Bayesian static parameter estimation using Multilevel...
Transcript of Bayesian static parameter estimation using Multilevel...
![Page 1: Bayesian static parameter estimation using Multilevel ...mcqmc2018.inria.fr/wp-content/uploads/2018/07/law_pmcmc-mcqmc18.pdf · normalizing constant [HSS13, DKST15, HTL16, BJLTZ17].](https://reader033.fdocuments.in/reader033/viewer/2022041617/5e3c7ea58de0a6468f3ca359/html5/thumbnails/1.jpg)
Introduction MLMC BIP Coupling PMCMC PMC(ML)MC Numerical simulations Summary
Bayesian static parameter estimation usingMultilevel Monte Carlo
Kody Law
joint with A. Jasra (NUS), K. Kamatani (Osaka), & Y. Zhou
MCQMC 2018Rennes, FR
July 4, 2018
![Page 2: Bayesian static parameter estimation using Multilevel ...mcqmc2018.inria.fr/wp-content/uploads/2018/07/law_pmcmc-mcqmc18.pdf · normalizing constant [HSS13, DKST15, HTL16, BJLTZ17].](https://reader033.fdocuments.in/reader033/viewer/2022041617/5e3c7ea58de0a6468f3ca359/html5/thumbnails/2.jpg)
Introduction MLMC BIP Coupling PMCMC PMC(ML)MC Numerical simulations Summary
Outline
1 Introduction
2 Multilevel Monte Carlo sampling
3 Bayesian inference problem
4 Approximate coupling
5 Particle Markov chain Monte Carlo
6 Particle Markov chain Multilevel Monte Carlo
7 Numerical simulations
8 Summary
![Page 3: Bayesian static parameter estimation using Multilevel ...mcqmc2018.inria.fr/wp-content/uploads/2018/07/law_pmcmc-mcqmc18.pdf · normalizing constant [HSS13, DKST15, HTL16, BJLTZ17].](https://reader033.fdocuments.in/reader033/viewer/2022041617/5e3c7ea58de0a6468f3ca359/html5/thumbnails/3.jpg)
Introduction MLMC BIP Coupling PMCMC PMC(ML)MC Numerical simulations Summary
Outline
1 Introduction
2 Multilevel Monte Carlo sampling
3 Bayesian inference problem
4 Approximate coupling
5 Particle Markov chain Monte Carlo
6 Particle Markov chain Multilevel Monte Carlo
7 Numerical simulations
8 Summary
![Page 4: Bayesian static parameter estimation using Multilevel ...mcqmc2018.inria.fr/wp-content/uploads/2018/07/law_pmcmc-mcqmc18.pdf · normalizing constant [HSS13, DKST15, HTL16, BJLTZ17].](https://reader033.fdocuments.in/reader033/viewer/2022041617/5e3c7ea58de0a6468f3ca359/html5/thumbnails/4.jpg)
Introduction MLMC BIP Coupling PMCMC PMC(ML)MC Numerical simulations Summary
Orientation
Aim: Approximate posterior expectations of the state path andstatic parameters associated to a discretely observed diffusion,which must be finitely approximated.Solution: Apply an approximate coupling strategy so that themultilevel Monte Carlo (MLMC) method can be used within aparticle marginal Metropolis-Hastings (PMMH) algorithm [B02,AR08, ADH10].
MLMC methods reduce cost to error= O(ε) [H00, G08];Recently this methodology has been applied to inference,mostly in cases where target can be evaluated up to anormalizing constant [HSS13, DKST15, HTL16, BJLTZ17].Here it cannot, but using PMMH we are able to sampleconsistently from an approximate coupling of successivetargets [JKLZ18].
![Page 5: Bayesian static parameter estimation using Multilevel ...mcqmc2018.inria.fr/wp-content/uploads/2018/07/law_pmcmc-mcqmc18.pdf · normalizing constant [HSS13, DKST15, HTL16, BJLTZ17].](https://reader033.fdocuments.in/reader033/viewer/2022041617/5e3c7ea58de0a6468f3ca359/html5/thumbnails/5.jpg)
Introduction MLMC BIP Coupling PMCMC PMC(ML)MC Numerical simulations Summary
Outline
1 Introduction
2 Multilevel Monte Carlo sampling
3 Bayesian inference problem
4 Approximate coupling
5 Particle Markov chain Monte Carlo
6 Particle Markov chain Multilevel Monte Carlo
7 Numerical simulations
8 Summary
![Page 6: Bayesian static parameter estimation using Multilevel ...mcqmc2018.inria.fr/wp-content/uploads/2018/07/law_pmcmc-mcqmc18.pdf · normalizing constant [HSS13, DKST15, HTL16, BJLTZ17].](https://reader033.fdocuments.in/reader033/viewer/2022041617/5e3c7ea58de0a6468f3ca359/html5/thumbnails/6.jpg)
Introduction MLMC BIP Coupling PMCMC PMC(ML)MC Numerical simulations 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)NT}Ni=1, NT = T/h.
