Multilevel sequential Monte Carlo samplers and...
Transcript of Multilevel sequential Monte Carlo samplers and...
![Page 1: Multilevel sequential Monte Carlo samplers and …ims.nus.edu.sg/events/2018/bay/files/law.pdfMultilevel sequential Monte Carlo samplers and other MLMC methods Kody Law & A. Jasra](https://reader033.fdocuments.net/reader033/viewer/2022053002/5f067e297e708231d4184391/html5/thumbnails/1.jpg)
Introduction MLMC BIP SMC MLSMC Other Summary
Multilevel sequential Monte Carlo samplersand other MLMC methods
Kody Law & A. Jasra (NUS) & others
Bayes Comp IMS 2018,Singapore
September 3, 2018
![Page 2: Multilevel sequential Monte Carlo samplers and …ims.nus.edu.sg/events/2018/bay/files/law.pdfMultilevel sequential Monte Carlo samplers and other MLMC methods Kody Law & A. Jasra](https://reader033.fdocuments.net/reader033/viewer/2022053002/5f067e297e708231d4184391/html5/thumbnails/2.jpg)
Introduction MLMC BIP SMC MLSMC Other Summary
Outline
1 Introduction
2 Multilevel Monte Carlo sampling
3 Bayesian inference problem
4 Sequential Monte Carlo samplers
5 Multilevel Sequential Monte Carlo (MLSMC) samplers
6 Other MLMC algorithms for inferenceImportance sampling (IS) strategyCoupled algorithm (CA) strategyApproximate coupling (AC) strategy
7 Summary
![Page 3: Multilevel sequential Monte Carlo samplers and …ims.nus.edu.sg/events/2018/bay/files/law.pdfMultilevel sequential Monte Carlo samplers and other MLMC methods Kody Law & A. Jasra](https://reader033.fdocuments.net/reader033/viewer/2022053002/5f067e297e708231d4184391/html5/thumbnails/3.jpg)
Introduction MLMC BIP SMC MLSMC Other Summary
Outline
1 Introduction
2 Multilevel Monte Carlo sampling
3 Bayesian inference problem
4 Sequential Monte Carlo samplers
5 Multilevel Sequential Monte Carlo (MLSMC) samplers
6 Other MLMC algorithms for inferenceImportance sampling (IS) strategyCoupled algorithm (CA) strategyApproximate coupling (AC) strategy
7 Summary
![Page 4: Multilevel sequential Monte Carlo samplers and …ims.nus.edu.sg/events/2018/bay/files/law.pdfMultilevel sequential Monte Carlo samplers and other MLMC methods Kody Law & A. Jasra](https://reader033.fdocuments.net/reader033/viewer/2022053002/5f067e297e708231d4184391/html5/thumbnails/4.jpg)
Introduction MLMC BIP SMC MLSMC Other Summary
Inverse Problems
Data Parameter
y = G ( u ) + e
forward model (PDE) observation/model errors
y ∈ RM
u ∈ EG : E → RM
Data y may be limited in number, noisy, and indirect.Parameter u often a function, discretized.Needs to be approximated.
![Page 5: Multilevel sequential Monte Carlo samplers and …ims.nus.edu.sg/events/2018/bay/files/law.pdfMultilevel sequential Monte Carlo samplers and other MLMC methods Kody Law & A. Jasra](https://reader033.fdocuments.net/reader033/viewer/2022053002/5f067e297e708231d4184391/html5/thumbnails/5.jpg)
Introduction MLMC BIP SMC MLSMC Other Summary
Bayesian inversion
Goal of Bayesian inference: given observed data y , find theposterior distribution
P(du|y) =P(y |u)P0(du)∫E P(y |u)P0(du)
,
and estimate quantities of interest, such as, for g : E → R,expected value,
∫E g(u)P(du|y);
variance,∫
E g(u)2P(du|y)− (∫
E g(u)P(du|y))2;probability of exceeding some value
∫u∈E ;g(u)>R P(du|y).
There exists a connection with a classical inverse problems:identify most probable value, supu∈E limδ→0
P(Bδ(u)|y)P(Bδ(v)|y) .
![Page 6: Multilevel sequential Monte Carlo samplers and …ims.nus.edu.sg/events/2018/bay/files/law.pdfMultilevel sequential Monte Carlo samplers and other MLMC methods Kody Law & A. Jasra](https://reader033.fdocuments.net/reader033/viewer/2022053002/5f067e297e708231d4184391/html5/thumbnails/6.jpg)
Introduction MLMC BIP SMC MLSMC Other Summary
Orientation
Aim: Approximate expectations with respect to a probabilitydistribution η∞, which needs to be approximated by some ηL,and can only be evaluated up to a normalizing constant.Solution: The multilevel Monte Carlo (MLMC) method isutilised with Sequential Monte Carlo (SMC) samplers, yieldingthe MLSMC sampler for Bayesian inference problems.
MLMC methods reduce cost to error= O(ε), can be used inthe case that ηL can be sampled from directly [H00,G08].Here it is assumed that ηL cannot be sampled from directly,but can be evaluated up to a normalizing constant (e.g.Bayesian inference problems) [HSS13, DKST15, HTL16,BJLTZ17].SMC samplers are a general class of algorithms which areeffective for sampling from such distributions [N01, C02,DDJ06].
