1. No category

Island particle algorithms and application to rare
event analysis
Christelle Vergé
(joint work with P. Del Moral, E. Moulines, J. Morio, J. Olsson,
C. Dubarry)
08/04/2015 - MASCOT-NUM
Introduction to island particle models
An example for rare event analysis
Plan
Introduction to island particle models : a way to parallelize sequential Monte
Carlo (SMC) algorithms
An example of island particle algorithm for rare event analysis
Conclusion
Conclusion
Introduction to island particle models
An example for rare event analysis
Plan
Introduction to island particle models : a way to parallelize sequential Monte
Carlo (SMC) algorithms
An example of island particle algorithm for rare event analysis
Conclusion
3/25
Island particle algorithms and their application to rare event analysis
Christelle Vergé 08/04/2015
Conclusion
Introduction to island particle models
An example for rare event analysis
Conclusion
Feynman-Kac models
{Qn }n∈N : a set of unnormalized transition kernels on the measurable
space (X, X )
η0 : a probability distribution
Definition :
A sequence {ηn }n∈N of Feynman-Kac measures is defined, for all h ∈ Fb (X) by
R
ηn h =
···
R
R
h(xn ) η0 (dx0 )
···
R
η0 (dx0 )
Qn−1
p=0
Qn−1
p=0
Qp (xp , dxp+1 )
Qp (xp , dxp+1 )
(n ∈ N).
I The Feynman-Kac measures satisfy the nonlinear recursion
ηn+1 = ηn Qn /ηn Qn 1X .
I Numerous applications (nonlinear filtering, rare event sampling, hidden
Markov chain parameter estimation, stochastic control problems, financial
mathematics...)
4/25
Island particle algorithms and their application to rare event analysis
Christelle Vergé 08/04/2015
Introduction to island particle models
An example for rare event analysis
Conclusion
Interacting particle systems (IPS)
Approximation of Feynman-Kac measures {ηn }n∈N :
Kalman filter : exact simulation for linear Gaussian models.
Otherwise, SMC methods : empirical approximation of ηn thanks to an
interacting particle systems {(ξnN (i), ωnN (i))}N
i=1 .
The weighted sample evolves recursively according to selection and
mutation steps :
{(ξnN (i), ωnN (i))}N
i=1
Selection
−−−−−→
{(ξnN (InN (i)), ωnN (InN (i)))}N
i=1
Mutation
−−−−−→
InN (i) ∈ J1, NK
N
N
{(ξn+1
(i), ωn+1
(i))}N
i=1
N
(i) ∼ Rn (ξnN (InN (i)), ·)
ξn+1
N
N
ωn+1
(i) = ωnN (InN (i)) wn (ξnN (InN (i)), ξn+1
(i))
where Rn is a proposal kernel and wn is the importance function such that
Qn (x , dy ) = wn (x , y )Rn (x , dy ) ((x , y ) ∈ X2 ).
Different kinds of selection may be considered :
I
I
5/25
systematic resampling : ∀i, InN (i) ∼ Mult({ωnN (k)}N
k=1 ) # bootstrap
algorithm [Ref : N.J. Gordon, D. J. Salmond, A. F. M. Smith (1993)]
resampling of all the particles only when their weights are skewed
(Effective Sample size, coefficient of variation) [Ref : J. Liu, R. Chen (1993)]
Island particle algorithms and their application to rare event analysis
Christelle Vergé 08/04/2015
Introduction to island particle models
An example for rare event analysis
Conclusion
The bootstrap algorithm
/* Initialization
for i ← 1 to N do
ξ0N (i) ∼ η0 ;
ξ1N (i) ∼ R0 (ξ0N (i), ·)
end
for p ← 1 to n − 1 do
*/
ω1N (i) ← w0 (ξ0N (i), ξ1N (i));
/* Selection step
*/
N
N
Sample {IpN (i)}N
i=1 ∼i.i.d Mult({ωp (k)}k=1 ) ;
/* Mutation step
*/
for i ← 1 to N do
Sample conditionally independently
N (i) ∼ R (ξ N (I N (i)), ·);
ξp+1
p p p
N (i) ← w (ξ N (I N (i)), ξ N (i));
Update the weights ωp+1
p p p
p+1
end
end
6/25
Island particle algorithms and their application to rare event analysis
Christelle Vergé 08/04/2015
Introduction to island particle models
An example for rare event analysis
Conclusion
Analysis of the bootstrap algorithm
Proposition :
[Ref : P. Del Moral, Feynman-Kac formulae, 2004] [Ref : R. Douc and E. Moulines, 2008]
PN
ω N (i)h(ξnN (i))/
For all n ∈ N and h ∈ Fb (X), set ηnN h ,
i=1 n
Then,