![Page 7: Bayesian static parameter estimation using Multilevel ...mcqmc2018.inria.fr/wp-content/uploads/2018/07/law_pmcmc-mcqmc18.pdf · normalizing constant [HSS13, DKST15, HTL16, BJLTZ17].](https://reader033.fdocuments.in/reader033/viewer/2022041617/5e3c7ea58de0a6468f3ca359/html5/thumbnails/7.jpg)
Introduction MLMC BIP Coupling PMCMC PMC(ML)MC Numerical simulations Summary
Multilevel Monte Carlo [H00, G08, CGST11]
Assume hl = 2−l , Xl approximates XT with stepsize hl , andthere are α, β, ζ > 0 such that
(i) weak error |E[g(Xl)− g(X )]| = O(hαl ).
(ii) strong error E|g(Xl)− g(X )|2 = O(hβl )⇒ Vl = O(hβl ),(iii) computational cost for a realization of g(Xl)− g(Xl−1),
Cl ∝ h−ζl .Then, it is possible to choose L and {Nl}Ll=0 so as to achieve amean squared error of O(ε2) at a cost
COST ≤ C
ε−2, if β > γ,
ε−2| log(ε)|2, if β = γ,
ε−(2+ γ−βα ), if β < γ.
Compared with O(ε−2−γ/α) for single level.
![Page 8: Bayesian static parameter estimation using Multilevel ...mcqmc2018.inria.fr/wp-content/uploads/2018/07/law_pmcmc-mcqmc18.pdf · normalizing constant [HSS13, DKST15, HTL16, BJLTZ17].](https://reader033.fdocuments.in/reader033/viewer/2022041617/5e3c7ea58de0a6468f3ca359/html5/thumbnails/8.jpg)
Introduction MLMC BIP Coupling PMCMC PMC(ML)MC Numerical simulations Summary
Outline
1 Introduction
2 Multilevel Monte Carlo sampling
3 Bayesian inference problem
4 Approximate coupling
5 Particle Markov chain Monte Carlo
6 Particle Markov chain Multilevel Monte Carlo
7 Numerical simulations
8 Summary
![Page 9: Bayesian static parameter estimation using Multilevel ...mcqmc2018.inria.fr/wp-content/uploads/2018/07/law_pmcmc-mcqmc18.pdf · normalizing constant [HSS13, DKST15, HTL16, BJLTZ17].](https://reader033.fdocuments.in/reader033/viewer/2022041617/5e3c7ea58de0a6468f3ca359/html5/thumbnails/9.jpg)
Introduction MLMC BIP Coupling PMCMC PMC(ML)MC Numerical simulations Summary
Parameter inference
Estimate the posterior expectation of a function ϕ of the jointpath X1:T and parameters θ, of an intractable SDE
dX = fθ(X )dt + σθ(X )dW , X0 ∼ µθ ,
given noisy partial observations
Yn ∼ gθ(Xn, ·) , n = 1, . . . ,T .
Aim: estimate E[ϕ(θ,X0:T )|y1:T ], where y1:T := {y1, . . . , yT}.
The hidden process {Xn} is a Markov chain.
Discretize with resolution h and denote the transition kernelFθ,h
(xp−1, dxp
)– this can be simulated from, but its density
cannot be evaluated.
![Page 10: Bayesian static parameter estimation using Multilevel ...mcqmc2018.inria.fr/wp-content/uploads/2018/07/law_pmcmc-mcqmc18.pdf · normalizing constant [HSS13, DKST15, HTL16, BJLTZ17].](https://reader033.fdocuments.in/reader033/viewer/2022041617/5e3c7ea58de0a6468f3ca359/html5/thumbnails/10.jpg)
Introduction MLMC BIP Coupling PMCMC PMC(ML)MC Numerical simulations Summary
Return to ML
The joint measure is
Πh(dθ, dx0:n) ∝ Π(dθ)µθ(dx0)n∏
p=1
gθ(xp, yp)Fθ,h(xp−1, dxp) ,
For +∞ > h0 > · · · > hL > 0, we would like to compute
EΠhL[ϕ(θ,X0:n)] =
L∑l=0
{EΠhl
[ϕ(θ,X0:n)]− EΠhl−1[ϕ(θ,X0:n)]
}where EΠh−1
[·] := 0.