![Page 7: Multilevel sequential Monte Carlo samplers and …ims.nus.edu.sg/events/2018/bay/files/law.pdfMultilevel sequential Monte Carlo samplers and other MLMC methods Kody Law & A. Jasra](https://reader033.fdocuments.net/reader033/viewer/2022053002/5f067e297e708231d4184391/html5/thumbnails/7.jpg)
Introduction MLMC BIP SMC MLSMC Other Summary
Outline
1 Introduction
2 Multilevel Monte Carlo sampling
3 Bayesian inference problem
4 Sequential Monte Carlo samplers
5 Multilevel Sequential Monte Carlo (MLSMC) samplers
6 Other MLMC algorithms for inferenceImportance sampling (IS) strategyCoupled algorithm (CA) strategyApproximate coupling (AC) strategy
7 Summary
![Page 8: Multilevel sequential Monte Carlo samplers and …ims.nus.edu.sg/events/2018/bay/files/law.pdfMultilevel sequential Monte Carlo samplers and other MLMC methods Kody Law & A. Jasra](https://reader033.fdocuments.net/reader033/viewer/2022053002/5f067e297e708231d4184391/html5/thumbnails/8.jpg)
Introduction MLMC BIP SMC MLSMC Other Summary
Example: expectation for SDE [G08]
Estimation of expectation of solution of intractable stochasticdifferential equation (SDE).
dX = f (X )dt + σ(X )dW , X0 = x0 .
Aim: estimate E(g(XT )).We need to(1) Approximate, e.g. by Euler-Maruyama method with
resolution h:
Xn+1 = Xn + hf (Xn) +√
hσ(Xn)ξn, ξn ∼ N(0,1).
(2) Sample X (i)NTNi=1, NT = T/h.
![Page 9: Multilevel sequential Monte Carlo samplers and …ims.nus.edu.sg/events/2018/bay/files/law.pdfMultilevel sequential Monte Carlo samplers and other MLMC methods Kody Law & A. Jasra](https://reader033.fdocuments.net/reader033/viewer/2022053002/5f067e297e708231d4184391/html5/thumbnails/9.jpg)
Introduction MLMC BIP SMC MLSMC Other Summary
Single level Monte Carlo
Aim: Approximate η∞(g) := Eη∞(g) for g : E → R.
Monte Carlo approachDiscretize the space⇒ approximate distribution ηL.
Sample U(i)L ∼ ηL i.i.d., and approximate
ηL(g) := EηL(g) ≈ Y NLL :=
1NL
NL∑i=1
g(U(i)L ).
Mean square error (MSE) EY NLL − Eη∞ [g(U)]2 splits into
EY NLL − EηL [g(U)]2︸ ︷︷ ︸variance=O(N−1
L )
+EηL [g(U)]− Eη∞ [g(U)]︸ ︷︷ ︸bias
2
Cost to achieve MSE= O(ε2) is Cost(U(i)L )× ε−2.
![Page 10: Multilevel sequential Monte Carlo samplers and …ims.nus.edu.sg/events/2018/bay/files/law.pdfMultilevel sequential Monte Carlo samplers and other MLMC methods Kody Law & A. Jasra](https://reader033.fdocuments.net/reader033/viewer/2022053002/5f067e297e708231d4184391/html5/thumbnails/10.jpg)
Introduction MLMC BIP SMC MLSMC Other Summary
Multilevel Monte Carlo I
Introduce a hierarchy of discretization levels ηlLl=1 and defineYl = Eηl [g(U)]− Eηl−1 [g(U)], with η−1 := 0.Observe the telescopic sum
EηL [g(U)] =L∑
l=0
Yl .
Each term can be unbiasedly approximated by
Y Nll =
1Nl
Nl∑i=1
g(U(i)l )− g(U(i)
l−1)
where g(U(i)−1) := 0.
![Page 11: Multilevel sequential Monte Carlo samplers and …ims.nus.edu.sg/events/2018/bay/files/law.pdfMultilevel sequential Monte Carlo samplers and other MLMC methods Kody Law & A. Jasra](https://reader033.fdocuments.net/reader033/viewer/2022053002/5f067e297e708231d4184391/html5/thumbnails/11.jpg)
Introduction MLMC BIP SMC MLSMC Other Summary
Multilevel Monte Carlo II
Multilevel Monte Carlo approach:Sample i.i.d. (Ul ,Ul−1)(i) ∼ ηl , such that∫ηldul−1,l = ηl,l−1, and approximate
ηL(g) ≈ YL,Multi :=L∑
l=0
Y Nll .
Mean square error (MSE) given by
EYL,Multi − Eη∞ [g(U)]2 =
EYL,Multi − EηL [g(U)]2︸ ︷︷ ︸variance=
∑Ll=0 Vl/Nl
+EηL [g(U)]− Eη∞ [g(U)]︸ ︷︷ ︸bias
2 .
Fix bias by choosing L. Minimize cost C =∑L
l=0 ClNl as afunction of NlLl=0 for fixed variance⇒ Nl ∝
√Vl/Cl .