ηnN h −−−−−→ ηn h almost surely,
PN
k=1
ωnN (k).
N→+∞
√
N(ηnN h
2
V0 (h) = η0 {(h−η0 h) }
D
− ηn h) −−−−−→ N(0, Vn (h)),
where
N→+∞
and
Vn (h) =
n−1
X
η` R` {w 2 Q`+1 · · · Qn−1 (h − ηn h)2 }
`
`=0
(η` Q` · · · Qn−1 1X )2
N E ηnN h − ηn h −−−−−→ Bn (h).
N→+∞
7/25
The precision of the estimation depends upon the size N of the particle
swarm ⇒ critical for online applications
Develop new SMC methods to reduce the size of the particle swarm,
while ensuring good estimates ⇒ parallelization of SMC methods.
Island particle algorithms and their application to rare event analysis
Christelle Vergé 08/04/2015
.
Introduction to island particle models
An example for rare event analysis
Conclusion
Parallelization of SMC methods
Spread the total number N , N1 N2 of particles into N1
batches of N2 particles each.
Each batch is called an island. Each island evolves
independently as a standard SMC algorithm with N2 particles.
The N1 islands may be considered in a parallel architecture or
may interact through a selection step on the island level, when
assigning as island weight, the average of the particle weights
in an island.
I
I
I
8/25
N1 independent bootstraps
Double bootstrap (B2 )
Double bootstrap with adaptive selection on the island level
(B2ASIL)
Island particle algorithms and their application to rare event analysis
Christelle Vergé 08/04/2015
Introduction to island particle models
An example for rare event analysis
Conclusion
The B2ASIL algorithm
/* Initialization
*/
For all i ∈ J1, N1 K and j ∈ J1, N2 K,
ξ0N (i, j) ∼ η0 ; ξ1N (i, j) ∼ R0 (ξ0N (i, j), ·) ;
ΩN
1 (i) ←
PN2
j=1
ω1N (i, j) ← w0 (ξ0N (i, j), ξ1N (i, j)) ;
ω1N (i, j)/N2 ;
for p ← 1 to n − 1 do
/* Island selection
N
1
if CV2 ({ΩN
p (i)}i=1 ) , N1
else
end
P N1
i=1
*/
PN1
ΩN
p (i)/
i 0 =1
N
0
ΩN
p (i )
2
− 1 > τ then
0
1
For all i ∈ J1, N1 K, I N (i) ∼ Mult({ΩN
p (i )}i 0 =1 ) ;
For all i ∈ J1, N1 K, I N (i) ← i;
/* Island mutation
*/
for i ← 1 to N1 do
/* Individual selection
*/
N
For all j ∈ J1, N2 K, J N (i, j) ∼ Mult({ωpN (I N (i), j 0 )} 02 ) ;
j =1
/* Mutation
*/
N
For all j ∈ J1, N2 K, ξp+1
(i, j) ∼ Rp (ξpN (I N (i), J N (i, j)), ·);
N
N
ωp+1
(i, j) ← wp (ξpN (I N (i), J N (i, j)), ξp+1
(i, j));
ΩN
p+1 (i) ←
PN2
j=1
N
ωp+1
(i, j)/N2 ;
end
end
9/25
Island particle algorithms and their application to rare event analysis
Christelle Vergé 08/04/2015
Introduction to island particle models
An example for rare event analysis
Conclusion
Analysis of the B2ASIL algorithm
Denote by η N
nh =
PN1
i=1
estimators returned by the
Theorem :
ΩN (i)
ω N (i, j)
N2
j=1
N (i 0 )
Ω
0
n
i =1
2
B ASIL algorithm.
PN1 n
PN2 n
P
j 0 =1
ωnN (i, j 0 )
h(ξnN (i, j)) the
[Ref : C. Vergé, P. Del Moral, E. Moulines, J. Olsson, preprint]
P
Let n ∈ N and h ∈ Fb (X). Then, η N
n h −−−−−→ ηn h.
N→+∞
Impose that for all β > 0, N1 exp(−βN2 ) −−−−−→ 0. Then, for all n ∈ N, the random
N→+∞
N1
variable 1{CV2 ({ΩN
n (i)}i=1 ) > τ } has a deterministic limit εn in probability. Moreover,
√
D
e
N(η N
n h − ηn h) −→ N(0, Vn (h) + Vn (h)),
e0 = 0, and Vn h =
V0 (h) = η0 {(h − η0 h)2 }, V
where
n−1
X
η` R` {w`2 Q`+1 · · · Qn−1 (h − ηn h)2 }
,
(η` Q` · · · Qn−1 1X )2
`=0
en h =
V
n−1 n−1
X
X
`=0 p=`+1
10/25
εp
η` R` {w`2 Q`+1 · · · Qn−1 (h − ηn h)2 }
(η` Q` · · · Qn−1 1X )2
Island particle algorithms and their application to rare event analysis
.
Christelle Vergé 08/04/2015
Introduction to island particle models
An example for rare event analysis
Conclusion
Proof (sketch)
Proof (sketch)
The proof is by induction on n.
We decompose one iteration of the B2ASIL algorithm into elementary operations
I selection on the island level
I selection on the particle level
I mutation,
and show that each of them preserves a law of large numbers, a Hoeffding-type
inequality and a central limit theorem.
This general framework allows to derive a law of large numbers and a central
