If T is the coalescence time of the Propp and Wilson [15], perfect simulation algorithm, the aim of this paper is to show that T depends on the second largest eigenvalue modulus of the transition matrix of the underlying Markov chain. This gives a relationship between the ordering based on the speed of convergence to stationarity in total variation distance and the ordering dened in terms of speed of coalescence in perfect simulation. Key words and phrases: Peskun ordering, Covariance ordering, Effciency ordering, MCMC, time-invariance estimating equations, asymptotic variance, continuous time Markov chains.

Leisen Fabrizio

Mira Antonietta

English

Research Papers in Economics

 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
UNIVERSITÀ DELL'INSUBRIA 
FACOLTÀ DI ECONOMIA 
 
http://eco.uninsubria.it 
 
F. Leisen, A. Mira 
 
Coalescence time and second largest 
eigenvalue modulus in the 
monotone reversible case 
 
   2006/11  
© Copyright  F. Leisen, A. Mira 
Printed in Italy in July 2006 
Università degli Studi dell'Insubria 
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All rights reserved. No part of this paper may be reproduced in 
any form without permission of the Author. 
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Authors only, and not necessarily the ones of the Economics 
Faculty of the University of Insubria. COALESCENCE TIME AND SECOND
LARGEST EIGENVALUE MODULUS IN THE
MONOTONE REVERSIBLE CASE
Fabrizio Leisenand Antonietta Miray
July 27, 2006
Abstract
If T is the coalescence time of the Propp and Wilson [15], perfect sim-
ulation algorithm, the aim of this paper is to show that T depends on the
second largest eigenvalue modulus of the transition matrix of the under-
lying Markov chain. This gives a relationship between the ordering based
on the speed of convergence to stationarity in total variation distance and
the ordering dened in terms of speed of coalescence in perfect simulation.
1 Introduction
In [15], Propp and Wilson dene an algorithm to sample "exactly" from a
probability distribution  dened on a nite set E. To this aim, we need
a Markov chain that has  as its unique stationary distribution. The
updating rule of the Markov chain can be considered as a function:
 : E  [0;1] ! E
(x;U) ! (x;U)
where U is a uniform random variable on the interval [0;1]. Let
P = fpijgi;j2E be the transition matrix of the Markov chain, then
pij = P((i;U) = j). Clearly, given a distribution of interest , the
choice of the Markov chain is not unique, and an open question is how
this choice can be made "optimally" in some sense. For Markov chains
reversible with respect to , we can dene an ordering, called "speed of
convergence", that is based on the second largest eigenvalue modulus
(SLEM throughout the paper) of the transition matrix (see [12] for a
survey on orderings for Markov chains). In this ordering convergence is
measured in terms of total variation distance. The purpose of this paper
Universit a di Modena e Reggio Emilia, Dipartimento di Matematica, Via Campi 213/b
Modena, E-mail: leisen.fabrizio@unimore.it
yUniversit a dell'Insubria, Dipartimento di Economia, Via Monte Generoso 71, 21100
Varese, E-mail: antonietta.mira@uninsubria.it
1is to show that, in monotone reversible cases, the speed of convergence
ordering implies an ordering on the coalescence times. In Section 2, we
review the Propp and Wilson perfect simulation algorithm and set the
notation. In Section 3 we give preliminary theorems and in Section 4 we
state the main result that will be used, in Section 5, to dene an
ordering for the coalescence times.
2 Propp and Wilson exact algorithm
In their seminal paper, [15], Propp and Wilson device a way to turn a
Markov chain Monte Carlo algorithm (MCMC) into a perfect simulation
algorithm. It is well know that in MCMC the crucial problem is how to