![Page 12: Multilevel sequential Monte Carlo samplers and …ims.nus.edu.sg/events/2018/bay/files/law.pdfMultilevel sequential Monte Carlo samplers and other MLMC methods Kody Law & A. Jasra](https://reader033.fdocuments.net/reader033/viewer/2022053002/5f067e297e708231d4184391/html5/thumbnails/12.jpg)
Introduction MLMC BIP SMC MLSMC Other Summary
Illustration of pairwise coupling
Pairwise coupling of trajectories of an SDE:
X 1n+1 = X 1
n +hf (X 1n )+√
hσ(X 1n )ξn, ξn ∼ N(0,1), n = 0, . . . ,N1
X 0n+1 = X 0
n +(2h)f (X 0n )+√
2hσ(X 0n )(ξ2n+ξ2n+1), n = 0, . . . , (N1−1)/2.
0.0 0.2 0.4 0.6 0.8 1.0t
0.2
0.0
0.2
0.4
0.6 W0
t
W1
t
(a) Wiener processW 1
n =√
h∑n
i=0 ξn, W 0n = W 1
2n.
0.0 0.2 0.4 0.6 0.8 1.0t
1.0
1.1
1.2
1.3
1.4
1.5
1.6
X
0
t
X
1
t
(b) Stochastic process driven byWiener process.
![Page 13: Multilevel sequential Monte Carlo samplers and …ims.nus.edu.sg/events/2018/bay/files/law.pdfMultilevel sequential Monte Carlo samplers and other MLMC methods Kody Law & A. Jasra](https://reader033.fdocuments.net/reader033/viewer/2022053002/5f067e297e708231d4184391/html5/thumbnails/13.jpg)
Introduction MLMC BIP SMC MLSMC Other Summary
Multilevel vs. Single level
Assume hl = 2−l and there are α, and β > ζ such that(i) weak error |E[g(Ul)− g(U)]| = O(hαl ).
(ii) strong error E|g(Ul)− g(U)|2 = O(hβl )⇒ Vl = O(hβl ),(iii) computational cost for a realization of g(Ul)− g(Ul−1),
Cl ∝ h−ζl .Both cases require hαL = O(ε)⇒ L ∝ | log ε|.
Single level cost C = O(ε−ζ/α−2) : cost per sample isCL ∝ ε−ζ/α, and fixed V ∝ ε2 ⇒ NL ∝ ε−2.Multilevel cost CML = O(ε−2) : Nl ∝ ε−2KLh(β+ζ)/2
l , soV ∝ ε2 and C ∝ ε−2K 2
L for KL =∑L
l=0 h(β−ζ)/2l = O(1)
[G08] – cost of simulating a scalar random variable.Example: Milstein solution of SDE
C = O(ε−3) vs. CML = O(ε−2).
![Page 14: Multilevel sequential Monte Carlo samplers and …ims.nus.edu.sg/events/2018/bay/files/law.pdfMultilevel sequential Monte Carlo samplers and other MLMC methods Kody Law & A. Jasra](https://reader033.fdocuments.net/reader033/viewer/2022053002/5f067e297e708231d4184391/html5/thumbnails/14.jpg)
Introduction MLMC BIP SMC MLSMC Other Summary
Outline
1 Introduction
2 Multilevel Monte Carlo sampling
3 Bayesian inference problem
4 Sequential Monte Carlo samplers
5 Multilevel Sequential Monte Carlo (MLSMC) samplers
6 Other MLMC algorithms for inferenceImportance sampling (IS) strategyCoupled algorithm (CA) strategyApproximate coupling (AC) strategy
7 Summary
![Page 15: Multilevel sequential Monte Carlo samplers and …ims.nus.edu.sg/events/2018/bay/files/law.pdfMultilevel sequential Monte Carlo samplers and other MLMC methods Kody Law & A. Jasra](https://reader033.fdocuments.net/reader033/viewer/2022053002/5f067e297e708231d4184391/html5/thumbnails/15.jpg)
Introduction MLMC BIP SMC MLSMC Other Summary
Inverse Problems
Data Parameter
y = G ( u ) + e
forward model (PDE) observation/model errors
y ∈ RM
u ∈ EG : E → RM
Data y may be limited in number, noisy, and indirect.Parameter u often a function, discretized.Needs to be approximated.
![Page 16: Multilevel sequential Monte Carlo samplers and …ims.nus.edu.sg/events/2018/bay/files/law.pdfMultilevel sequential Monte Carlo samplers and other MLMC methods Kody Law & A. Jasra](https://reader033.fdocuments.net/reader033/viewer/2022053002/5f067e297e708231d4184391/html5/thumbnails/16.jpg)
Introduction MLMC BIP SMC MLSMC Other Summary
Example forward problem
Let V := H1(Ω) ⊂ L2(Ω) ⊂ H−1(Ω) =: V ∗, Ω ⊂ Rd with ∂Ωconvex, and f ∈ V ∗. Consider
−∇ · (u∇p) = f , on Ω
p = 0, on ∂Ω,
where
u(x) = u(x) +K∑
k=1
ukσk Φk (x) .
Define u = ukKk=1 ∈ E :=∏K
k=1[−1,1], with uk ∼ U[−1,1]i.i.d. This determines the prior distribution for u. Assumeu,Φk ∈ C∞, and ‖Φk‖∞ = 1 for all k , and require
infx
u(x) ≥ infx
u(x)−K∑
k=1
σk ≥ u∗ > 0.