limit theorem for any algorithm that may be decomposed into these elementary
operations.
11/25
Island particle algorithms and their application to rare event analysis
Christelle Vergé 08/04/2015
Introduction to island particle models
An example for rare event analysis
Conclusion
A criterion to determine when islands should interact
N1 independent
bootstraps
B2ASIL
Bn
Bn
Bn + Bn
N
N2
N1 N2
Vn
Vn
Vn + Vn
N
N1 N2
N1 N2
Bootstrap
Bias
Variance
e
e
en , Vn and Ven can be found in :
Explicit expressions of Bn , B
[Ref : C. Vergé, C. Dubarry, P. Del Moral, E. Moulines, Statistics and Computing, 2015]
[Ref : C. Vergé, P. Del Moral, E. Moulines, J. Olsson, preprint]
Use the mean squared error to make a compromise between bias and variance : island
interaction is beneficial when
en
Vn
B2
Vn + V
+ n2 >
N1 N2
N1 N2
N2
⇔
N2 <
Bn2
en
V
N1 .
When N2 << N1 , the interaction is beneficial, but prevents a total parallelization.
12/25
Island particle algorithms and their application to rare event analysis
Christelle Vergé 08/04/2015
Introduction to island particle models
An example for rare event analysis
Conclusion
Stability of the double bootstrap (B2 )
Note that the B2 algorithm, which selects systematically the islands, is a
particular case of B2ASIL algorithm for which τ = 0 (and hence εn = 1 for all
n ∈ N∗ ). We may hence furnish the asymptotic variance of the B2 algorithm :
σn2 (h) =
n−1
X
`=0
Theorem
(n − `)
η` R` {w`2 Q`+1 · · · Qn−1 (h − ηn h)2 }
.
(η` Q` · · · Qn−1 1X )2
[Ref : C. Vergé, P. Del Moral, E. Moulines, J. Olsson, preprint]
Suppose the standard strong mixing conditions :
(i) There exist constants 0 < σ− < σ+ < ∞ and ϕ ∈ M1 (X) such that for all
p ∈ N, x ∈ X, and A ∈ X , σ− ϕ(A) ≤ Mp (x , A) ≤ σ+ ϕ(A).
(ii) w+ , supp∈N kwp k∞ < ∞.
(iii) c− , inf (p,x )∈N×X Qp 1X (x ) > 0.
Then for all n ∈ N and h ∈ Fb (X), σn2 (h) ≤ w+
osc2 (h)
, where
(1 − ρ)2 (1 − ρ2 )2 c−
ρ , 1 − σ− /σ+ .
13/25
Island particle algorithms and their application to rare event analysis
Christelle Vergé 08/04/2015
Introduction to island particle models
An example for rare event analysis
Plan
Introduction to island particle models : a way to parallelize sequential Monte
Carlo (SMC) algorithms
An example of island particle algorithm for rare event analysis
Conclusion
14/25
Island particle algorithms and their application to rare event analysis
Christelle Vergé 08/04/2015
Conclusion
Introduction to island particle models
An example for rare event analysis
Conclusion
Context
Consider a black box model :
Black box
A rare event is often modeled by the exceedence of a threshold S :
{φ(X ) > S} such that P(φ(X ) > S) < 10−4 .
I Risk analysis is not just evaluating a risk or a probability of failure, but
estimating the law of random phenomena that leads to a critic event.
Some parameters θ of the model or density parameters of the input
random variables X may be fixed by the experimenter and can influence
the output random variable Y .
I We want to determine the impact of such tuning of parameters on the
realization of the critic event, i.e. to compute the law of the parameters
Θ conditionally on the rare event, denoted by π , Law(Θ|φ(X ) > S).
15/25
Island particle algorithms and their application to rare event analysis
Christelle Vergé 08/04/2015
Introduction to island particle models
An example for rare event analysis
Conclusion
The splitting algorithm : a way to evaluate P(φ(X ) > S|Θ = θ)
[Ref : S-K. Au and J.L. Beck, Estimation of small failure probabilities in high dimensions by subset simulation, 2001]
Consider an increasing sequence of thresholds
−∞ , S0 < S1 < ... < Sm , S,
and decompose, using Bayes’ formula,
m−1
P(φ(X ) > S|Θ = θ) =
Y
P(φ(X ) > Sp+1 |φ(X ) > Sp , Θ = θ).
p=0
Approximation using an SMC with N2 particles :





