QMC Program: Trends and Advances in Monte Carlo Sampling Algorithms Workshop, About Infinite-Dimensional Geometric MCMC - Shiwei Lan, Dec 14, 2017
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This document discusses advanced techniques in geometric Markov Chain Monte Carlo (MCMC) methods for infinite-dimensional inverse problems, covering algorithms such as Hamiltonian Monte Carlo and Langevin dynamics. It presents theoretical foundations, including the Metropolis-Hastings algorithm and various proposals for dimension reduction, along with detailed mathematical formulations. The work culminates in a discussion on practical applications and the connections among different infinite-dimensional MCMC strategies.
![9
∞-dimensional MALA
(Beskos et al., 2008)
Consider the Langevin SDE
du
dt
= −
1
2
K(C−1
u + DΦ(u)) +
√
K
dW
dt
(3)
Semi-implicit scheme to discretize the above SDE
u − u
h
= −
1
2
KC−1
[(1 − θ)u + θu ] −
α
2
KDΦ(u)) +
K
h
ξ, ξ ∼ N(0, I)
which may be simplified to give
u = Aθu + Bθv, v =
√
Cξ −
α
2
√
hKCDΦ(u)
Aθ = (I +
θ
2
hKC−1
)−1
(I −
1 − θ
2
hKC−1
), Bθ = (I +
θ
2
hKC−1
)−1
√
hKC−1
(4)
where α = 1. α = 0 =⇒ pCN.
K = I (IA) or K = C (PIA).
S.Lan | ∞-Dimensional Geometric MCMC](https://image.slidesharecdn.com/2geom-infmcmc20171214-171215212152/85/QMC-Program-Trends-and-Advances-in-Monte-Carlo-Sampling-Algorithms-Workshop-About-Infinite-Dimensional-Geometric-MCMC-Shiwei-Lan-Dec-14-2017-10-320.jpg)
![14
∞-dimensional manifold MALA
(Beskos, 2014)
Theorem (Beskos (2014))
Denote ν(du, du ) = µ(du)Q(u, du ) with Q(u, du ) being the transition kernel of (9). Under Assumptions
1 and 2, ν νT
, and the acceptance probability has the following form
a(u, u ) = 1 ∧
dνT
dν
(u, u ),
dνT
dν
(u, u ) =
exp{−Φ(u )}λ(ρ−1
(u; u ); u )
exp{−Φ(u)}λ(ρ−1(u ; u); u)
(10)
where ρ−1
(u ; u) = [(1 + h/4)u − (1 − h/4)u]/
√
h and λ(w; u) is calculated as follows:
λ(w; u) =
dN (
√
h/2g(u), G(u)−1
)
dN (0, C)
=
dN (
√
h/2g(u), G(u)−1
)
dN (0, G(u)−1)
dN (0, G(u)−1
)
dN (0, C)
= exp
√
h
2
G
1
2 (u)g(u), G
1
2 (u)w −
h
8
|G
1
2 (u)g(u)|
2
·
exp −
1
2
|G
1
2 (u)w|
2
+
1
2
|C
− 1
2 w|
2
|C
1
2 G
1
2 (u)|
S.Lan | ∞-Dimensional Geometric MCMC](https://image.slidesharecdn.com/2geom-infmcmc20171214-171215212152/85/QMC-Program-Trends-and-Advances-in-Monte-Carlo-Sampling-Algorithms-Workshop-About-Infinite-Dimensional-Geometric-MCMC-Shiwei-Lan-Dec-14-2017-15-320.jpg)
![20
Dimension Reduction
intrinsic low-dimensional subspace
Particular choice of K(u) = G(u)−1
could be
K(u) = (C−1
+ H(u))−1
, H(u) = EY|u DΦ(u), DΦ(u) (14)
Computationally infeasible to update Fisher metric H(u) in each u
Given eigenbasis {φi(x)} we define projection operator Pr as follows:
Pr : X → Xr, u → ur
:=
r
i=1
φi φi, u (15)
Truncate H(u) on the r-dimensional subspace Xr ⊂ X (X = Xr + X⊥)
Hr(u)(v, w) = Prv, EY|u[DrΦ(u)DrΦ(u)
T
]Prw , ∀v, w ∈ X (16)
K(u)−1
can be approximated
K(u)−1
≈ Hr(u) + C−1
(17)
S.Lan | ∞-Dimensional Geometric MCMC](https://image.slidesharecdn.com/2geom-infmcmc20171214-171215212152/85/QMC-Program-Trends-and-Advances-in-Monte-Carlo-Sampling-Algorithms-Workshop-About-Infinite-Dimensional-Geometric-MCMC-Shiwei-Lan-Dec-14-2017-21-320.jpg)
![23
Laminar Jet
formulation of the inverse problem
We consider the following 2d Navier-Stokes equation:
Momentum equation:
− div ν( u + uT
) + u · u + p = 0, (23)
where u = (u, v) is the velocity field and p the pressure;
Continuity equation:
div u = 0. (24)
Denote σn = −pn + ν( u + uT
) · n as the boundary traction.