![Page 17: Multilevel sequential Monte Carlo samplers and …ims.nus.edu.sg/events/2018/bay/files/law.pdfMultilevel sequential Monte Carlo samplers and other MLMC methods Kody Law & A. Jasra](https://reader033.fdocuments.net/reader033/viewer/2022053002/5f067e297e708231d4184391/html5/thumbnails/17.jpg)
Introduction MLMC BIP SMC MLSMC Other Summary
Bayesian inverse problem
Let gm ∈ V ∗ for m = 1, . . . ,M and define
G(p) = [g1(p), · · · ,gM(p)]> .
Let p(·; u) denote the weak solution for parameter value u.
DATA : y = G(p(·; u)) + e, e ∼ N(0, Γ), e ⊥ u.
The unnormalized density of u|y over u ∈ E is given by
κ(u) = e−Φ[G(p(·;u))] ; Φ(G) = 12 |G − y |2Γ .
TARGET : η(u) =κ(u)
Z, Z =
∫Eκ(u)du.
![Page 18: Multilevel sequential Monte Carlo samplers and …ims.nus.edu.sg/events/2018/bay/files/law.pdfMultilevel sequential Monte Carlo samplers and other MLMC methods Kody Law & A. Jasra](https://reader033.fdocuments.net/reader033/viewer/2022053002/5f067e297e708231d4184391/html5/thumbnails/18.jpg)
Introduction MLMC BIP SMC MLSMC Other Summary
Outline
1 Introduction
2 Multilevel Monte Carlo sampling
3 Bayesian inference problem
4 Sequential Monte Carlo samplers
5 Multilevel Sequential Monte Carlo (MLSMC) samplers
6 Other MLMC algorithms for inferenceImportance sampling (IS) strategyCoupled algorithm (CA) strategyApproximate coupling (AC) strategy
7 Summary
![Page 19: Multilevel sequential Monte Carlo samplers and …ims.nus.edu.sg/events/2018/bay/files/law.pdfMultilevel sequential Monte Carlo samplers and other MLMC methods Kody Law & A. Jasra](https://reader033.fdocuments.net/reader033/viewer/2022053002/5f067e297e708231d4184391/html5/thumbnails/19.jpg)
Introduction MLMC BIP SMC MLSMC Other Summary
SMC sampler algorithm
Distributions ηl dictated by an accuracy parameter hl (hereFEM mesh diameter)∞ > h0 > h1 · · · > h∞ = 0. ApproximateEηL [g(U)] = ηL(g) =
∫E g(u)ηL(u)du.
Idea: interlace sequential importance resampling (selection)along the hierarchy, and mutation by MCMC kernels.
Initialize i.i.d. U i0 ∼ η0, i = 1, . . . ,N. For l ∈ 0, . . . ,L− 1:
Resample U il Ni=1 according to the weights w i
l Ni=1,
w il = Gi
l/∑N
j=1 Gjl , Gi
l = (κl+1/κl)(U il ).
Draw U il+1 ∼ Ml+1(U i
l , ·), where Ml+1 is an MCMC kernelsuch that ηl+1Ml+1 = ηl+1.
For g : E → R, l ∈ 0, . . . ,L, we have the following estimators
Eηl [g(U)] ≈ ηNl (g) :=
1N
N∑i=1
g(U il ) .
![Page 20: Multilevel sequential Monte Carlo samplers and …ims.nus.edu.sg/events/2018/bay/files/law.pdfMultilevel sequential Monte Carlo samplers and other MLMC methods Kody Law & A. Jasra](https://reader033.fdocuments.net/reader033/viewer/2022053002/5f067e297e708231d4184391/html5/thumbnails/20.jpg)
Introduction MLMC BIP SMC MLSMC Other Summary
Outline
1 Introduction
2 Multilevel Monte Carlo sampling
3 Bayesian inference problem
4 Sequential Monte Carlo samplers
5 Multilevel Sequential Monte Carlo (MLSMC) samplers
6 Other MLMC algorithms for inferenceImportance sampling (IS) strategyCoupled algorithm (CA) strategyApproximate coupling (AC) strategy
7 Summary
![Page 21: Multilevel sequential Monte Carlo samplers and …ims.nus.edu.sg/events/2018/bay/files/law.pdfMultilevel sequential Monte Carlo samplers and other MLMC methods Kody Law & A. Jasra](https://reader033.fdocuments.net/reader033/viewer/2022053002/5f067e297e708231d4184391/html5/thumbnails/21.jpg)
Introduction MLMC BIP SMC MLSMC Other Summary
MLSMC sampler
Notice
EηL [g(U)] = Eη0 [g(U)] +L∑
l=1
Eηl [g(U)]− Eηl−1 [g(U)]
= Eη0 [g(U)] +L∑
l=1
Eηl−1
[(κl(U)Zl−1
κl−1(U)Zl− 1)
g(U)]. †
Idea: Approximate † using SMC sample hierarchy.
Key: Subsample (U1:N00 , . . . ,U1:NL−1
L−1 ) as in single level SMC, butwith +∞ > N0 ≥ N1 · · · ≥ NL−1 ≥ 1 appropriately chosen.
