Perfect simulation algorithm of a trajectory under a Feynman-Kac law
Perfect simulation algorithm of a trajectory under
a Feynman-Kac law
University of Warwick, Recent Advances in Sequential Monte
Carlo - Sep 19-21, 2012
C. Andrieu, A. Doucet, S. Rubenthaler
University of Bristol, University of Oxford, Université de Nice-Sophia Antipolis
Perfect simulation algorithm of a trajectory under a Feynman-Kac law
Introduction
Notations
I
X ,X
is a Markov chain in E with initial law M
M (say E = Rd or Zd )
G , G , · · · : E → R+ are potentials
total time : P
1
2
,...
transition
I
I
1
2
One is interested in the law :
Q
E(f (X1 , . . . , XP ) Pi =−11 Gi (Xi ))
.
π(f ) =
Q
E( Pi =−11 Gi (Xi ))
1
and
Perfect simulation algorithm of a trajectory under a Feynman-Kac law
Metropolis-like algorithm on extended space
Simple branching system
I Start with
N
1
particles.
Xni (i -th particle at time n) has Ain+ osprings
i
i
with law P(An+ = j ) = fn+ (Gn (Xn ), j ) (independant of other
I The particle
1
1
1
particles).
I Total number of particles :
Nn+
1
=
PNn i
i =1 An+1 .
Density :
N1
P
Y
Y
i
i
i
q0 (N1 , . . . , NP , (An ), (Xn )) = M1 (X1 )
n=2
i =1


NY
n−1
Y

fn (Gn−1 (Xni −1 ), Ain ) M (Xni −1 , Xnj )
i =1
j ∈...
Perfect simulation algorithm of a trajectory under a Feynman-Kac law
Metropolis-like algorithm on extended space
Extension
Take a trajectory and draw a branching system conditionned to
contain this trajectory.
0
+1
+2
Perfect simulation algorithm of a trajectory under a Feynman-Kac law
Metropolis-like algorithm on extended space
Extension
Take a trajectory and draw a branching system conditionned to
contain this trajectory.
0
+2
0
+1
+1
+2
0
+1
0
-1
-2
1
0
Perfect simulation algorithm of a trajectory under a Feynman-Kac law
Metropolis-like algorithm on extended space
Extension
When a particle is blue at position
x
at time
n − 1, the number of
children is chosen with law :
P(j
osprings)
=b
fn (Gn−1 (x ), j ) =
1
fn (Gn−1 (x ), j )
.
− fn (Gn−1 (x ), 0)
Perfect simulation algorithm of a trajectory under a Feynman-Kac law
Metropolis-like algorithm on extended space
Extension
When a particle is blue at position
x
at time
n − 1, the number of
children is chosen with law :
P(j
osprings)
=b
fn (Gn−1 (x ), j ) =
1
fn (Gn−1 (x ), j )
.
− fn (Gn−1 (x ), 0)
fn such that ffnn ((gg ,,jj )) = kGngk∞ (∀n, j , g ) (take
fn (g , 0) = 1 − kGngk∞ , fn (g , j ) = kn kGgn k∞ , 1 ≤ j ≤ kn ).
We choose
b
Perfect simulation algorithm of a trajectory under a Feynman-Kac law
Metropolis-like algorithm on extended space
Proposal
P and
q on the space of (size of
Take a branching system like above, select a particle at time
its ancestral line. We get some density
each generation)×(numbers of osprings)×(positions)×(special
trajectory).
0
+1
+1
0
+2
-2
0
0
-1
1
0
Perfect simulation algorithm of a trajectory under a Feynman-Kac law
Metropolis-like algorithm on extended space
Proposal
P and
q on the space of (size of
Take a branching system like above, select a particle at time
its ancestral line. We get some density
each generation)×(numbers of osprings)×(positions)×(special
trajectory).
0
+1
+1
0
+2
-2
0
0
-1
1
0
Perfect simulation algorithm of a trajectory under a Feynman-Kac law
Metropolis-like algorithm on extended space
Proposal
P and
q on the space of (size of
Take a branching system like above, select a particle at time
its ancestral line. We get some density
each generation)×(numbers of osprings)×(positions)×(special
trajectory).
0
+1
+1
0
+2
-2
0
0
-1
1
0
Perfect simulation algorithm of a trajectory under a Feynman-Kac law
Metropolis-like algorithm on extended space
Proposal
P and
q on the space of (size of
Take a branching system like above, select a particle at time
its ancestral line. We get some density
each generation)×(numbers of osprings)×(positions)×(special
trajectory).