![Page 11: Bayesian static parameter estimation using Multilevel ...mcqmc2018.inria.fr/wp-content/uploads/2018/07/law_pmcmc-mcqmc18.pdf · normalizing constant [HSS13, DKST15, HTL16, BJLTZ17].](https://reader033.fdocuments.in/reader033/viewer/2022041617/5e3c7ea58de0a6468f3ca359/html5/thumbnails/11.jpg)
Introduction MLMC BIP Coupling PMCMC PMC(ML)MC Numerical simulations Summary
Outline
1 Introduction
2 Multilevel Monte Carlo sampling
3 Bayesian inference problem
4 Approximate coupling
5 Particle Markov chain Monte Carlo
6 Particle Markov chain Multilevel Monte Carlo
7 Numerical simulations
8 Summary
![Page 12: Bayesian static parameter estimation using Multilevel ...mcqmc2018.inria.fr/wp-content/uploads/2018/07/law_pmcmc-mcqmc18.pdf · normalizing constant [HSS13, DKST15, HTL16, BJLTZ17].](https://reader033.fdocuments.in/reader033/viewer/2022041617/5e3c7ea58de0a6468f3ca359/html5/thumbnails/12.jpg)
Introduction MLMC BIP Coupling PMCMC PMC(ML)MC Numerical simulations Summary
Approximate coupling
Consider a single pair EΠh [ϕ(θ,X0:n)]− EΠh′ [ϕ(θ,X0:n)], h < h′.
Let z = (x , x ′) and let Qθ,h,h′(z, dz) be a coupling of(Fθ,h(x , dx),Fθ,h′(x ′, dx ′)).
Let Gp,θ(z) = max{gθ(x , yp),gθ(x ′, yp)}.
We will sample from the joint coarse/fine filter
Πh,h′(dθ, dz0:n) ∝ Π(dθ)νθ(dz0)n∏
p=1
Gp,θ(zp)Qθ,h,h′(zp−1, dzp),
where νθ is the initial coupling
νθ(d(x , x ′)) = µθ(dx)δx (dx ′).
![Page 13: Bayesian static parameter estimation using Multilevel ...mcqmc2018.inria.fr/wp-content/uploads/2018/07/law_pmcmc-mcqmc18.pdf · normalizing constant [HSS13, DKST15, HTL16, BJLTZ17].](https://reader033.fdocuments.in/reader033/viewer/2022041617/5e3c7ea58de0a6468f3ca359/html5/thumbnails/13.jpg)
Introduction MLMC BIP Coupling PMCMC PMC(ML)MC Numerical simulations Summary
Change of measure
We haveEΠh [ϕ(θ,X0:n)]− EΠh′ [ϕ(θ,X ′0:n)] =
EΠh,h′ [ϕ(θ,X0:n)H1,θ(θ,Z0:n)]
EΠh,h′ [H1,θ(θ,Z0:n)]−
EΠh,h′ [ϕ(θ,X ′0:n)H2,θ(θ,Z0:n)]
EΠh,h′ [H2,θ(θ,Z0:n)]
where
H1,θ(θ, z0:n) =n∏
p=1
gθ(xp, yp)
Gp,θ(zp)
H2,θ(θ, z0:n) =n∏
p=1
gθ(x ′p, yp)
Gp,θ(zp).
![Page 14: Bayesian static parameter estimation using Multilevel ...mcqmc2018.inria.fr/wp-content/uploads/2018/07/law_pmcmc-mcqmc18.pdf · normalizing constant [HSS13, DKST15, HTL16, BJLTZ17].](https://reader033.fdocuments.in/reader033/viewer/2022041617/5e3c7ea58de0a6468f3ca359/html5/thumbnails/14.jpg)
Introduction MLMC BIP Coupling PMCMC PMC(ML)MC Numerical simulations Summary
Outline
1 Introduction
2 Multilevel Monte Carlo sampling
3 Bayesian inference problem
4 Approximate coupling
5 Particle Markov chain Monte Carlo
6 Particle Markov chain Multilevel Monte Carlo
7 Numerical simulations
8 Summary
![Page 15: Bayesian static parameter estimation using Multilevel ...mcqmc2018.inria.fr/wp-content/uploads/2018/07/law_pmcmc-mcqmc18.pdf · normalizing constant [HSS13, DKST15, HTL16, BJLTZ17].](https://reader033.fdocuments.in/reader033/viewer/2022041617/5e3c7ea58de0a6468f3ca359/html5/thumbnails/15.jpg)
Introduction MLMC BIP Coupling PMCMC PMC(ML)MC Numerical simulations Summary
Particle filter, for fixed θ
Let M ≥ 1 and θ be fixed, and introduce a0:n−1 ∈ {1, . . . ,M}n.The bootstrap particle filter [Del04] approximates
Πh,h′(dz0:n|θ) ∝ νθ(dz0)n∏
p=1
Gp,θ(zp)Qθ,h,h′(zp−1, dzp)
by sampling from
P(a1:M0:n−1, dz1:M
0:n |θ) =( M∏
i=1
νθ(dz i0)) n∏
p=1
M∏i=1
( Gp−1,θ(zai
p−1p−1 )∑M
j=1 Gp−1,θ(z jp−1)
Qθ,h,h′(zai
p−1p−1 , dz i
p)),
where G0,θ := 1,
i.e. zai
p−1p−1 is resampled with the probability
Gp−1,θ(zaip−1
p−1 )∑Mj=1 Gp−1,θ(z j
p−1).