![Page 22: Multilevel sequential Monte Carlo samplers and …ims.nus.edu.sg/events/2018/bay/files/law.pdfMultilevel sequential Monte Carlo samplers and other MLMC methods Kody Law & A. Jasra](https://reader033.fdocuments.net/reader033/viewer/2022053002/5f067e297e708231d4184391/html5/thumbnails/22.jpg)
Introduction MLMC BIP SMC MLSMC Other Summary
MLSMC estimator
The MLSMC consistent estimator of ηL(g) is given by
Y := ηN00 (g) +
L∑l=1
ηNl−1l−1 (gGl−1)
ηNl−1l−1 (Gl−1)
− ηNl−1l−1 (g)
.
i) the L + 1 terms above are not unbiased estimates ofEηl [g(U)]− Eηl−1 [g(U)], so decompose MSE as:
E[Y − Eη∞ [g(U)]2
]≤
2E[Y − EηL [g(U)]2
]+ 2 EηL [g(U)]− Eη∞ [g(U)]2 .
ii) the same L + 1 estimates are not independent, so a morecomplex error analysis will be required to characterizeE[Y − EηL [g(U)]2].
![Page 23: Multilevel sequential Monte Carlo samplers and …ims.nus.edu.sg/events/2018/bay/files/law.pdfMultilevel sequential Monte Carlo samplers and other MLMC methods Kody Law & A. Jasra](https://reader033.fdocuments.net/reader033/viewer/2022053002/5f067e297e708231d4184391/html5/thumbnails/23.jpg)
Introduction MLMC BIP SMC MLSMC Other Summary
Assumptions
(A1) There exist 0 < C < C < +∞ such that
sup1≤l≤L
supu∈E
Gl(u) ≤ C ,
inf1≤l≤L
infu∈E
Gl(u) ≥ C .
(A2) There exist a ρ ∈ (0,1) such that for any 1 ≤ p ≤ L− 1,(u, v) ∈ E2, A ∈ σ(E),∫
AMp(u,du′) ≥ ρ
∫A
Mp(v ,du′).
(A3) There is a β > 0 such that
Vl := ‖Zl−1Zl
Gl−1 − 1‖2∞ = O(hβl ) .
![Page 24: Multilevel sequential Monte Carlo samplers and …ims.nus.edu.sg/events/2018/bay/files/law.pdfMultilevel sequential Monte Carlo samplers and other MLMC methods Kody Law & A. Jasra](https://reader033.fdocuments.net/reader033/viewer/2022053002/5f067e297e708231d4184391/html5/thumbnails/24.jpg)
Introduction MLMC BIP SMC MLSMC Other Summary
Main result
Theorem (BJLTZ16 )Assume (A1-3). For any g : E → R bounded
E[Y − EηL [g(U)]2
]=
V2
.1
N0+
L∑l=1
(Vl
Nl+(Vl
Nl
)1/2 L∑q=l+1
V 1/2q
Nq
).
In particular, for β > ζ, L and NlLl=0 can be chosen such thatMSE= O(ε2) for computational cost= O(ε−2), the optimal case.
![Page 25: Multilevel sequential Monte Carlo samplers and …ims.nus.edu.sg/events/2018/bay/files/law.pdfMultilevel sequential Monte Carlo samplers and other MLMC methods Kody Law & A. Jasra](https://reader033.fdocuments.net/reader033/viewer/2022053002/5f067e297e708231d4184391/html5/thumbnails/25.jpg)
Introduction MLMC BIP SMC MLSMC Other Summary
Runtime cost as a function of error
215
220
225
230
2−25 2−20 2−15 2−10
MSE (𝜀2)
Runtim
ecost∝∑𝐿 𝑙=0𝑁𝑙ℎ−1 𝑙
Algorithm MLSMC SMC
![Page 26: Multilevel sequential Monte Carlo samplers and …ims.nus.edu.sg/events/2018/bay/files/law.pdfMultilevel sequential Monte Carlo samplers and other MLMC methods Kody Law & A. Jasra](https://reader033.fdocuments.net/reader033/viewer/2022053002/5f067e297e708231d4184391/html5/thumbnails/26.jpg)
Introduction MLMC BIP SMC MLSMC Other Summary
Outline
1 Introduction
2 Multilevel Monte Carlo sampling
3 Bayesian inference problem
4 Sequential Monte Carlo samplers
5 Multilevel Sequential Monte Carlo (MLSMC) samplers
6 Other MLMC algorithms for inferenceImportance sampling (IS) strategyCoupled algorithm (CA) strategyApproximate coupling (AC) strategy
7 Summary
![Page 27: Multilevel sequential Monte Carlo samplers and …ims.nus.edu.sg/events/2018/bay/files/law.pdfMultilevel sequential Monte Carlo samplers and other MLMC methods Kody Law & A. Jasra](https://reader033.fdocuments.net/reader033/viewer/2022053002/5f067e297e708231d4184391/html5/thumbnails/27.jpg)
Introduction MLMC BIP SMC MLSMC Other Summary
(IS) MLSMC sampler for normalizing constants
Two unbiased estimators proposed which provide the optimalrate with a logarithmic penalty on the cost: MSE O(ε2) for costO(| log ε|ε−2) [DJLZ16]. Here one can also construct rMLMCestimators of Rhee&Glynn type.