Xn (1)
.
.
.
.
.
.
Xn (N2 )
16/25





 −−−Selection

−
−
−
−
−
→
1
 {.≥Sn+1 } 

b
Xn (1)
.
.
.
.
.
.
b
Xn (N2 )



 Mutation 
 −−−−−→ 




Xn+1 (1)
.
.
.
.
.
.
Xn+1 (N2 )





Island particle algorithms and their application to rare event analysis
Christelle Vergé 08/04/2015
Introduction to island particle models
An example for rare event analysis
Conclusion
The splitting algorithm : a way to evaluate P(φ(X ) > S|Θ = θ)
[Ref : S-K. Au and J.L. Beck, Estimation of small failure probabilities in high dimensions by subset simulation, 2001]
Consider an increasing sequence of thresholds
−∞ , S0 < S1 < ... < Sm , S,
and decompose, using Bayes’ formula,
m−1
P(φ(X ) > S|Θ = θ) =
Y
P(φ(X ) > Sp+1 |φ(X ) > Sp , Θ = θ).
p=0
Sm
Approximation using an SMC with N2 particles :





Xn (1)
.
.
.
.
.
.
Xn (N2 )





 −−−Selection

−
−
−
−
−
→
1
 {.≥Sn+1 } 

b
Xn (1)
.
.
.
.
.
.
b
Xn (N2 )

S2


 Mutation 
 −−−−−→ 




Xn+1 (1)
.
.
.
.
.
.
Xn+1 (N2 )





S1
x0
T
16/25
Island particle algorithms and their application to rare event analysis
Christelle Vergé 08/04/2015
Introduction to island particle models
An example for rare event analysis
Conclusion
The splitting algorithm : a way to evaluate P(φ(X ) > S|Θ = θ)
[Ref : S-K. Au and J.L. Beck, Estimation of small failure probabilities in high dimensions by subset simulation, 2001]
Consider an increasing sequence of thresholds
−∞ , S0 < S1 < ... < Sm , S,
and decompose, using Bayes’ formula,
m−1
P(φ(X ) > S|Θ = θ) =
Y
P(φ(X ) > Sp+1 |φ(X ) > Sp , Θ = θ).
p=0
Sm
Approximation using an SMC with N2 particles :





Xn (1)
.
.
.
.
.
.
Xn (N2 )





 −−−Selection

−
−
−
−
−
→
1
 {.≥Sn+1 } 

b
Xn (1)
.
.
.
.
.
.
b
Xn (N2 )

S2


 Mutation 
 −−−−−→ 




Xn+1 (1)
.
.
.
.
.
.
Xn+1 (N2 )





S1
x0
T
16/25
Island particle algorithms and their application to rare event analysis
Christelle Vergé 08/04/2015
Introduction to island particle models
An example for rare event analysis
Conclusion
The splitting algorithm : a way to evaluate P(φ(X ) > S|Θ = θ)
[Ref : S-K. Au and J.L. Beck, Estimation of small failure probabilities in high dimensions by subset simulation, 2001]
Consider an increasing sequence of thresholds
−∞ , S0 < S1 < ... < Sm , S,
and decompose, using Bayes’ formula,
m−1
P(φ(X ) > S|Θ = θ) =
Y
P(φ(X ) > Sp+1 |φ(X ) > Sp , Θ = θ).
p=0
Sm
Approximation using an SMC with N2 particles :