detect when the Markov chain has reached its stationary regime. Many
convergence diagnostics have been designed that aim at detecting failure
of convergence based on the simulated path of the Markov chain. All
these convergence diagnostics can at most give negative answers but will
never reassure the user on the fact that their Markov chains have reached
stationarity. Propp and Wilson propose a clever way to simulate a Markov
chain so that at the end of the perfect simulation algorithm an exact
sample from the stationary distribution, , is obtained.
2.1 The exact simulation algorithm
Let :
 E = fx1;x2;;xNg be a nite states space
 U 1;U 2;U 3; be a sequence of i.i.d. random variables
  = (x;U) x 2 E be the updating rule of an ergodic Markov chain
with stationary distribution  with the property that for all i;j 2 E
P((i;Ut) = j) = pij.
2Consider the following exact simulation algorithm:
step -1 Compute :
1 = (x1;U 1)
2 = (x2;U 1)
...
...
n = (xn;U 1)
If 1 = 2 =  = n = k for some k 2 E then return k.
else go to step -2
step -2 Compute :
1 = ((x1;U 2);U 1)
2 = ((x2;U 2);U 1)
...
...
n = ((xn;U 2);U 1)
If 1 = 2 =  = n = k then return k.
else go to step -3
.......
.......
.......
step  t Compute :
1 = ((((x1;U t);U t+1);U 2);U 1)
2 = ((((x2;U t);U t+1);U 2);U 1)
...
...
n = ((((xn;U t);U t+1);U 2);U 1)
If 1 = 2 =  = n = k then return k.
else go to step  t   1
.......
Propp and Wilson, [15], proved that this algorithm ends with probability
one and that the value k that is returned is distributed as . The time
step, T, where 1 = 2 =  = n = k is called "coalescence time" and
it depends on U 1;U 2;U 3;, so T is a random variable.
2.2 Monotone case and Coupling inequality
Let E be a partial ordering on the states space E and assume that E
admits maximum and minimum with respect to this ordering. Through-
out the paper we denote the maximal and minimal state by ^ 1 and ^ 0.
If the updating rule of an ergodic Markov chain with stationary distribu-
tion  satises the property:
8x;y 2 E such that x E y we have (x;U) E (y;U)
then the chain is called monotone. This property is very important for the
Propp and Wilson algorithm because it guaranties that it is sucient to
run only two chains, starting at ^ 0 (minimal chain) and ^ 1 (maximal chain)
respectively, and to verify coalescence only for these two chains. The
monotonicity property ensures that if the maximal and minimal chain
3have coalesced, then all other chains (started from any other possible
value in the state space) will also coalesce because they will be \trapped"
between the maximal and the minimal chain. Computationally, running
only the maximal and minimal chain this is more convenient then running
all possible chains.
The question is: if 1;2 are two monotone updating rules of two ergodic
Markov chains with stationary distribution  and transitions matrix P
and Q respectively, which one coalesces before? In this paper the above
question will be addressed. For monotone Markov chains we have two
inequalities called "coupling inequalities":
P(T > t)  ld(t)
P(T > t)  d(t)  d(t)
where l is the length of the longest chain in the totally ordered space E
and :
d(t) = max
x;y2E
jjP
t(x;)   P
t(y;)jjTV ;
d(t) = max
x2E
jjP
t(x;)   ()jjTV :
3 Preliminaries
For each xed M 2 N, we dene two Markov chains on the state space E
in the following way :
X
M
0 = ^ 0 X
M
1 = (^ 0;U M) X
M
2 = (X
M
1 ;U M+1)X
M
M = (X
M
M 1;U 1);
Y
M
0 = ^ 1 Y
M
1 = (^ 1;U M) Y
M
2 = (Y
M
1 ;U M+1)Y
M
M = (Y
M
M 1;U 1):
If T is the coalescence time, we have:
X
T
T ;Y
T
T  :
We now construct a new Markov chain on the enlarged state space EE,
in the following way:
Z
M
0 = (^ 0;^ 1)
Z
M
1 = ^ ((^ 0;^ 1);U M) = ((^ 0;U M);(^ 1;U M)) = (X
M
1 ;Y
M
1 )
Z
M
2 = ^ (Z
M
1 ;U M+1) = ((X
M
1 ;U M+1);(Y
M
1 ;U M+1)) = (X
M
2 ;Y
M
2 )
  
  
  