![Page 16: Bayesian static parameter estimation using Multilevel ...mcqmc2018.inria.fr/wp-content/uploads/2018/07/law_pmcmc-mcqmc18.pdf · normalizing constant [HSS13, DKST15, HTL16, BJLTZ17].](https://reader033.fdocuments.in/reader033/viewer/2022041617/5e3c7ea58de0a6468f3ca359/html5/thumbnails/16.jpg)
Introduction MLMC BIP Coupling PMCMC PMC(ML)MC Numerical simulations Summary
Unbiased estimator, for fixed θ
Draw J with probability proportional to Gn,θ(z in) for i = 1, . . . ,M.
Let zn = z jn, and trace its ancestral lineage
zn−1 = zaj
n−1n−1 , zn−2 = z
aajn−1
n−2n−2 , and so on .
Define pMh,h′(y0:n|θ) =
∏np=1
(1M∑M
j=1 Gp,θ(z jp))
, let E denote
expectation w.r.t. (A1:M0:n−1,Z
1:M0:n , J), and note that [Del04]
E[ϕ(z0:n)pM
h,h′(y0:n|θ)]
=
∫Zn+1
ϕ(z0:n)νθ(dz0)n∏
p=1
Gp,θ(zp)Qθ,h,h′(zp−1, dzp) .
Therefore, one can run a MH chain {θi , z0:n(θi)} targeting∝ pM
h,h′(y0:n|θ)Π(dθ), and it is consistent with respect toΠh,h′(dz0:n, dθ) [B03, AR08, ADH10].
![Page 17: Bayesian static parameter estimation using Multilevel ...mcqmc2018.inria.fr/wp-content/uploads/2018/07/law_pmcmc-mcqmc18.pdf · normalizing constant [HSS13, DKST15, HTL16, BJLTZ17].](https://reader033.fdocuments.in/reader033/viewer/2022041617/5e3c7ea58de0a6468f3ca359/html5/thumbnails/17.jpg)
Introduction MLMC BIP Coupling PMCMC PMC(ML)MC Numerical simulations Summary
Particle marginal MH (PMMH) [ADH10]
1 Sample θ0 ∼ π(dθ) and (a1:M0:n−1, z
1:M0:n ) from particle filter
P(da1:M0:n−1, dz1:M
0:n |θ0), and store pMh,h′(y0:n|θ0).
2 Select a path z00:n: draw z j
n with probability proportional to Gn,θ0 (z jn), let
z0n = z j
n, and trace back its ancestral lineage
z0n−1 = z
ajn−1
n−1 , z0n−2 = z
aajn−1
n−2n−2 , and so on ; Set i = 1 .
3 Sample θ∗|θi−1 according to R(dθ∗|θi−1) = r(θ∗|θi−1)dθ∗, then samplefrom particle filter P(da1:M
0:n−1, dz1:M0:n |θ∗). Select one path z∗0:n as above.
4 Set θi = θ∗, z i0:n = z∗0:n with probability:
min
{1,
pMh,h′(y0:n|θ∗)
pMh,h′(y0:n|θi−1)
π(θ∗)r(θi−1|θ∗)π(θi−1)r(θ∗|θi−1)
}
otherwise θi = θi−1, z i0:n = z i−1
0:n .5 Set i = i + 1 and return to the start of 3.