215
220
225
230
2−30 2−25 2−20 2−15
MSE (𝜀2)
Runtim
ecost∝∑𝐿 𝑙=0𝑁𝑙ℎ−1 𝑙
Algorithm MLSMC ( 𝛾𝑁0∶𝑙−2𝑙 (1)) MLSMC (𝛾𝑁0∶𝑙−1𝑙 (1)) SMC (𝛾𝑁0∶𝑙−1𝑙 (1))
![Page 28: Multilevel sequential Monte Carlo samplers and …ims.nus.edu.sg/events/2018/bay/files/law.pdfMultilevel sequential Monte Carlo samplers and other MLMC methods Kody Law & A. Jasra](https://reader033.fdocuments.net/reader033/viewer/2022053002/5f067e297e708231d4184391/html5/thumbnails/28.jpg)
Introduction MLMC BIP SMC MLSMC Other Summary
(IS) MLSMC samplers with DILI mutations
Posterior over function-space, levels include refinement inparameter and model ηl(u0:l).Covariance-based LIS (cLIS) introduced and incorporatedin DILI proposals [CLM16, BJLMZ17] – substantialreduction in cost.
220
230
240
250
260
2−10 2−5 20
Error
rmse
Algorithm mlsmc (dili) mlsmc (pcn) smc ∝ 𝜀−2 ∝ 𝜀−3
1
![Page 29: Multilevel sequential Monte Carlo samplers and …ims.nus.edu.sg/events/2018/bay/files/law.pdfMultilevel sequential Monte Carlo samplers and other MLMC methods Kody Law & A. Jasra](https://reader033.fdocuments.net/reader033/viewer/2022053002/5f067e297e708231d4184391/html5/thumbnails/29.jpg)
Introduction MLMC BIP SMC MLSMC Other Summary
(CA) ML particle filter (MLPF) for SDE
Filtering involves a sequence of Bayesian inversions,separated by propagation in time (through an SDE).Let η0,m, . . . , ηL,m, . . . , η∞,m denote the time m filteringdistribution at a hierarchy of levels (time discretization).Coupled traditional SMC algorithms (particle filters) can beused for each level.Mutation M` is now coupled propagation of a pair of initialconditions through an SDE discretized at two successivemesh-refinements, for ` = 0, . . . ,L.Selection is performed by novel pairwise coupledresampling which preserves marginals (maximal coupling).MLMC results carry over with somewhat weaker rateβ → β/2 [JKLZ15].New version with improved rate using L2 optimal couplinginstead.
![Page 30: Multilevel sequential Monte Carlo samplers and …ims.nus.edu.sg/events/2018/bay/files/law.pdfMultilevel sequential Monte Carlo samplers and other MLMC methods Kody Law & A. Jasra](https://reader033.fdocuments.net/reader033/viewer/2022053002/5f067e297e708231d4184391/html5/thumbnails/30.jpg)
Introduction MLMC BIP SMC MLSMC Other Summary
(CA) MLPF numerical experiments
MSE Bias2 Variance
10−8
10−7
10−6
10−5
10−4
10−10
10−9
10−8
10−7
10−6
10−7
10−6
10−5
10−4
10−3
10−8
10−7
10−6
10−5
10−4
OUGBM
LangevinNLM
102 103 104 105 106 107 102 103 104 105 106 107 102 103 104 105 106 107Cost
Error
Algorithm MLPF PF
![Page 31: Multilevel sequential Monte Carlo samplers and …ims.nus.edu.sg/events/2018/bay/files/law.pdfMultilevel sequential Monte Carlo samplers and other MLMC methods Kody Law & A. Jasra](https://reader033.fdocuments.net/reader033/viewer/2022053002/5f067e297e708231d4184391/html5/thumbnails/31.jpg)
Introduction MLMC BIP SMC MLSMC Other Summary
(CA) ML ensemble Kalman filter (MLEnKF) for S(P)DE
EnKF uses sample covariance from an ensemble ofparticles to approximate a linear Gaussian Bayesianupdate, given by an affine transformation of particles.Multilevel approximation of the covariance improves costfor MSE O(ε2) [HLT16].
Best theoretical bound (at step n): O(| log ε|2nε−2).Numerically (uniformly in n): O(ε−2).
New SPDE extension: same n-dependence, much morebroadly applicable [CHLNT17].
10−1 100 101 102 103 104
Runtime [s]
10−6
10−5
10−4
10−3
RM
SE
10−1 100 101 102 103 104
Runtime [s]
EnKF
MLEnKF
cs−1/3
cs−1/2
10−1 100 101 102 103 104
Runtime [s]
![Page 32: Multilevel sequential Monte Carlo samplers and …ims.nus.edu.sg/events/2018/bay/files/law.pdfMultilevel sequential Monte Carlo samplers and other MLMC methods Kody Law & A. Jasra](https://reader033.fdocuments.net/reader033/viewer/2022053002/5f067e297e708231d4184391/html5/thumbnails/32.jpg)
Introduction MLMC BIP SMC MLSMC Other Summary
(AC) Parameter estimation for SDE with pMCMC
Aim: estimate E[ϕ(θ)|y ], where y is a finite set of partialobservations of the SDE X θ
t on [0,T ], parameterized by θ.particle MCMC: Iterate
propose θ′ ∼ q(θ, θ′),simulate X θ′,iM
i=1 ≈ π(X |θ′, y) with particle filter,compute non-negative and unbiased estimatorpM(y |θ′) =
∏np=1( 1
M
∑Mi=1 gp(X θ′,i
p )),accept/reject according to
1 ∧ pM(y |θ′)π(θ′)q(θ′, θ)
pM(y |θ)π(θ)q(θ, θ′).