Z
M
M = ^ (Z
M
M 1;U 1) = ((X
M
M 1;U 1);(Y
M
M 1;U M)) = (X
M
M ;Y
M
M )
The transition matrix of the Markov chain Z dened above is:
~ P = fp(xi1;xi2)!(xj1;xj2)g(xi1;xi2);(xj1;xj2)2EE
where :
p(xi1;xi2)!(xj1;xj2) = pxi1xj1pxi2xj2:
4It is easy to check that this is indeed a stochastic matrix in fact, for any
(xi1;xi2) we have that:
X
(xj1;xj2)2EE
p(xi1;xi2)!(xj1;xj2) =
X
xj12E
X
xj22E
pxi1xj1pxi2xj2 = 1:
We note that the matrix ~ P can be written in the convenient form:
~ P =
2
6
6
6
6
6
6
4
px1x1P px1x2P  px1xNP
px2x1P px2x2P  px2xNP
   
   
   
pxNx1P pxNx2P  pxNxNP
3
7
7
7
7
7
7
5
and it's easy to prove that :
~ P
n =
2
6
6
6
6 6
6
6
4
p
(n)
x1x1P
n p
(n)
x1x2P
n  p
(n)
x1xNP
n
p
(n)
x2x1P
n p
(n)
x2x2P
n  p
(n)
x2xNP
n
   
   
   
p
(n)
xNx1P
n p
(n)
xNx2P
n  p
(n)
xNxNP
n
3
7
7
7
7 7
7
7
5
Theorem 3.1 If P is reversible w.r.t.  then ~ P is also reversible w.r.t.
~ , where ~  is dened, for all i;j 2 E  E, as:
~ (i;j) = (i)(j):
Proof:
We show that for all (xi1;xi2);(xj1;xj2) 2 E  E the following identity
holds:
~ (xi1;xi2)p(xi1;xi2)!(xj1;xj2) = ~ (xj1;xj2)p(xj1;xj2)!(xi1;xi2):
The above identity follows from:
~ (xi1;xi2)p(xi1;xi2)!(xj1;xj2) = (xi1)(xi2)pxi1xj1pxi2xj2
= (xi1)pxi1xj1(xi2)pxi2xj2
= pxj1xi1(xj1)pxj2xi2(xj2)
= pxj1xi1pxj2xi2(xj1)(xj2)
= ~ (xj1;xj2)p(xj1;xj2)!(xi1;xi2) 
The next theorem establishes a connection between the Z chain and the
two chains that start from ^ 0 and ^ 1. We use this notation: ^ 0, ^ 1 for the
measures on E, concentrated, respectively, on ^ 0, ^ 1, and (^ 0;^ 1) for the
5measure on E  E concentrated on (^ 0;^ 1). Note that,
(^ 0;^ 1)(x;y) = ^ 0(x)^ 1(y), so we can write (^ 0;^ 1) in this compact form:
(^ 0;^ 1) =
2
6
6
6
6
6
6
4
^ 0(x1)^ 1
^ 0(x2)^ 1



^ 0(xN)^ 1
3
7
7
7
7
7
7
5
Vectors are considered as column and we use the superscript t to denote
the transpose.
Theorem 3.2 Let T be a random variable with values in f1;2;g. The
following two statements are equivalent:
1. X
T
T   and Y
T
T  
2. Z
T
T  ~ 
Proof:
"(1 ) 2)" If X
T
T  ,Y
T
T   then
P(X
T
T = i) = (i) = (^ 0P
T)(i) = p
(T)
^ 0i
P(Y
T
T = i) = (i) = (^ 1P
T)(i) = p
(T)
^ 1i
So

t
(^ 0;^ 1)
~ P
T = [^ 0(^ 0)
t
^ 1p
(T)
^ 0x1
~ P
T;^ 0(^ 0)
t
^ 1p
(T)
^ 0x2
~ P
T;;^ 0(^ 0)
t
^ 1p
(T)
^ 0xN
~ P
T]
= [(x1)
t;(x2)
t;;(xN)
t]
= ~ 
"(2 ) 1)" If Z
T
T  ~  then
jj(^ 0;^ 1)P
T   ~ jjTV = 0
Since:
jj
t
(^ 0;^ 1)P
T   ~ jjTV =
1
2
PN
i=1
PN
j=1 j(
t
(^ 0;^ 1)
~ P
T)(xi;xj)   ~ (xi;xj)j
=
1
2
PN
i=1
PN
j=1 jp
(T)
^ 0xip
(T)
^ 1xj   (xi)(xj)j
= (we apply the triangular inequality)

