Introduction
Island bootstrap approximation
The double bootstrap algorithm
Extensions
On the convergence of Island particle models
C. Dubarry, P. Del Moral, E. Moulines
Institut Mines-Télécom, Télécom ParisTech/ Télécom SudParis, INRIA Bordeaux
June 14, 2012
C. Dubarry, P. Del Moral, E. Moulines
1/32
Introduction
Island bootstrap approximation
The double bootstrap algorithm
Extensions
Outline
1
Introduction
2
Island bootstrap approximation
3
4
The double bootstrap algorithm
Algorithm description
Bias and variance of the double bootstrap
Numerical application
Extensions
C. Dubarry, P. Del Moral, E. Moulines
1/32
Introduction
Island bootstrap approximation
The double bootstrap algorithm
Extensions
Outline
1
Introduction
2
Island bootstrap approximation
3
4
The double bootstrap algorithm
Algorithm description
Bias and variance of the double bootstrap
Numerical application
Extensions
C. Dubarry, P. Del Moral, E. Moulines
2/32
Introduction
Island bootstrap approximation
The double bootstrap algorithm
Extensions
Notations
(Xn , Xn )n≥0 is a sequence of measurable sets.
Bb (Xn , Xn ) is the Banach space of all bounded and measurable functions
on (Xn , Xn ).
(Xn )n≥0 is a non-homogenous Markov chain with initial distribution η0 ,
and Markov kernels (Mn )n≥1 .
Feynman-Kac ow
def
ηn (fn ) = γn(fn )/γn (1) ,

Y
def
γn (fn ) = E fn (Xn )
gp (Xp ) .
0≤p<n
C. Dubarry, P. Del Moral, E. Moulines
3/32
Introduction
Island bootstrap approximation
The double bootstrap algorithm
Extensions
Feynman-Kac ow
Dene by P(Xn , Xn ) the set of probabilities on (Xn , Xn ).
The sequence of probabilities (ηn )n≥0 satises the following recursion:
ηn+1 = Ψn (ηn )Mn+1 ,
where Ψn : P(Xn , Xn ) → P(Xn , Xn ) is dened by:
def
Ψn (ηn )(An ) =
1
ηn (gn )
Z
C. Dubarry, P. Del Moral, E. Moulines
gn (xn ) ηn (dxn ) ,
An ∈ Xn .
An
4/32
Introduction
Island bootstrap approximation
The double bootstrap algorithm
Extensions
Outline
1
Introduction
2
Island bootstrap approximation
3
4
The double bootstrap algorithm
Algorithm description
Bias and variance of the double bootstrap
Numerical application
Extensions
C. Dubarry, P. Del Moral, E. Moulines
5/32
Introduction
Island bootstrap approximation
The double bootstrap algorithm
Extensions
Particle approximation
⊗N1
1
Xp , X p ) = (XN
Let N1 be an integer. For any integer p we set (X
).
p , Xp
Xn , X n ) to (X
Xn+1 , X n+1 ) as the
Dene the Markov kernel M n+1 from (X
product measure
def
def
M n+1 (xn , An+1 ) =
Y
Ψn (m(xn , ·))Mn+1 (Ain+1 ) ,
1≤i≤N1
where m(xn , ·) stands for the empirical measure of xn given for any
An ∈ Xn by
def
m(xn , An ) =
N1
1 X
δ i (An ) .
N1 i=1 xn
The particles are multinomially resampled with probabilities proportional
1
to their potential {gn (xin )}N
i=1 ; new particle positions are then sampled
from the Markov kernel Mn+1 .
C. Dubarry, P. Del Moral, E. Moulines
6/32
Introduction
Island bootstrap approximation
The double bootstrap algorithm
Extensions
Particle approximation
Dene a Markov chain {X n }n≥0 where for each n ∈ N,
X n = (ξn1 , . . . , ξnN1 ) ∈ Xn
with initial distribution η 0 = η0⊗N1 and transition kernel M n+1 .
N1 -particle approximations
def
def
ηnN1 (fn ) = m(X n , fn )
Y N
def
γnN1 (fn ) = ηnN1 (fn )
ηp 1 (gp ) .
0≤p<n
C. Dubarry, P. Del Moral, E. Moulines
7/32
Introduction
Island bootstrap approximation
The double bootstrap algorithm
Extensions
Unbiasedness of the particle approximation
Theorem (Del Moral, 199x)
For any
fn ∈ Bb (Xn , Xn ), γnN1 (fn ) is an unbiased estimator of γn (fn ):


h
i
Y
E γnN1 (fn ) = E ηnN1 (fn )
ηpN1 (gp )
0≤p<n

= E fn (Xn )