![Page 18: Bayesian static parameter estimation using Multilevel ...mcqmc2018.inria.fr/wp-content/uploads/2018/07/law_pmcmc-mcqmc18.pdf · normalizing constant [HSS13, DKST15, HTL16, BJLTZ17].](https://reader033.fdocuments.in/reader033/viewer/2022041617/5e3c7ea58de0a6468f3ca359/html5/thumbnails/18.jpg)
Introduction MLMC BIP Coupling PMCMC PMC(ML)MC Numerical simulations Summary
PMMH increment estimator
1N∑N
i=1 ϕ(θi , x i0:n)H1,θ(θi , z i
0:n)1N∑N
i=1 H1,θ(θi , z i0:n)
−1N∑N
i=1 ϕ(θi , x ′i0:n)H2,θ(θi , z i0:n)
1N∑N
i=1 H2,θ(θi , z i0:n)
.
−→N→∞
EΠh,h′ [ϕ(θ,X0:n)H1,θ(θ,Z0:n)]
EΠh,h′ [H1,θ(θ,Z0:n)]−
EΠh,h′ [ϕ(θ,X ′0:n)H2,θ(θ,Z0:n)]
EΠh,h′ [H2,θ(θ,Z0:n)]
=
EΠh [ϕ(θ,X0:n)]− EΠh′ [ϕ(θ,X ′0:n)]
![Page 19: Bayesian static parameter estimation using Multilevel ...mcqmc2018.inria.fr/wp-content/uploads/2018/07/law_pmcmc-mcqmc18.pdf · normalizing constant [HSS13, DKST15, HTL16, BJLTZ17].](https://reader033.fdocuments.in/reader033/viewer/2022041617/5e3c7ea58de0a6468f3ca359/html5/thumbnails/19.jpg)
Introduction MLMC BIP Coupling PMCMC PMC(ML)MC Numerical simulations Summary
Outline
1 Introduction
2 Multilevel Monte Carlo sampling
3 Bayesian inference problem
4 Approximate coupling
5 Particle Markov chain Monte Carlo
6 Particle Markov chain Multilevel Monte Carlo
7 Numerical simulations
8 Summary
![Page 20: Bayesian static parameter estimation using Multilevel ...mcqmc2018.inria.fr/wp-content/uploads/2018/07/law_pmcmc-mcqmc18.pdf · normalizing constant [HSS13, DKST15, HTL16, BJLTZ17].](https://reader033.fdocuments.in/reader033/viewer/2022041617/5e3c7ea58de0a6468f3ca359/html5/thumbnails/20.jpg)
Introduction MLMC BIP Coupling PMCMC PMC(ML)MC Numerical simulations Summary
Multilevel estimatorConsider
L∑l=0
ENll (ϕ) , ENl
l (ϕ) = ENll (ϕ)− El(ϕ) , (1)
where
ENll (ϕ) =
1Nl
∑Nli=1 ϕ(θi , , x i
0:n)H1,θ(θi , z i0:n)
1Nl
∑Nli=1 H1,θ(θi , z i
0:n)−
1Nl
∑Nli=1 ϕ(θi , x ′i0:n)H2,θ(θi , z i
0:n)
1Nl
∑Nli=1 H2,θ(θi , z i
0:n)
is a consistent estimator of
El(ϕ) := EΠhl[ϕ(θ,X0:n)]− EΠhl−1
[ϕ(θ,X0:n)] .
⇒ (1) is a consistent estimator of EΠhL[ϕ(θ,X0:n)].
One must bound
E[(L∑
l=0
ENll (ϕ))2] =
L∑l=0
E[ENll (ϕ)2] .
![Page 21: Bayesian static parameter estimation using Multilevel ...mcqmc2018.inria.fr/wp-content/uploads/2018/07/law_pmcmc-mcqmc18.pdf · normalizing constant [HSS13, DKST15, HTL16, BJLTZ17].](https://reader033.fdocuments.in/reader033/viewer/2022041617/5e3c7ea58de0a6468f3ca359/html5/thumbnails/21.jpg)
Introduction MLMC BIP Coupling PMCMC PMC(ML)MC Numerical simulations Summary
Assumptions
(A1) ∀ y ∈ T, ∃ C > 0 such that ∀ x ∈ S, θ ∈ Θ,
C ≤ gθ(x , y) ≤ C−1.
And ∀ y ∈ T, gθ(x , y) is globally Lipschitz on S×Θ.
(A2) ∀ 0 ≤ k ≤ n, ∃ β > 0 such that ∀ϕ ∈ Bb(Θ× Sk+1) ∩ Lip(Θ× Sk+1) ∃ C > 0∫
Θ×S2k+2|ϕ(θ, x0:k )−ϕ(θ, x ′0:k )|2Π(dθ)νθ(dz0)
k∏p=1
Qθ,h,h′(zp−1, dzp) ≤ C(h′)β .