1
2
PN
i=1 jp
(T)
^ 0xi   (xi)j
= jj
t
^ 0P
T   jjTV
it follows:
0  jj
t
^ 0P
T   jjTV  jj
t
(^ 0;^ 1)P
T   ~ jjTV = 0:
6Analogously, we can show that:
jj
t
^ 1P
T   jjTV = 0
We can thus conclude that:
X
T
T ;Y
T
T  

We now want to derive the eigenvalues of the matrix ~ P from the
eigenvalues of the matrix P. We recall that a reversible matrix is always
diagonalizable. In particular let R be the matrix:
R = diag((x1);(x2);;(xN))
then the matrix
S = R
1
2 PR
  1
2
is symmetric. So there exists an orthogonal matrix O such that:
D = O
tSO
where D is a diagonal matrix that has the same eigenvalues of P. Thus,
if 1;2;;N are the eigenvalues of P, then
D = diag(1;2;;N):
Theorem 3.3 If 1;2;;N are the eigenvalues of P then the set:
fij : i;j = 1;;Ng
is the set of all the eigenvalues of ~ P.
Proof: We use the preceding technique. Let
~ R = diag(~ (x1;x1); ~ (x1;x2);; ~ (xN;xN))
If S = fsxixjgxi;xj2E from a direct calculation it follows:
~ S =
2
6
6
6 6
6
6
4
sx1x1S sx1x2S  sx1xNS
sx2x1S sx2x2S  sx2xNS
   
   
   
sxNx1S sxNx2S  sxNxNS
3
7
7
7 7
7
7
5
And if O = foxixjgxi;xj2E then we can choose ~ O:
~ O =
2
6
6
6 6
6
6
4
ox1x1O ox1x2O  ox1xNO
ox2x1O ox2x2O  ox2xNO
   
   
   
oxNx1O oxNx2O  oxNxNO
3
7
7
7 7
7
7
5
7This is an orthogonal matrix and ~ D = ~ O
t ~ S ~ O is a diagonal matrix with
the same eigenvalues of ~ P. This matrix has the form:
~ D =
2
6
6
6 6
6
6
4
1D 0  0
0 2D  0
   
   
   
0 0  ND
3
7
7
7 7
7
7
5

The following two technical lemmas will be used in Section 4.
Lemma 3.1 Let P be a reversible irreducible transition matrix on the
nite state space E with stationary distribution . Then, for n  1,all
i 2 E, and all A  E
j
t
iP
n(A)   
t(A)j  (
1   (i)
(i)
)
1
2 min((A)
1
2;
1
2
)
n
where  is the SLEM of P.
An immediate consequence of the Lemma (3.1) is that if A = fig then
j
t
iP
n(i)   
t(i)j  
n
For a proof of Lemma (3.1), see Theorem 3.3, p.209 of [3].
If Q = fqijgi;j2E is a stochastic matrix then we can dene the
"ergodicity coecient of Dobrushin, " as:
(Q) = max
x;y
1
2
X
i2E
jqxi   qyij:
So obviously:
d(t) = (P
t)
where d(t) is the norm dened in Section 2.2. We recall a result on
stochastic matrices and the ergodicity coecient from [17] pp. 81-82:
Theorem 3.4 Let w = fwig be an arbitrary vector and P = fpijg a
stochastic matrix indexed by E. If z = Pw, z = fzig, then, for any two
indexes h;h
0:
zh   zh0 
1
2
X
j2E
jphj   ph0jjfmax
j2E
wj   min
j2E
wjg
and
fmax
j2E
zj   min
j2E
zjg  (P)fmax
j2E
wj   min
j2E
wjg
or equivalently:
max
h;h02E
jzh   zh0j  (P)f max
j;j02E
jwj   wj0jg: (1)
An immediate consequence of this Theorem is the following Lemma.
8Lemma 3.2 If P is a stochastic matrix with  as its second largest
eigenvalues modulus, then
d(t)  
t
Proof: If w = fwigi2E is an eigenvector with corresponding eigenvalue
;jj < 1. Then:
Pw = w
P
tw = 
tw:
If z = P
tw then
zh= P
tw(h) = 
twh
zh0= P
tw(h
0) = 
twh0
maxh;h02E jzh   zh0j= maxh;h02E j
twh   
twh0j = jj
t maxh;h02E jwh   wh0j
So by applying (1) we obtain:
jj
t max
h;h02E
jwh   wh0j  (P
t)f max
j;j02E
jwj   wj0jg
But:
maxh;h02E jwh   wh0j = maxj;j02E jwj   wj0j
d(t) = (P
t)
So if  = jj is the SLEM of P we obtain:
d(t)  
t