Y
gp (Xp ) .
0≤p<n
C. Dubarry, P. Del Moral, E. Moulines
8/32
Introduction
Island bootstrap approximation
The double bootstrap algorithm
Extensions
The island Feynman-Kac model
N1
1
For xn = (x1n , · · · , xN
n ) ∈ Xn dene the sample averaged potential
def
g n (xn ) = m(xn , gn ) =
N1
1 X
gn (xin ) .
N1 i=1
Feynman-Kac model
γ n (1)
η n (f n ) = γ n (f n )/γ


Y
γ n (f n ) = E f n (Xn )
g p (Xp ) ,
0≤p<n
C. Dubarry, P. Del Moral, E. Moulines
9/32
Introduction
Island bootstrap approximation
The double bootstrap algorithm
Extensions
The island Feynman-Kac model
Since g n (X p ) = ηnN1 (gp )P
, the unbiasedness property implies that for any f n of
i
1
the form f n (xn ) = N1−1 N
i=1 fn (xn )


E fn (Xn )
Y

gp (Xp ) = E f n (Xn )
0≤p<n

Y
g p (Xp ) ,
0≤p<n
or equivalently
γ n (f n ) = γn (fn )
and η n (f n ) = ηn (fn ) .
C. Dubarry, P. Del Moral, E. Moulines
10/32
Introduction
Island bootstrap approximation
The double bootstrap algorithm
Extensions
The island Feynman-Kac model
From now on, a population of particles X n is called an island.
Idea: we may apply the interacting particle system approximation of the
Feynman-Kac semigroups both within each island but also across island.
To be more specic, we will now describe the so-called double bootstrap
algorithm where the bootstrap algorithm is applied both within an island
but also across the islands.
Of course, many other options are available (more to come !)
C. Dubarry, P. Del Moral, E. Moulines
11/32
Introduction
Island bootstrap approximation
The double bootstrap algorithm
Extensions
Outline
1
Introduction
2
Island bootstrap approximation
3
4
The double bootstrap algorithm
Algorithm description
Bias and variance of the double bootstrap
Numerical application
Extensions
C. Dubarry, P. Del Moral, E. Moulines
12/32
Introduction
Island bootstrap approximation
The double bootstrap algorithm
Extensions
Algorithm description
Feynman-Kac at the island level
Xn , X n ) the set of probabilities measures on (X
X n , X n ).
Dene by P(X
The sequence of measures (η n )n≥0 satises the following recursion
η n+1 = Ψn (η n )M n+1 ,
Xn , X n ) → P(X
Xn , X n ) is dened by
where Ψn : P(X
def
Ψn (η n )(An ) =
1
η n (g n )
C. Dubarry, P. Del Moral, E. Moulines
Z
g n (x) η n (dx) ,
An ∈ X n .
An
13/32
Introduction
Island bootstrap approximation
The double bootstrap algorithm
Extensions
Algorithm description
The double bootstrap algorithm
selection i mutation ξin −−−−−
−−→ b
ξn −−−−−
−−→ ξin+1
Let N2 be the number of interacting islands.
i
During the selection stage, we select randomly N2 islands b
ξn
1≤i≤N2
2
among the current islands ξin 1≤i≤N2 ∈ X N
n with probability
proportional to the empirical mean of the individuals in each island
g n (ξin ) = N1−1
N1
X
gn (ξni,j ) , 1 ≤ i ≤ N2 .
j=1
i
2
During the mutation transition, selected islands (b
ξ n )N
i=1 evolve randomly
i
to a new conguration ξn+1 according to the Markov transition M n+1 .
C. Dubarry, P. Del Moral, E. Moulines
14/32
Introduction
Island bootstrap approximation
The double bootstrap algorithm
Extensions
Algorithm description
The double bootstrap
⊗N2
⊗N2
2
2
2
XN
XN
Dene the Markov kernel LN
) to (X
n ,Xn
n+1 from (X
n+1 , X n+1 ) for
N2
1
N2
N2
1
N2
any (xn , . . . , xn ) ∈ X n and (An+1 , . . . , An+1 ) ∈ X n by
1
N2
1
N2
2
LN
n+1 (xn , . . . , xn , An+1 × · · · × An+1 )
Y
def
i
2
=
Ψn (m(x1n , . . . , xN
n , ·))M n+1 (An+1 ) ,
1≤i≤N2
where
stands for the empirical measure of the islands
given for any An ∈ X n by
2
m(x1n , . . . , xN
n , ·)
1
2
(xn , . . . , xN
n )
def
2
m(x1n , . . . , xN
n , An ) =
C. Dubarry, P. Del Moral, E. Moulines
N2
1 X
δ i (An ) .
N2 i=1 xn
15/32
Introduction
Island bootstrap approximation
The double bootstrap algorithm
Extensions
Algorithm description
The double bootstrap algorithm
p from 0 to n − 1 do
N2
multinomially with proba. prop. to
Sample I p = (Ipi )i=1
N2
P
N1
i,j
1
.
j=1 gp (ξp )
N1
1:
2:
for
3:
4:
i from 1 to N2 do
1
Sample J ip = (Jpi,j )N
j=1 multinomially with proba. prop. to
N1
i
I ,j
gp (ξpp )
.
i=1
for
j=1
i,j
according to
For 1 ≤ j ≤ N1 , sample independently ξp+1
5:
j
I i ,Jp
Mp+1 (ξpp
6:
7:
, ·).
end for
end for
C. Dubarry, P. Del Moral, E. Moulines
16/32