u · n = −θ(y), σn × n = 0 on I := x = 0, y ∈ (−
1
2
Ly,
1
2
Ly)
σn + β(u · n)− u = 0 on O := x = Lx, y ∈ (−
1
2
Ly,
1
2
Ly)
u · n = 0, σn × n = 0 on B := x ∈ (0, Lx), y = ±
1
2
Ly
where (u · n)− = u·n−|u·n|
2 , and β ∈ (0, 1] is the backflow stabilization
parameter
S.Lan | ∞-Dimensional Geometric MCMC](https://image.slidesharecdn.com/2geom-infmcmc20171214-171215212152/85/QMC-Program-Trends-and-Advances-in-Monte-Carlo-Sampling-Algorithms-Workshop-About-Infinite-Dimensional-Geometric-MCMC-Shiwei-Lan-Dec-14-2017-24-320.jpg)
![29
Connection to DILI
(Cui et al., 2016)
By considering the approximation H(v) ≈ VrΛrV∗
r , we have the
posterior covariance projected to r-dimensional subspace:
Kr = Covµ[V∗
r v] = V∗
r (I + H(v))−1
Vr ≈ Dr := (Ir + Λr)−1
(29)
While DILI computes the posterior covariance in the subspace
Kr := Covµ[V∗
r v] empirically.
They both have the following approximate posterior covariance
Kv = Covµ[v] ≈ VrKrV∗
r + I − VrV∗
r (30)
Since we directly work with (29), empirical calculation of Kr is avoided;
The approximation (29) is already in diagonal form thus the rotation
Ψr = VrWr in DILI is not needed.
They capture the similar geometric feature of subspace.
S.Lan | ∞-Dimensional Geometric MCMC](https://image.slidesharecdn.com/2geom-infmcmc20171214-171215212152/85/QMC-Program-Trends-and-Advances-in-Monte-Carlo-Sampling-Algorithms-Workshop-About-Infinite-Dimensional-Geometric-MCMC-Shiwei-Lan-Dec-14-2017-30-320.jpg)
![30
Elliptic Inverse Problem
formulation of the inverse problem
Consider the elliptic inverse problem as in (DILI Cui et al., 2016)
defined on the unit square domain Ω = [0, 1]2
:
− · (k(s) p(s)) = f (s), s ∈ Ω
k(s) p(s), n(s) = 0, s ∈ ∂Ω
∂Ω
p(s)dl(s) = 0
(31)
where k(s) is the transmissivity field, p(s) is the potential function,
f (s) is the forcing term, and n(s) is the outward normal to the
boundary.
The inverse problem is to infer u = log k from observations {yn}.
25 observations arise from the solutions (solved on a 80 × 80 mesh)
contaminated by additive Gaussian error:
yn = p(xn) + εn, εn ∼ N(0, σ2
y), SNR := max
s
{u(s)}/σy = 100
S.Lan | ∞-Dimensional Geometric MCMC](https://image.slidesharecdn.com/2geom-infmcmc20171214-171215212152/85/QMC-Program-Trends-and-Advances-in-Monte-Carlo-Sampling-Algorithms-Workshop-About-Infinite-Dimensional-Geometric-MCMC-Shiwei-Lan-Dec-14-2017-31-320.jpg)
QMC Program: Trends and Advances in Monte Carlo Sampling Algorithms Workshop, About Infinite-Dimensional Geometric MCMC - Shiwei Lan, Dec 14, 2017
- 1. Department of Computing + Mathematical Sciences California Institute of Technology About ∞-Dimensional Geometric MCMC a b Shiwei Lan DEC 14, 2017 SAMSI, DUKE aBeskos, Alexandros, Mark Girolami, Shiwei Lan, Patrick E. Farrell, and Andrew M. Stuart (2017), Geometric MCMC for Infinite-Dimensional Inverse Problems. Journal of Computational Physics, 335:327–351. bHolbrook, Andrew, Shiwei Lan, Jeffrey Streets, and Babak Shahbaba. The non-parametric Fisher information geometry and the chi-square process density prior. arXiv:1707.03117.