(A3) Suppose that ∀ n > 0, ∃ ξ ∈ (0,1) and ν ∈ P(W) such thatfor each w ∈W, ϕ ∈ Bb(W) ∩ Lip(W), h,h′:∫
Wϕ(w ′)K (w ,dw ′) ≥ ξ
∫Wϕ(v)ν(dv).
K is η-reversible, where η is the joint on the extended space.
![Page 22: Bayesian static parameter estimation using Multilevel ...mcqmc2018.inria.fr/wp-content/uploads/2018/07/law_pmcmc-mcqmc18.pdf · normalizing constant [HSS13, DKST15, HTL16, BJLTZ17].](https://reader033.fdocuments.in/reader033/viewer/2022041617/5e3c7ea58de0a6468f3ca359/html5/thumbnails/22.jpg)
Introduction MLMC BIP Coupling PMCMC PMC(ML)MC Numerical simulations Summary
Main result
Theorem (JKLZ18)Assume (A1-3). Then ∀ n > 0, ∃ β > 0 such that ∀ϕ ∈ Bb(Θ× Sn+1) ∩ Lip(Θ× Sn+1) ∃ C > 0 such that
E[ENll (ϕ)2] ≤
ChβlNl
,
and β is from (A2).
![Page 23: Bayesian static parameter estimation using Multilevel ...mcqmc2018.inria.fr/wp-content/uploads/2018/07/law_pmcmc-mcqmc18.pdf · normalizing constant [HSS13, DKST15, HTL16, BJLTZ17].](https://reader033.fdocuments.in/reader033/viewer/2022041617/5e3c7ea58de0a6468f3ca359/html5/thumbnails/23.jpg)
Introduction MLMC BIP Coupling PMCMC PMC(ML)MC Numerical simulations Summary
(A4) ∃ γ, α,C > 0 such that the cost to simulate ENll is controlled by
C(ENll ) ≤ CNlh
−γl , and the bias is controlled by
|EΠhL(ϕ(θ,X0:n))− EΠ0 (ϕ(θ,X0:n))| ≤ ChαL .
Corollary
Assume (A1-4). ∀ n > 0 and ϕ ∈ Bb(Θ× Sn+1) ∩ Lip(Θ× Sn+1) ∃C > 0 such that ∀ ε > 0 one can choose (L, {Nl}L
l=0) so
E
[|
L∑l=0
ENll (ϕ)− EΠ0 (ϕ(θ,X0:n))|2
]≤ Cε2 ,
with a total cost (per time step)
COST ≤ C
ε−2, if β > γ,
ε−2| log(ε)|2, if β = γ,
ε−(2+ γ−βα ), if β < γ.
![Page 24: Bayesian static parameter estimation using Multilevel ...mcqmc2018.inria.fr/wp-content/uploads/2018/07/law_pmcmc-mcqmc18.pdf · normalizing constant [HSS13, DKST15, HTL16, BJLTZ17].](https://reader033.fdocuments.in/reader033/viewer/2022041617/5e3c7ea58de0a6468f3ca359/html5/thumbnails/24.jpg)
Introduction MLMC BIP Coupling PMCMC PMC(ML)MC Numerical simulations Summary
Outline
1 Introduction
2 Multilevel Monte Carlo sampling
3 Bayesian inference problem
4 Approximate coupling
5 Particle Markov chain Monte Carlo
6 Particle Markov chain Multilevel Monte Carlo
7 Numerical simulations
8 Summary
![Page 25: Bayesian static parameter estimation using Multilevel ...mcqmc2018.inria.fr/wp-content/uploads/2018/07/law_pmcmc-mcqmc18.pdf · normalizing constant [HSS13, DKST15, HTL16, BJLTZ17].](https://reader033.fdocuments.in/reader033/viewer/2022041617/5e3c7ea58de0a6468f3ca359/html5/thumbnails/25.jpg)
Introduction MLMC BIP Coupling PMCMC PMC(ML)MC Numerical simulations Summary
Ornstein-Uhlenbeck process
dXt = θ(µ− Xt )dt + σdWt , X0 = x0 ,
Yk |Xδk ∼ N (Xδk , τ2) ,
θ ∼ G(1,1) ,
σ ∼ G(1,0.5) .
N (m, τ2) denotes the Normal with mean m and varianceτ2.G(a,b) denotes the Gamma with shape a and scale b.x0 = 0, µ = 0, δ = 0.5, and τ2 = 0.2.100 observations simulated with θ = 1 and σ = 0.5.