Introduction
Island bootstrap approximation
The double bootstrap algorithm
Extensions
Bias and variance of the double bootstrap
Bootstrap approximation: bias and variance
Theorem
For any time horizon
n≥0
and any bounded function
fn ∈ Bb (Xn , Xn ),
we
have
h
i
lim N1 E ηnN1 (fn ) − ηn (fn ) = Bn (fn ) ,
N1 →∞
lim N1 Var ηnN1 (fn ) = Vn (fn ) ,
N1 →∞
where
Bn (fn )
and
Vn (fn )
can be computed explicitly.
C. Dubarry, P. Del Moral, E. Moulines
17/32
Introduction
Island bootstrap approximation
The double bootstrap algorithm
Extensions
Bias and variance of the double bootstrap
Double bootstrap approximation: bias and variance
Theorem
n ≥ 0 and any fn ∈ Bb (Xn , Xn ), we have
h
i
2
e
lim lim N1 N2 E η N
n (m(·, fn )) − η n (m(·, fn )) = Bn (fn ) + Bn (fn ) ,
N1 →∞ N2 →∞
2
e
lim lim N1 N2 Var η N
n (m(·, fn )) = Vn (fn ) + Vn (fn ) ,
For any time horizon
N1 →∞ N2 →∞
where
en (fn ), Vn (fn ), Ven (fn )
Bn (fn ), B
can be computed explicitly.
The rate of the interacting island (N2 islands each with N1 individuals) is
the same as the one of the single island model with N1 N2 particles.
Even though the constant terms may be worst in the interacting island
model, it allows to use parallel implementations.
C. Dubarry, P. Del Moral, E. Moulines
18/32
Introduction
Island bootstrap approximation
The double bootstrap algorithm
Extensions
Bias and variance of the double bootstrap
Independent islands
Theorem
n ≥ 0 and any fn ∈ Bb (Xn , Xn ), we have
i
o
n h
2
eN
lim N1 E η
n (m(·, fn )) − ηn (fn ) = Bn (fn ) ,
N1 →∞
2
eN
lim N1 N2 Var η
n (m(·, fn )) = Vn (fn ) ,
For any time horizon
N1 →∞
where
Bn (fn )
and
Vn (fn )
are the same than for the single island model.
Although the variance of the particle approximation is inversely proportional to
N1 N2 , the bias is independent of N2 and is inversely proportional to N1 .
C. Dubarry, P. Del Moral, E. Moulines
19/32
Introduction
Island bootstrap approximation
The double bootstrap algorithm
Extensions
Bias and variance of the double bootstrap
Example
1
Linear Gaussian Model
Xp+1 = φXp + σu Up ,
Yp = Xp + σv Vp ,
Computing the predictive distribution of the state Xn given the
observations Y0:n−1 = y0:n−1 up to time n − 1 can be cast into the
framework of Feynman-Kac model by setting for all p ≥ 0
1
Mp+1 (xp , dxp+1 ) = √
exp −(xp+1 − φxp )2 /(2σu2 ) dxp+1 ,
2πσu
1
gp (xp ) = √
exp −(yp − xp )2 /(2σv2 ) .
2πσv
C. Dubarry, P. Del Moral, E. Moulines
20/32
Introduction
Island bootstrap approximation
The double bootstrap algorithm
Extensions
Bias and variance of the double bootstrap
How to choose between interacting and independent islands?
Independent islands
Interacting islands
Squared bias
Variance
Sum
2
Bn (fn )2
N12
en (fn )
Bn (fn ) + B
Vn (fn )
N1 N2
Vn (fn ) + Ven (fn )
N1 N2
Bn (fn )2
Vn (fn )
+
N1 N2
N12
Vn (fn ) + Ven (fn )
N1 N2
Vn (fn )
Bn (fn )2
Vn (fn ) + Ven (fn )
+
<
N1 N2
N12
N1 N2
C. Dubarry, P. Del Moral, E. Moulines
N12 N22
⇔
N1 >
Bn (fn )2
N2 .
Ven (fn )
21/32
Introduction
Island bootstrap approximation
The double bootstrap algorithm
Extensions
Numerical application
Numerical application: Linear Gaussian Model
The model is dened by
Xp+1 = φXp + σu Up ,
Yp = Xp + σv Vp .
n + 1 = 11 observations were generated with φ = 0.9, σu = 0.6 and
σv = 1 .
We have E [Xn |Y0:n−1 = y0:n−1 ] = ηn (Id).
We compare interacting to independent islands through
M
h
E
100 ×
2 2 i
2
2
eN
ηN
−E η
n (Id) − ηn (Id)
n (Id) − ηn (Id)
.
2 2
eN
E η
n (Id) − ηn (Id)
C. Dubarry, P. Del Moral, E. Moulines
22/32
Introduction
Island bootstrap approximation
The double bootstrap algorithm
Extensions
Numerical application
Results for the LGSS model
Figure: Interacting versus independent island renormalized estimators.
C. Dubarry, P. Del Moral, E. Moulines
23/32
Introduction
Island bootstrap approximation
The double bootstrap algorithm
Extensions
Outline
1
Introduction
2
Island bootstrap approximation
3
4
The double bootstrap algorithm
Algorithm description
Bias and variance of the double bootstrap
Numerical application
Extensions
C. Dubarry, P. Del Moral, E. Moulines
24/32
Introduction
Island bootstrap approximation
The double bootstrap algorithm
Extensions
Eective Sample Size Interaction
Dene
Θn,α