Xn (1)
.
.
.
.
.
.
Xn (N2 )





 −−−Selection

−
−
−
−
−
→
1
 {.≥Sn+1 } 

b
Xn (1)
.
.
.
.
.
.
b
Xn (N2 )

S2


 Mutation 
 −−−−−→ 




Xn+1 (1)
.
.
.
.
.
.
Xn+1 (N2 )





S1
x0
T
16/25
Island particle algorithms and their application to rare event analysis
Christelle Vergé 08/04/2015
Introduction to island particle models
An example for rare event analysis
Conclusion
The SMC2 algorithm
I One way to sample from π , Law(Θ|φ(X ) > S), is to use a standard SMC
1
algorithm with N1 particles {θn (i)}N
i=1 .
I For that purpose, we create a dynamic introducing intermediary thresholds
S1 < ... < Sm , S, and defining the distributions {πn }n∈N∗ by
πn (dθ) , Law(Θ|φ(X ) > Sn ).
I Instead of trying to sample directly from π, sample successively from
π1 , . . . , πm , π.
I The particles {θn (i)}N1 evolve according to usual selection and mutation steps :
i=1
Selection : Multinomial resampling with weights proportional to :
1
{P(φ(X ) > Sn+1 |φ(X ) > Sn , Θ = θn (i))}N
i=1 ,
Mutation : An acceptance / rejection step involving the probabilities
P(φ(X ) > S|θn (i)), which are not computable.
I For each parameter θn (i), we run a splitting with N2 particles {Xn (i, j)}N2 , in
j=1
order to replace every incalculable quantity by an unbiased estimator.
We then have 2 embedded SMC algorithms # the SMC2 algorithm.
[Ref : N. Chopin, P. Jacob and O. Papaspiliopoulos, JRSSB, 2013]
[Ref : C. Vergé, J. Morio, P. Del Moral, preprint]
17/25
Island particle algorithms and their application to rare event analysis
Christelle Vergé 08/04/2015
Introduction to island particle models
An example for rare event analysis
Conclusion
Analysis of SMC2
Theorem :
The SMC2 algorithm converges to the target law π when N1 → ∞, for any fixed N2 .
Sketch of proof : The SMC2 algorithm can be viewed as an SMC algorithm on an
extended state space.
Toy case : threshold exceedence for a
Gaussian tail. We can compute explicitly
Law(Θ|X ≥ 5) when X ∼ N (Θ, 1) and
Θ ∼ N (0, 1).
In this simulation, we use 2000 × 20
particles for the SMC2 algorithm.
Figure: Law(Θ|X ≥ 5)
18/25
Island particle algorithms and their application to rare event analysis
Christelle Vergé 08/04/2015
Introduction to island particle models
An example for rare event analysis
Conclusion
Application of SMC2 algorithm to the fallout zone of a launch vehicle
40
40
35
35
30
30
25
25
histogram
histogram
We apply SMC2 algorithm where φ simulates the distance between the true
position of the fallout zone of a stage rocket and its predicted position.
X is a Gaussian vector with covariance matrix equal to I4 and mean
θ = (θ1 , θ2 , θ3 , θ4 )t with a Gaussian prior i.e. for all i ∈ J1, 4K, θi ∼ N (0, 1).
The critic event is when the output distance φ(X ) exceeds 0.72 km.
20
15
10
5
−3
−2
−1
0
Θ1
1
2
3
0
−4
4
−3
−2
−1
0
Θ2
1
2
3
4
Law(θ1 |critic event)
Law(θ2 |critic event)
40
40
35
35
30
30
25
25
histogram
histogram
15
10
5
0
−4
20
20
15
10
15
10
5
0
−4
20
5
−3
−2
−1
0
Θ3
1
2
3
4
Law(θ3 |critic event)
0
−4
−3
−2
−1
0
Θ4
1
2
3
4
Law(θ4 |critic event)
θ1 : wind direction ; θ4 : descent angle.
19/25
Island particle algorithms and their application to rare event analysis
Christelle Vergé 08/04/2015
Introduction to island particle models
An example for rare event analysis
Conclusion
Influence of the parameters on the probability of interest
Monte Carlo estimates for different sets of parameters :
θ1
0
1
−1
−1
I
θ2
0
0
0
0
θ3
0
0
0
0
θ4
0
1
1
−1
ˆ (φ(X ) > S|θ)
P
8.5 10−4
1.05 10−2
1.02 10−2
1.14 10−2
A bad tuning of the parameters can imply a large increase of
the probability of the critic event and an underestimation of
the associated risk ⇒ security matter.
[Ref : C. Vergé, J. Morio, P. Del Moral, preprint]
20/25
Island particle algorithms and their application to rare event analysis
Christelle Vergé 08/04/2015
Introduction to island particle models
An example for rare event analysis
Plan
Introduction to island particle models : a way to parallelize sequential Monte