![Page 26: Bayesian static parameter estimation using Multilevel ...mcqmc2018.inria.fr/wp-content/uploads/2018/07/law_pmcmc-mcqmc18.pdf · normalizing constant [HSS13, DKST15, HTL16, BJLTZ17].](https://reader033.fdocuments.in/reader033/viewer/2022041617/5e3c7ea58de0a6468f3ca359/html5/thumbnails/26.jpg)
Introduction MLMC BIP Coupling PMCMC PMC(ML)MC Numerical simulations Summary
Langevin SDE
dXt =12∇ log π(Xt )dt + σdWt , X0 = x0
Yk |Xk ∼ N (0, τ2 exp Xk ) ,
θ ∼ G(1,1) ,
σ ∼ G(1,0.5) .
π(x) denotes the probability density function of a Student’st-distribution with θ degrees of freedom.x0 = 0.1,000 observations simulated with θ = 10, σ = 1, andτ2 = 1.
![Page 27: Bayesian static parameter estimation using Multilevel ...mcqmc2018.inria.fr/wp-content/uploads/2018/07/law_pmcmc-mcqmc18.pdf · normalizing constant [HSS13, DKST15, HTL16, BJLTZ17].](https://reader033.fdocuments.in/reader033/viewer/2022041617/5e3c7ea58de0a6468f3ca359/html5/thumbnails/27.jpg)
Introduction MLMC BIP Coupling PMCMC PMC(ML)MC Numerical simulations Summary
Langevin SDE Ornstein-Uhlenbech process
0 10 20 30 0 10 20 30
0.00
0.25
0.50
0.75
1.00
Lag
ACF
Parameter 𝜎 𝜃
Figure: Autocorrelation of a typical PMCMC chain.
![Page 28: Bayesian static parameter estimation using Multilevel ...mcqmc2018.inria.fr/wp-content/uploads/2018/07/law_pmcmc-mcqmc18.pdf · normalizing constant [HSS13, DKST15, HTL16, BJLTZ17].](https://reader033.fdocuments.in/reader033/viewer/2022041617/5e3c7ea58de0a6468f3ca359/html5/thumbnails/28.jpg)
Introduction MLMC BIP Coupling PMCMC PMC(ML)MC Numerical simulations Summary
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
CostAlgorithm ML-PMCMC PMCMC
Figure: Cost vs. MSE for the 2 parameters for each of the 2 SDEs.
![Page 29: Bayesian static parameter estimation using Multilevel ...mcqmc2018.inria.fr/wp-content/uploads/2018/07/law_pmcmc-mcqmc18.pdf · normalizing constant [HSS13, DKST15, HTL16, BJLTZ17].](https://reader033.fdocuments.in/reader033/viewer/2022041617/5e3c7ea58de0a6468f3ca359/html5/thumbnails/29.jpg)
Introduction MLMC BIP Coupling PMCMC PMC(ML)MC Numerical simulations Summary
Model Parameter ML-PMCMC PMCMCOU θ −1.022 −1.463
σ −1.065 −1.522Langevin θ −1.060 −1.508
σ −1.023 −1.481
Table: Estimated rates of convergence of MSE with respect to cost forvarious parameters, fitted to the curves.
![Page 30: Bayesian static parameter estimation using Multilevel ...mcqmc2018.inria.fr/wp-content/uploads/2018/07/law_pmcmc-mcqmc18.pdf · normalizing constant [HSS13, DKST15, HTL16, BJLTZ17].](https://reader033.fdocuments.in/reader033/viewer/2022041617/5e3c7ea58de0a6468f3ca359/html5/thumbnails/30.jpg)
Introduction MLMC BIP Coupling PMCMC PMC(ML)MC Numerical simulations Summary
Outline
1 Introduction
2 Multilevel Monte Carlo sampling
3 Bayesian inference problem
4 Approximate coupling
5 Particle Markov chain Monte Carlo
6 Particle Markov chain Multilevel Monte Carlo
7 Numerical simulations
8 Summary
![Page 31: Bayesian static parameter estimation using Multilevel ...mcqmc2018.inria.fr/wp-content/uploads/2018/07/law_pmcmc-mcqmc18.pdf · normalizing constant [HSS13, DKST15, HTL16, BJLTZ17].](https://reader033.fdocuments.in/reader033/viewer/2022041617/5e3c7ea58de0a6468f3ca359/html5/thumbnails/31.jpg)
Introduction MLMC BIP Coupling PMCMC PMC(ML)MC Numerical simulations Summary
Summary
New approximate coupling strategy can be used to applyMLMC to PMCMC for static parameter estimation[JKLZ18.i].Same strategy can be employed for multi-index MCMC[JKLZ18.ii].In progress: MISMC2 [JLX18.s]PhD and postdocs wanted – please inquire !