2
PN1 i
i




w
g
(x
)
n
i=1 n n
1
1
N1
N1
= xn = (xn , wn , . . . , xn , wn ) ∈ X n PN1
≥ αN1 .
2


i=1 (wni gn (xin ))


Dene m(xn , ·) stands for the empirical measure of xn given for any
An ∈ Xn by
1
def
m(xn , An ) = PN1
i=1
C. Dubarry, P. Del Moral, E. Moulines
N1
X
wni
wni δxin (An ) ,
i=1
25/32
Introduction
Island bootstrap approximation
The double bootstrap algorithm
Extensions
Eective Sample Size Interaction
Consider the Markov kernel M n+1
M n+1 (xn , An+1 ) =
(Q
N1
i
i
i
i g (xi ) (Bn+1 )Mn+1 (xn , An+1 )
i=1 δwn
n
n
QN
i
i
1
i=1 δ1 (Bn+1 )Ψn (m(xn , ·))Mn+1 (An+1 )
xn ∈ Θn,α
xn 6∈ Θn,α
Dene a Markov chain {X n }n≥0 where for each n ∈ N,
i
h
X n = (ξn1 , ωn1 ), . . . , (ξnN1 , ωnN1 ) ∈ X n ,
C. Dubarry, P. Del Moral, E. Moulines
26/32
Introduction
Island bootstrap approximation
The double bootstrap algorithm
Extensions
ESS: particle approximation
N1 -particle approximations of the measures ηn and γn
1
def
ηnN1 (fn ) = m(X n , fn ) = PN1
N1
X
i
i=1 ωn i=1
def
Y
γnN1 (fn ) = ηnN1 (fn )
ωni fn ξni ,
ηpN1 (gp ) .
0≤p<n
Theorem
fn ∈ Bb (Xn , Xn ), γnN1 (fn ) is an unbiased estimator of γn (fn ):