- 2. 1 Table of contents 1. Introduction 2. Geometric Monte Carlo on Infinite Dimensions 3. Dimension Reduction 4. ∞-dimensional Spherical Hamiltonian Monte Carlo 5. Conclusion S.Lan | ∞-Dimensional Geometric MCMC
- 3. 2 Geometric Monte Carlo powerful RWM θ 1 -2 -1 0 1 2 θ 2 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 HMC θ 1 -2 -1 0 1 2 θ 2 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 RHMC θ 1 -2 -1 0 1 2 θ 2 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 LMC θ 1 -2 -1 0 1 2 θ 2 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 S.Lan | ∞-Dimensional Geometric MCMC
- 4. 3 Dimension-Independent MCMC robust 20 40 60 80 100 120 140 160 180 200 dimension 0 0.2 0.4 0.6 0.8 1 acceptancerate HMC LMC ∞-HMC ∞-mHMC S.Lan | ∞-Dimensional Geometric MCMC
- 5. 4 Bayesian Inverse Problems Let X, Y be separable Banach spaces, equipped with Borel σ-algebra, G : X → Y measurable. Want to find u from y y = G(u) + η Prior: u ∼ µ0 on X. Noise: η ∼ Q0 on Y and η ⊥ u. Assume y|u ∼ Qu Q0 for u µ0-a.s. Define likelihood dQu dQ0 (y) = exp(−Φ(u; y)) Theorem (Bayes’ Theorem) Assume for y Q0-a.s.: Z := X exp(−Φ(u; y))µ0(du) > 0. Then u|y exists under ν, denoted by µy . Furthermore, µy µ0 and for y ν-a.s. dµy dµ0 (u) = 1 Z exp(−Φ(u; y)). S.Lan | ∞-Dimensional Geometric MCMC
- 6. 5 Metropolis-Hastings in general state space (Tierney, 1998) target µ(du), e.g. µy (du) ∝ exp(−Φ(u; y))µ0(du), µ0 = N(0, C). Denote Q(u, du ) as the proposal probability kernel Denote the transition measure and its transpose as follows ν(du, du ) = µ(du)Q(u, du ), νT (du, du ) = ν(du , du) When ν νT (i.e. ν νT and νT ν) with density r(u, u ) = dν dνT (u, u ) The acceptance probability is a(u, u ) = 1 ∧ r(u, u ) S.Lan | ∞-Dimensional Geometric MCMC
- 7. 6 Random Walk S.Lan | ∞-Dimensional Geometric MCMC
- 8. 7 preconditioned Crank-Nicolson (Cotter et al., 2013) A finite difference method used for numerically solving PDE (Richtmyer and Morton, 1994) Modified Random Walk Metropolis (RWM) Given u, sample ξ ∼ N(0, C) Make proposal u = 1 − β2u + βξ (1) Accept u with probability a(u, u ) = 1 ∧ r(u, u ), r(u, u ) = exp(−Φ(u ) + Φ(u)) (2) Independent of dimension, mixing faster than RWM S.Lan | ∞-Dimensional Geometric MCMC
- 9. 8 + Gradient S.Lan | ∞-Dimensional Geometric MCMC
- 10. 9 ∞-dimensional MALA (Beskos et al., 2008) Consider the Langevin SDE du dt = − 1 2 K(C−1 u + DΦ(u)) + √ K dW dt (3) Semi-implicit scheme to discretize the above SDE u − u h = − 1 2 KC−1 [(1 − θ)u + θu ] − α 2 KDΦ(u)) + K h ξ, ξ ∼ N(0, I) which may be simplified to give u = Aθu + Bθv, v = √ Cξ − α 2 √ hKCDΦ(u) Aθ = (I + θ 2 hKC−1 )−1 (I − 1 − θ 2 hKC−1 ), Bθ = (I + θ 2 hKC−1 )−1 √ hKC−1 (4) where α = 1. α = 0 =⇒ pCN. K = I (IA) or K = C (PIA). S.Lan | ∞-Dimensional Geometric MCMC
- 11. 10 ∞-dimensional HMC (Beskos et al., 2011) Consider the Hamiltonian differential equation d2 u dt2 + K(C−1 u + DΦ(u)) = 0, (v := du dt ) t=0 ∼ N(0, K) (5) Let K = C, f (u) := −DΦ(u). Störmer-Verlet/splitting scheme (Verlet, 1967; Neal, 2010) is used to discretize (5) v− = v + t 2 Cf (u) u v+ = cos t sin t − sin t cos t u v− v = v+ + t 2 Cf (u ) (6) This defines a mapping Ψt : (u, v) → (u , v ) S.Lan | ∞-Dimensional Geometric MCMC
- 12. 11 + Metric S.Lan | ∞-Dimensional Geometric MCMC