0
+1
+1
0
+2
-2
0
0
-1
1
0
Perfect simulation algorithm of a trajectory under a Feynman-Kac law
Metropolis-like algorithm on extended space
Acception/rejection
Target law : trajectory with the law
π,
to which we add a
(conditionned) branching system. We have a law
π
b(. . . )
=
q (. . . )
with
Z
Q
:= E( Pn=−11 Gn (Xn ))
NP
QP −1
i =1 kGi k∞ ,
NZ
1
(partition function).
π
b
on forests.
Perfect simulation algorithm of a trajectory under a Feynman-Kac law
Metropolis-like algorithm on extended space
Markov chain
I start with trajectory, extend it into a forest
Perfect simulation algorithm of a trajectory under a Feynman-Kac law
Metropolis-like algorithm on extended space
Markov chain
I start with trajectory, extend it into a forest
I when at point
X
(in forests): propose
X,
Perfect simulation algorithm of a trajectory under a Feynman-Kac law
Metropolis-like algorithm on extended space
Markov chain
I start with trajectory, extend it into a forest
I when at point
X
(in forests): propose
I accept with probability
π
b(X )q (X )
π
b(X )q (X )
P
=N
NP ,
X,
Perfect simulation algorithm of a trajectory under a Feynman-Kac law
Metropolis-like algorithm on extended space
Markov chain
I start with trajectory, extend it into a forest
I when at point
X
(in forests): propose
I accept with probability
π
b(X )q (X )
π
b(X )q (X )
X,
P
=N
NP ,
I then prune the forest to have only the colored trajectory.
Perfect simulation algorithm of a trajectory under a Feynman-Kac law
Metropolis-like algorithm on extended space
Markov chain
I start with trajectory, extend it into a forest
I when at point
X
(in forests): propose
I accept with probability
π
b(X )q (X )
π
b(X )q (X )
X,
P
=N
NP ,
I then prune the forest to have only the colored trajectory.
We have here a Markov process on the trajectory space whose
invariant law is
π.
Perfect simulation algorithm of a trajectory under a Feynman-Kac law
Metropolis-like algorithm on extended space
Markov chain
Perfect simulation algorithm of a trajectory under a Feynman-Kac law
Coupling from the past
Coupling from the past in a nutshell
Transition of a Markov chain
(Uk ):
Zk +
Suppose
(Zk )
n ≥ 0, set
T
such that
1
expressed with i.i.d. variables
= h(Zk , Uk +1 ).
has invariant law
Z−z n = z , Z−z n+
If
1
(Zk )
π.
For a starting point
= h(Z−z n , Un ) , . . . ,
Z z = Z z 0 , ∀z , z 0 : Z−z T
0
0
= z,
z
and
Z z = h(Z−z , U
0
Z−z 0T
1
= z 0,
1
then
).
Z z ∼ π.
0
Perfect simulation algorithm of a trajectory under a Feynman-Kac law
Coupling from the past
Coupling from the past algorithm
Perfect simulation algorithm of a trajectory under a Feynman-Kac law
Coupling from the past
Coupling from the past algorithm
Perfect simulation algorithm of a trajectory under a Feynman-Kac law
Coupling from the past
Detection of a coupling time
Look for a time such that the red proposal is accepted for all
possible blue trajectories.
0
0
+1
+2
+1
0
?
1
-2
?
?
?
?
?
?
?
Bound the number of squares at the bottom and you bound the
acceptation ratio.
Perfect simulation algorithm of a trajectory under a Feynman-Kac law
Examples
Directed polymers in
U (i , j ) i.i.d.
Z
i ∈ N, j ∈ Z). Take (Xn ) the
simple random walk in Z with X = 0. Set Gi (j ) = exp(−β U (i , j )).
To draw a trajectory of length n , the cost is O (n ).
Draw
of Bernoulli law (
0
2
Figure: 500 trajectories
Perfect simulation algorithm of a trajectory under a Feynman-Kac law
Examples
Directed polymers in
Z
Figure: 100 trajectories
Perfect simulation algorithm of a trajectory under a Feynman-Kac law
Examples
Radar detection
Transition (in
R)
:
M (x , dy ) = √ πb
1
Gn (x ) = √ πc exp − (x −cYn )
Yn = Xn + c n (n ∼ N (0, 1)).
1
2
2
2
2 2
2
2
exp
, where
ax )
− (y −
2b 2
(Xn )
2
,
has transition
You can bound the number of ospring of any trajectory by
discretizing the space (works for
: X
: Y
:
tion
perfect simula-
a ∈ [−1, 1]).
M
and