![Page 32: Bayesian static parameter estimation using Multilevel ...mcqmc2018.inria.fr/wp-content/uploads/2018/07/law_pmcmc-mcqmc18.pdf · normalizing constant [HSS13, DKST15, HTL16, BJLTZ17].](https://reader033.fdocuments.in/reader033/viewer/2022041617/5e3c7ea58de0a6468f3ca359/html5/thumbnails/32.jpg)
Introduction MLMC BIP Coupling PMCMC PMC(ML)MC Numerical simulations Summary
References[G08]: Giles. "Multilevel Monte Carlo path simulation." Op.Res., 56, 607-617 (2008).[H00] Heinrich. "Multilevel Monte Carlo methods." LSSCproceedings (2001).[CGST11] Cliffe, Giles, Scheichl, & Teckentrup. "MLMCand applications to elliptic PDEs." Computing andVisualization in Science, 14(1), 3 (2011).[D04]: Del Moral. "Feynman-Kac Formulae." Springer:New York (2004).[B02] Beaumont. "Estimation of population growth..."Genetics 164(3) 1139-1160 (2003).[AR08] Andrieu & Roberts. "The pseudo-marginalapproach." Annals of Stat. 37(2) 697-725 (2009).[ADH10] Andrieu, Doucet, and Holenstein. "ParticleMCMC methods." JRSSB 72(3) 269-342 (2010).
![Page 33: Bayesian static parameter estimation using Multilevel ...mcqmc2018.inria.fr/wp-content/uploads/2018/07/law_pmcmc-mcqmc18.pdf · normalizing constant [HSS13, DKST15, HTL16, BJLTZ17].](https://reader033.fdocuments.in/reader033/viewer/2022041617/5e3c7ea58de0a6468f3ca359/html5/thumbnails/33.jpg)
Introduction MLMC BIP Coupling PMCMC PMC(ML)MC Numerical simulations Summary
References[JKLZ18.i]: Jasra, Kamatani, Law, Zhou. "MLMC for staticBayesian parameter estimation." SISC 40, A887-A902(2018).[JKLZ18.ii]: Jasra, Kamatani, Law, Zhou. "A Multi-IndexMarkov Chain Monte Carlo Method." IJUQ 8(1), 61-73(2018).[JLX18.s]: Jasra, Law, Xu. "Multi-index SMC2." Submitted.[BJLTZ15]: Beskos, Jasra, Law, Tempone, Zhou."Multilevel Sequential Monte Carlo samplers." SPA 127:5,1417–1440 (2017).[HSS13]: Hoang, Schwab, Stuart. Inverse Prob., 29,085010 (2013).[DKST13]: Dodwell, Ketelsen, Scheichl, Teckentrup. "Ahierarchical MLMCMC algorithm." SIAM/ASA JUQ 3(1)1075-1108 (2015).
![Page 34: Bayesian static parameter estimation using Multilevel ...mcqmc2018.inria.fr/wp-content/uploads/2018/07/law_pmcmc-mcqmc18.pdf · normalizing constant [HSS13, DKST15, HTL16, BJLTZ17].](https://reader033.fdocuments.in/reader033/viewer/2022041617/5e3c7ea58de0a6468f3ca359/html5/thumbnails/34.jpg)
Introduction MLMC BIP Coupling PMCMC PMC(ML)MC Numerical simulations Summary
References[CHLNT17] Chernov, Hoel, Law, Nobile, Tempone."Multilevel ensemble Kalman filtering for spatio-temporalprocesses." arXiv:1608.08558 (2017).[JLS17] Jasra, Law, Suciu. "Advanced Multilevel MonteCarlo Methods." arXiv:1704.07272 (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 Inverse UQMLMC Algorithms for an Elliptic Nonlocal Equation." IJUQ6(6), 501–514 (2016).[HLT15]: Hoel, Law, Tempone. "Multilevel ensembleKalman filter." SINUM 54(3), 1813–1839 (2016).
![Page 35: Bayesian static parameter estimation using Multilevel ...mcqmc2018.inria.fr/wp-content/uploads/2018/07/law_pmcmc-mcqmc18.pdf · normalizing constant [HSS13, DKST15, HTL16, BJLTZ17].](https://reader033.fdocuments.in/reader033/viewer/2022041617/5e3c7ea58de0a6468f3ca359/html5/thumbnails/35.jpg)
Introduction MLMC BIP Coupling PMCMC PMC(ML)MC Numerical simulations Summary
Thank you