Carlo (SMC) algorithms
An example of island particle algorithm for rare event analysis
Conclusion
21/25
Island particle algorithms and their application to rare event analysis
Christelle Vergé 08/04/2015
Conclusion
Introduction to island particle models
An example for rare event analysis
Conclusion
Conclusions and perspectives
Conclusions :
I Introduction to island particle models,
I Definition of operations on islands,
I Establishment of a criterion to determine when islands should be
considered in parallel or may interact,
I Study of asymptotic properties of island particle models,
I Transposition of an existing island particle model for rare event analysis.
Application :
I Reliability analysis of a launch vehicle stage fallout.
Perspectives and future application :
I Application to reliability analysis for collision between a space debris and
a satellite,
I Study of the SMC2 algorithm.
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Island particle algorithms and their application to rare event analysis
Christelle Vergé 08/04/2015
Introduction to island particle models
An example for rare event analysis
Conclusion
Publications
Book :
P. Del Moral and C. Vergé, Algorithmes Stochastiques : Modèles et Applications, Springer Series : Maths & Applications,
SMAI, vol.75, 2014, 487 pages, DOI = 10.1007/978-3-64254616-7 (published).
Book chapters :
Contribution to two chapters of : Estimation of rare event
probabilities in complex (and other) systems - a practical approach, J. Morio and M. Balesdent, Elsevier-Woodhead Publishing (August 2015).
I Chapter 5 : Simulation techniques
I Chapter 11 : Estimation of collision probability
between a space debris and a satellite
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Island particle algorithms and their application to rare event analysis
Christelle Vergé 08/04/2015
Introduction to island particle models
An example for rare event analysis
Conclusion
Publications
Journal publications :
I
Published
1. C. Vergé, C. Dubarry, P. Del Moral, E. Moulines, On parallel implementation of Sequential Monte
Carlo methods : the island particle model, Statistics and Computing, vol. 25, Issue 2, Mars 2015,
pp. 243-260, DOI = 10.1007/s11222-013-9429-x.
2. J. Morio, M. Balesdent, D. Jacquemart, C. Vergé, A survey of rare event estimation methods for
static input-output models, Simulation Modelling Practice and Theory, vol. 49, pp 287-304, 2014.
I
Submitted
1. C. Vergé, P. Del Moral, E. Moulines, J. Olsson, Asymptotic properties of weighted archipelagos
with application to particle island methods.
2. C. Vergé, J. Morio, P. Del Moral, An island Particle Markov Chain Monte Carlo algorithm for
safety analysis.
3. C. Vergé, C. Ichard, Introduction to labeled island particle models and their asymptotic properties.
Conference publication :
1. P. Del Moral, G. W. Peters, C. Vergé, An introduction to particle integration methods : with applications
to risk and insurance, Monte Carlo and Quasi-Monte Carlo Methods 2012, Springer Proceedings in
Mathematics & Statistics, volume 65, 2013, p.39-81, DOI = 10.1007/978-3-642-41095-6_3.
2. C. Vergé, E. Moulines, J. Olsson, Asymptotic properties of particle island models with application to the
double bootstrap filter, Signal Processing Conference (EUSIPCO), Proceedings of the 22nd European,
IEEE (submitted, march 2015).
3. C. Vergé, E. Moulines, J. Olsson, Fluctuation analysis of island particle models, 18th INFORMS Applied
Probability Conference, Istanbul University, Turkey (submitted, march 2015).
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Island particle algorithms and their application to rare event analysis
Christelle Vergé 08/04/2015
Introduction to island particle models
An example for rare event analysis
Thank you for your attention !
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Island particle algorithms and their application to rare event analysis
Christelle Vergé 08/04/2015
Conclusion