . ,Φ˜T,min_N and Φ˜T,max = ˜ ΦT,max₁ , . . . ,Φ˜N T ,max

the minimal, respectively, maximal processes started at time−T; set

˜ ΦT ,min_−i (n) =[ j<i ˜ ΦT ,min_j (n)[ j>i ˜

ΦT ,min_j (n−1) ; χT,min_i (n) =1_{U(Φ˜T ,min

−i (n))⊃Ai}. (3.10) ˜ ΦT,max_−i (n) =[ j<i ˜ ΦT ,max_j (n)[ j>i ˜

ΦT,max_j (n−1) ; χT,max_i (n) = 1_{U(Φ˜T ,max

−i (n))⊃Ai}. (3.11) Furthermore forn= 0,−1,−2, . . .andi= 1, . . . , N let(X(n, i), X0(n, i))denote a tuple of Pois- son processes onEAi such thatX

0_{(n, i)is conditioned to contain at least one germ andX(n, i) ⊆}

X0(n, i). It is relatively straightforward to simulate such a tuple: Algorithm 3.6 (Simulation of(X(n, i), X0(n, i))).

Sett = 1; drawYi ∼Poisson(λ)onEAi. ifYi 6=∅: setX(n, i) = X0(n, i) =Yi;t = 0

else: setX(n, i) =∅. whilet 6= 0:

ifYi 6=∅: setX0(n, i) = Yi;t= 0.

else: sett=t+ 1. return(X(n, i), X0(n, i)). Denoting the integer part of T₂ bybT

2c, the exact Gibbs sampler is summarized below.

Algorithm 3.7 (Exact Gibbs: Conditional Boolean Model). FixT >0.

whileT >0:

fori= 1, . . . , N:

construct realizations of(X(n, i), X0(n, i))forn ∈

−T + 1, . . . ,−bT

2c .

InitializeΦ˜T ,min(−T) = x,Φ˜T ,max(−T) =x. forn∈ {−T + 1, . . . ,0}:

fori∈ {1, . . . , N}:

ifχT ,max_i (n) = 1: setΦ˜T ,min_i (n) = X(n, i). else: setΦ˜T,min_i (n) = X0(n, i).

ifχT ,min_i (n) = 1: setΦ˜T ,max_i (n) = X(n, i). else: setΦ˜T,max_i (n) = X0(n, i).

ifΦ˜T ,min_i (0) = ˜ΦT ,max_i (0)for alli∈ {1, . . . , N}(ie. coalescence): setT =−1. else: setT = 2T.

returnΦ˜T ,min₍₀₎

As noted in Algorithm 3.4, when extending the(X(n, i), X0(n, i))backwards over some interval in one iteration of the algorithm, it is vital to re-use the same realizations for all subsequent iterations.

Properties of the Exact Gibbs Sampler

Algorithm 3.7 is now shown to satisfy the following relationships, which are analogues of the results for the Cai & Kendall Algorithm 3.4. The following Lemmas will be used to prove that its output is indeed a conditional Boolean model (Theorem 3.2).

Lemma 3.6 (Sandwiching). The processesΦ˜T ,min_andΦ˜T ,max_{of Algorithm 3.7 satisfy}

˜

ΦT ,min_i (n)₄Φ˜T ,max_i (n), for alli∈ {1, . . . , N}, n∈ {−T, . . . ,0}. (3.12) 76

Proof. The initialization procedure ensures that Eq. (3.12) holds at time −T. Suppose that this relationship holds at some timen−1>−T. The tuple(X(n, i), X0(n, i))is such thatX(n, i)⊆

X0(n, i)for alli = 1, . . . , N andn = 0,−1, . . . (cf. Algorithm 3.6). The update at timen for the upper process is determined by the current state of the lower process and vice versa; this ‘cross-over’ ensures that: Φ˜T,max_i is set equal toX(n, i) only ifΦ˜T ,min_i is; and Φ˜T ,min_i is set equal to X0(n, i)

only if Φ˜T ,max_i is. The result then follows via induction along the sequence of times{−T, . . . ,0}, since the maximal and minimal processes are initialized so as to satisfy Eq. (3.12).

Lemma 3.7 (Coalescence). If Φ˜T ,min(n∗) = ˜ΦT,max(n∗) then Φ˜T,min(n) = ˜ΦT ,max(n) for all

n∈ {n∗, . . . ,0}.

Proof. At timen∗there is no longer any distinction between the upper and lower processes. So from timen∗the updates forΦ˜T ,min _andΦ˜T,max_{will be identical for each of theN components.}

Lemma 3.8 (Funnelling). For all−S≤ −T ≤n≤0andi= 1, . . . , N

˜

ΦT,min_i (n)₄Φ˜S,min_i (n)₄Φ˜S,max_i (n)₄Φ˜T ,max_i (n). (3.13)

Proof. By definition of ‘₄’, ifx= (x1, . . . , xN),y= (y1, . . . , yN)andxi ⊆yi for alli= 1, . . . , N

thenx₄ y. The initializing procedure ensures that Eq. (3.13)holds at time−T. Suppose that this relationship holds up timen−1>−T; consider updating the first component at timen.

By virtue of Eq. (3.13) holding at time n−1, Φ˜T,min_i (n−1) ⊆ Φ˜S,min_i (n−1). Therefore if

χT,min₁ (n) = 1 then χS,min₁ (n) = 1, hence the ‘cross-over’ trick ensures that Φ˜T,max₁ (n) will be set equal toX(n,1)only ifΦ˜S,max₁ (n)also is. Similarly,Φ˜T ,min₁ (n)is set equal toX0(n,1)only if

˜

ΦS,min₁ (n)also is. Either way

˜

Φ₁T ,min(n)⊆Φ˜S,min₁ (n)⊆Φ˜S,max₁ (n)⊆Φ˜T,max₁ (n)

whereΦ˜T,min₁ (n)⊆Φ˜T ,max₁ (n)is established by Lemma 3.6. Induction alongi= 1, . . . , N gives

˜

ΦT,min_i (n)⊆Φ˜S,min_i (n)⊆Φ˜S,max_i (n)⊆Φ˜_iT ,max(n), for alli= 1, . . . , N.

The results follows by induction along the sequence of times{−T, . . . ,0}.

Proof. IfX(n, i) = X0(n, i), for alliand somen <0, then, forT >−n,Φ˜T,min_i (n) = ˜ΦT ,max_i (n); moreover Φ˜T,min_i (0) = ˜ΦT ,max_i (0) for all i and hence the algorithm terminates. The event that

X(n, i) = X0(n, i) has probability 1−e−λm2[E_Ai] _{(cf. Algorithm 3.6), with m}

2 denoting

Lebesgue measure. Thus

P[X(n, i) = X0(n, i) for alli] =

Y

i

1−e−λm2[E_Ai]_>0

and independent of n. The event {X(n, i) = X0(n, i) for alli} is independent of events in the past. So, applying the second Borel-Cantelli Lemma, {X(n, i) = X0(n, i) for alli} happens al- most surely for infinitely manyn. The algorithm hence terminates in finite time.

Theorem 3.2. Algorithm 3.7 terminates in finite time, and the output of the algorithm has distribu- tionπ_λC, the distribution of a Poisson process of intensityλconditioned to cover the setC.

Proof. Lemma 3.9 establishes the first part. Lemmas 3.6, 3.7 & 3.8 and the arguments of Theorem 3.1 then completes the proof.