![Page 33: Multilevel sequential Monte Carlo samplers and …ims.nus.edu.sg/events/2018/bay/files/law.pdfMultilevel sequential Monte Carlo samplers and other MLMC methods Kody Law & A. Jasra](https://reader033.fdocuments.net/reader033/viewer/2022053002/5f067e297e708231d4184391/html5/thumbnails/33.jpg)
Introduction MLMC BIP SMC MLSMC Other Summary
MLMC version [JKLZ16]:Construct approximate coupling πl,l−1(θ,X l ,X l−1): usualcoupled forward kernel, and coupled selection functionGp,θ(X l ,X l−1) = maxgp,θ(X l ),gp,θ(X l−1).Let Hl (θ,X l ,X l−1) =
∏np=1 gp,θ(X l )/Gp,θ(X l ,X l−1). Then
Eπl [ϕ(θ)]− Eπl−1 [ϕ(θ)] =Eπl,l−1 [ϕ(θ)Hl (θ,X l ,X l−1)]
Eπl,l−1 [Hl (θ,X l ,X l−1)]− Eπl,l−1 [ϕ(θl−1)Hl−1(θ,X l ,X l−1)]
Eπl,l−1 [Hl−1(θ,X l ,X l−1)].
Optimal results hold with same rate as forward.
Lagenvin SDE
𝜎
Lagenvin SDE
𝜃
Ornstein-Uhlenbech process
𝜎
Ornstein-Uhlenbech process
𝜃
2−8 2−7 2−6 2−5 2−4 2−3 2−2 2−5 2−4 2−3 2−2 2−1 20
2−8 2−7 2−6 2−5 2−4 2−3 2−6 2−5 2−4 2−3 2−2 2−122242628210
22242628210
22242628210
22242628210
Error
Cost
Algorithm ML-PMCMC PMCMC
![Page 34: Multilevel sequential Monte Carlo samplers and …ims.nus.edu.sg/events/2018/bay/files/law.pdfMultilevel sequential Monte Carlo samplers and other MLMC methods Kody Law & A. Jasra](https://reader033.fdocuments.net/reader033/viewer/2022053002/5f067e297e708231d4184391/html5/thumbnails/34.jpg)
Introduction MLMC BIP SMC MLSMC Other Summary
(AC) Multi-index Markov chain Monte Carlo (MIMCMC)
If spatio-temporal approximation dimension d > 1, thenMIMC is preferable to MLMC [HNT15]. α ∈ Nd
∆iEα(ϕ(u)) = Eα(ϕ(u))− Eα−ei (ϕ(u)), ∆ = ∆d · · ·∆1,
E(ϕ(u)) =∑α
∆Eα(ϕ(u)) ≈∑α∈I
∆Eα(ϕ(u))
Approximate coupling can be applied to the 2d
probability measures in each summand.
![Page 35: Multilevel sequential Monte Carlo samplers and …ims.nus.edu.sg/events/2018/bay/files/law.pdfMultilevel sequential Monte Carlo samplers and other MLMC methods Kody Law & A. Jasra](https://reader033.fdocuments.net/reader033/viewer/2022053002/5f067e297e708231d4184391/html5/thumbnails/35.jpg)
Introduction MLMC BIP SMC MLSMC Other Summary
Optimal results hold for appropriate regularity [JKLZ17]
210
220
230
240
2−4 2−2 20 22
Error
Cost
Algorithm mcmc mimcmc
1
![Page 36: Multilevel sequential Monte Carlo samplers and …ims.nus.edu.sg/events/2018/bay/files/law.pdfMultilevel sequential Monte Carlo samplers and other MLMC methods Kody Law & A. Jasra](https://reader033.fdocuments.net/reader033/viewer/2022053002/5f067e297e708231d4184391/html5/thumbnails/36.jpg)
Introduction MLMC BIP SMC MLSMC Other Summary
Outline
1 Introduction
2 Multilevel Monte Carlo sampling
3 Bayesian inference problem
4 Sequential Monte Carlo samplers
5 Multilevel Sequential Monte Carlo (MLSMC) samplers
6 Other MLMC algorithms for inferenceImportance sampling (IS) strategyCoupled algorithm (CA) strategyApproximate coupling (AC) strategy
7 Summary
![Page 37: Multilevel sequential Monte Carlo samplers and …ims.nus.edu.sg/events/2018/bay/files/law.pdfMultilevel sequential Monte Carlo samplers and other MLMC methods Kody Law & A. Jasra](https://reader033.fdocuments.net/reader033/viewer/2022053002/5f067e297e708231d4184391/html5/thumbnails/37.jpg)
Introduction MLMC BIP SMC MLSMC Other Summary
Summary
MLSMC sampler can perform as well as MLMC.For our example β > ζ. If β ≤ ζ, cost is somewhat higher,analogous to standard MLMC.If ζ > 2α then the optimal cost is ε−ζ/α, the cost of a singlesimulation at the finest level.New importance sampling: MLSMC with DILI mutations.Coupled algorithms: MLPF strong error is effectivelyreduced by coupled resampling β → β/2.Coupled algorithms: MLEnKF has a spuriousn-dependent logarithmic penalty | log ε|2n on cost.New coupled algorithms: MLMCMC.Friday approximate couplings: ML PMCMC for SDEparameter estimation preserves strong error β.New approximate couplings: MIMCMC and MISMC2.Looking for students/postdocs with similar interests.