Now recall that
d(t)  d(t)  2d(t)
so
2d(t)  
t:
We note that, given two transition matrices, P1 and P2, such that
1 < 2, then ~ 1 < ~ 2, where ~ 1 and ~ 2 are the SLEM of the enlarged
chain, ~ P1 and ~ P2 respectively.
4 Main result
Let us begin with a remark. If T is a coalescence time we have that:
jj
t
(^ 0;^ 1) ~ P
T   ~ jjTV
Furthermore, for monotone Markov chains, we have that:
d(T) = max
(xi;xj)2(EE)
jj
t
(xi;xj) ~ P
T   ~ jjTV = 0:
9This is due to the fact that, if the minimal and maximal chains (i.e. the
chains starting from ^ 0 and ^ 1 respectively), coalesce then, thanks to the
monotonicity property, chains starting from any other initial state also
coalesce. We are now ready to state the main result.
Theorem 4.1 Let 1;2 be two monotone updating rules of Markov
chains with transitions matrix P1 and P2 reversible with respect to . If
T1;T2 are the coalescence times respectively of 1, 2 and
1 < 2
where 1 = SLEM (P1);2 = SLEM (P2),then
T1 < T2; a.s. :
Proof: If 1 < 2 then
~ 1 < ~ 2
where ~ 1 = SLEM ( ~ P1); ~ 2 = SLEM ( ~ P2):
If T1,T2 are coalescence times then:
jj
t
(^ 0;^ 1) ~ P1
T1   ~ jjTV = 0;
jj
t
(^ 0;^ 1) ~ P2
T2   ~ jjTV = 0:
From the rst we obtain:
1
2
X
(xi;xj)2EE
j(
t
(^ 0;^ 1) ~ P1
T1)(xi;xj)   ~ (xi;xj)j = 0
But this is a sum of positive terms, so every terms must be equal to
zero. In particular:
j(
t
(^ 0;^ 1) ~ P1
T1)(^ 0;^ 1)   ~ (^ 0;^ 1)j = 0: (2)
Suppose that T1  T2, then
jj
t
(^ 0;^ 1) ~ P2
T1   ~ jjTV = 0
but from the remark at the beginning of the section:
max
(xi;xj)2(EE)
jj
t
(xi;xj) ~ P2
T1   ~ jjTV = 0: (3)
By combining (2) with (3) and using the lemmas of the previous section,
we obtain:
0 = (2)   2  (3)  ~ 1
T1   ~ 2
T1:
We thus obtain:
~ 2  ~ 1;
which contradicts the hypothesis.

105 An ordering for coalescence time
Consider two transition matrices, P1 and P2, with the same stationary
distribution, that are monotone with respect to some partial ordering
dened on the state space. Let 1;2 be the SLEMs and T1;T2 be the
coalescence times of P1 and P2 respectively.
Denition 5.1 We say that P1 dominates P2 in terms of speed of con-
vergence and write P1 S P2, if
1  2:
This ordering was already dened in [12].
Denition 5.2 We say that P1 dominates P2 in terms of coalescence
time and write P1 C P2, if
T1  T2:
Theorem 5.1 If P1 S P2 then P1 C P2.
The result follows from Theorem 4.1 in Section 4.
6 Conclusions
We give a sucient condition for selecting an updating rule in perfect
simulation that is optimal in the sense of minimizing the coalescence time.
The results holds for nite state spaces and monotone chains. We plan to
investigate perfect sampling algorithm with continuous state spaces and
non-monotone updating mechanism in further research.
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12

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