h
i
Y N
Y
N1
N
E γn (fn ) = E ηn 1 (fn )
ηp 1 (gp ) = E fn (Xn )
gp (Xp ) .
For any
0≤p<n
C. Dubarry, P. Del Moral, E. Moulines
0≤p<n
27/32
Introduction
Island bootstrap approximation
The double bootstrap algorithm
Extensions
ESS: Feynman-Kac approximation
N1
1
For xn = (x1n , wn1 , · · · , xN
n , wn ) ∈ X n we set
1
def
g n (xn ) = m(xn , gn ) = PN1
i=1
N1
X
wni
wni gn xin .
i=1
The associated Feynman-Kac model {(η n , γ n )}n≥0 is
γ n (1)
η n (f n ) = γ n (f n )/γ


Y
γ n (f n ) = E f n (Xn )
g p (Xp ) ,
0≤p<n
C. Dubarry, P. Del Moral, E. Moulines
28/32
Introduction
Island bootstrap approximation
The double bootstrap algorithm
Extensions
ESS: Feynman-Kac approximation
Since g n (X n ) = ηnN1 (gn ), for any f n of the form
f n (xn ) =
P
N1
i=1
wni
−1 P

E fn (Xn )
N1
i=1
wni fn xin where fn ∈ Bb (Xn , Xn ),

Y

gp (Xp ) = E f n (Xn )
0≤p<n

Y
g p (Xp ) ,
0≤p<n
Therefore
γ n (f n ) = γn (fn )
η n (f n ) = ηn (fn ) .
C. Dubarry, P. Del Moral, E. Moulines
29/32
Introduction
Island bootstrap approximation
The double bootstrap algorithm
Extensions
1: for p from 0 to n − 1 do
2:
Selection step and weight actualization between islands:
2
2
PN2 i
eff = PN2 Ωi g (ξi , ω i )
i
3:
Set N
/
Ωp g p (ξi
p
p , ωp ) .
2
i=1 p p p
i=1
eff
4:
if N2 ≥ αIslands N2 then
i
i
i
i
5:
For 1 ≤ i ≤ N2 , set Ω
p+1 = Ωp g p (ξp , ω p ).
i N2
6:
Set I p = (I )
p i=1 = (1, 2, . . . , N2 ).
7:
else
N
2
8:
Set Ωp+1 =
Ωi
p+1 i=1 = (1, . . . , 1).
N
N2
i
i
i
i
2
9:
Sample I p = (I )
p i=1 multinomially with proba. prop. to Ωp g p (ξp , ω p ) i=1 .
10:
end if
11:
Island mutation step:
12:
for i from 1 to N2 do
13:
Particle selection and weight actualization within each island:
14:
same business as usual
15:
end for
16: end for
17:
Approximate
ηn (fn )
by
N1
N2
X
X
1
1
i
i,j
i,j
.
Ωn P
ωn fn ξn
PN2
N1
i,j
i
Ω
ω
i=1 n i=1
j=1 n j=1
C. Dubarry, P. Del Moral, E. Moulines
30/32
Introduction
Island bootstrap approximation
The double bootstrap algorithm
Extensions
Results for the ESS model
N2 = 100
N2 = 250
N2 = 500
N2 = 1000
N1 = 100
N1 = 250
N1 = 500
N1 = 1000
(1) ESS
(2) Bootstrap
(2) Independent
C. Dubarry, P. Del Moral, E. Moulines
Real value
31/32
Introduction
Island bootstrap approximation
The double bootstrap algorithm
Extensions
Number of interactions
Table: Number of interactions between islands for the ESS within ESS estimator as a
percentage of the one the ESS within bootstrap estimator in the LGM.
PP
P P N2
PP
N1
P
100
250
500
1000
100
250
500
1000
4.32
0.88
0.04
0
4.76
0.60
0.02
0
4.92
0.34
0
0
4.98
0.32
0
0
C. Dubarry, P. Del Moral, E. Moulines
32/32