![Page 38: Multilevel sequential Monte Carlo samplers and …ims.nus.edu.sg/events/2018/bay/files/law.pdfMultilevel sequential Monte Carlo samplers and other MLMC methods Kody Law & A. Jasra](https://reader033.fdocuments.net/reader033/viewer/2022053002/5f067e297e708231d4184391/html5/thumbnails/38.jpg)
Introduction MLMC BIP SMC MLSMC Other Summary
References
[BJLTZ15]: Beskos, Jasra, Law, Tempone, Zhou."Multilevel Sequential Monte Carlo samplers." SPA 127:5,1417–1440 (2017).[DJLZ16]: Del Moral, Jasra, Law, Zhou. "MultilevelSequential Monte Carlo samplers for normalizingconstants." ToMACS 27(3), 20 (2017).[JKLZ15]: Jasra, Kamatani, Law, Zhou. "Multilevel particlefilter." SINUM 55(6), 3068–3096 (2017).[JLZ16]: Jasra, Law, and Zhou. "Forward and InverseUncertainty Quantification using Multilevel Monte CarloAlgorithms for an Elliptic Nonlocal Equation." Int. J. Unc.Quant., 6(6), 501–514 (2016).[HLT15]: Hoel, Law, Tempone. "Multilevel ensembleKalman filter." SINUM 54(3), 1813–1839 (2016).
![Page 39: Multilevel sequential Monte Carlo samplers and …ims.nus.edu.sg/events/2018/bay/files/law.pdfMultilevel sequential Monte Carlo samplers and other MLMC methods Kody Law & A. Jasra](https://reader033.fdocuments.net/reader033/viewer/2022053002/5f067e297e708231d4184391/html5/thumbnails/39.jpg)
Introduction MLMC BIP SMC MLSMC Other Summary
References[G08]: Giles. Op. Res., 56, 607-617 (2008).[H00] Heinrich. LSSC proceedings (2001).[HNT15] Haji-Ali, Nobile, Tempone. NumerischeMathematik, 132, 767-806 (2016).[DDJ06]: Del Moral, Doucet, Jasra. J. R. Statist. Soc. B,68, 411-436 (2006).[C02]: Chopin. Biometrika 89:3 539–552 (2002).[D04]: Del Moral. "Feynman-Kac Formulae." Springer:New York (2004).[CLM16]: Cui, Law, Marzouk. J. Comp. Phys. 304,109-137 (2016).[HSS13]: Hoang, Schwab, Stuart. Inverse Prob., 29,085010 (2013).[KST13]: Ketelsen, Scheichl, Teckentrup. SIAM/ASA JUQ3(1) 1075-1108 (2015).
![Page 40: Multilevel sequential Monte Carlo samplers and …ims.nus.edu.sg/events/2018/bay/files/law.pdfMultilevel sequential Monte Carlo samplers and other MLMC methods Kody Law & A. Jasra](https://reader033.fdocuments.net/reader033/viewer/2022053002/5f067e297e708231d4184391/html5/thumbnails/40.jpg)
Introduction MLMC BIP SMC MLSMC Other Summary
References
[CHLNT17] Chernov, Hoel, Law, Nobile, Tempone."Multilevel ensemble Kalman filtering for spatio-temporalprocesses." arXiv:1608.08558 (2017).[JKLZ17] Jasra, Kamatani, Law, Zhou. "MLMC for staticBayesian parameter estimation." SISC 40(2), A887-A902(2018). See talk on Friday.[JKLZ17] Jasra, Kamatani, Law, Zhou. "A Multi-IndexMarkov Chain Monte Carlo Method." IJUQ 8(1) (2018).[JLS17] Jasra, Law, Suciu. "Advanced Multilevel MonteCarlo Methods." arXiv:1704.07272 (2017).[BJLMZ17]: Beskos, Jasra, Law, Marzouk, Zhou. "MLSMCsamplers with DILI mutations." SIAM/ASA JUQ 6(2),762-786 (2018).
![Page 41: Multilevel sequential Monte Carlo samplers and …ims.nus.edu.sg/events/2018/bay/files/law.pdfMultilevel sequential Monte Carlo samplers and other MLMC methods Kody Law & A. Jasra](https://reader033.fdocuments.net/reader033/viewer/2022053002/5f067e297e708231d4184391/html5/thumbnails/41.jpg)
Introduction MLMC BIP SMC MLSMC Other Summary
References
[JLX18.i] Jasra, Law, Xu. “Markov chain Simulation forMultilevel Monte Carlo.” arXiv:1806.09754 (2018).[JLX18.ii]: Jasra, Law, Xu. “MISMC2 for partially observedSPDE.” arXiv:1805.00415 (2018).
![Page 42: Multilevel sequential Monte Carlo samplers and …ims.nus.edu.sg/events/2018/bay/files/law.pdfMultilevel sequential Monte Carlo samplers and other MLMC methods Kody Law & A. Jasra](https://reader033.fdocuments.net/reader033/viewer/2022053002/5f067e297e708231d4184391/html5/thumbnails/42.jpg)
Introduction MLMC BIP SMC MLSMC Other Summary
Thank you