- 13. 12 ∞-dimensional manifold MALA (Beskos, 2014) Choose K = G(u)−1 in (3) (Girolami and Calderhead, 2011) du dt = − 1 2 G(u)−1 (C−1 u + DΦ(u)) + G(u)−1 dW dt (7) Define G : X → L(X, X) and g : X → X by g(u) = −G(u)−1 ((C−1 − G(u))u + DΦ(u)) (8) Similar semi-implicit scheme (θ = 1 2 ) yields u = a 1 2 u + b 1 2 v, v = √ h 2 g(u) + G(u)− 1 2 ξ a 1 2 = 1 − h/4 1 + h/4 , b 1 2 = √ h 1 + h/4 (9) Note a2 1 2 + b2 1 2 = 1 follows Crank-Nicolson scheme (Richtmyer and Morton, 1994). S.Lan | ∞-Dimensional Geometric MCMC
- 14. 13 ∞-dimensional manifold MALA (Beskos, 2014) Assumption (1) N(0, G(u)−1 ) N(0, C) for µ0-a.s. in u. Assumption (2) For µ0-a.s. in u, we have g(u) ∈ Im(C 1 2 ), i.e. N(g(u), C) N(0, C). S.Lan | ∞-Dimensional Geometric MCMC
- 15. 14 ∞-dimensional manifold MALA (Beskos, 2014) Theorem (Beskos (2014)) Denote ν(du, du ) = µ(du)Q(u, du ) with Q(u, du ) being the transition kernel of (9). Under Assumptions 1 and 2, ν νT , and the acceptance probability has the following form a(u, u ) = 1 ∧ dνT dν (u, u ), dνT dν (u, u ) = exp{−Φ(u )}λ(ρ−1 (u; u ); u ) exp{−Φ(u)}λ(ρ−1(u ; u); u) (10) where ρ−1 (u ; u) = [(1 + h/4)u − (1 − h/4)u]/ √ h and λ(w; u) is calculated as follows: λ(w; u) = dN ( √ h/2g(u), G(u)−1 ) dN (0, C) = dN ( √ h/2g(u), G(u)−1 ) dN (0, G(u)−1) dN (0, G(u)−1 ) dN (0, C) = exp √ h 2 G 1 2 (u)g(u), G 1 2 (u)w − h 8 |G 1 2 (u)g(u)| 2 · exp − 1 2 |G 1 2 (u)w| 2 + 1 2 |C − 1 2 w| 2 |C 1 2 G 1 2 (u)| S.Lan | ∞-Dimensional Geometric MCMC
- 16. 15 ∞-dimensional manifold HMC Let K = G(u)−1 in (5) (Girolami and Calderhead, 2011) d2 u dt2 + u = g(u), (v := du dt ) t=0 ∼ N(0, G(u)−1 ) (11) Discretize (11) by a splitting method of the form v− = v + t 2 g(u) u v+ = cos τ sin τ − sin τ cos τ u v− v = v+ + t 2 g(u ) (12) where τ(t)/t → 1 as t → 0. Evolving (u, v) for k-folds (12) defines Ψk t : (u0, v0) → (uk, vk). S.Lan | ∞-Dimensional Geometric MCMC
- 17. 16 ∞-dimensional manifold HMC Theorem Denote ν(du, du ) = µ(du)Q(u, du ) with Q(u, du ) being the transition kernel of Ψk t . Under Assumptions 1 and 2, ν νT , and the acceptance probability has the following form 1 ∧ exp(−∆H(u, v)) (13) where ∆H(u, v) = H(Ψ (k) t (u, v)) − H(u, v) =Φ(uk) − Φ(u0) + 1 2 (|(G(uk) − C −1 ) 1 2 vk| 2 − |(G(u0) − C −1 ) 1 2 v0| 2 − log |G(uk)| + log |G(u0)|) + t2 8 (|C − 1 2 g(u0)| 2 − |C − 1 2 g(uk)| 2 ) + t 2 k−1 =0 ( v , C −1 g(u ) + v +1, C −1 g(u +1) ) S.Lan | ∞-Dimensional Geometric MCMC
- 18. 17 Connections between ∞-dimensional MCMC Remark As a direct corollary, when I = 1, ∞-(m)HMC reduces to ∞-(m)MALA. ∞ − MALA position-dependent pre-conditioner K(u) −−−−−−−−−−−−−−−−−−−−→ ∞ − mMALA h=4 −−→ SN multiplesteps(k>1) −−−−−−−−−−→ multiplesteps(k>1) −−−−−−−−−−→ ∞ − HMC position-dependent pre-conditioner K(u) −−−−−−−−−−−−−−−−−−−−→ ∞ − mHMC S.Lan | ∞-Dimensional Geometric MCMC
- 19. 18 Geometric Monte Carlo weight (computation) control RWM θ 1 -2 -1 0 1 2 θ 2 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 HMC θ 1 -2 -1 0 1 2 θ 2 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 RHMC θ 1 -2 -1 0 1 2 θ 2 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 LMC θ 1 -2 -1 0 1 2 θ 2 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 S.Lan | ∞-Dimensional Geometric MCMC
- 20. 19 Dimension Reduction intrinsic low-dimensional subspace S.Lan | ∞-Dimensional Geometric MCMC
- 21. 20 Dimension Reduction intrinsic low-dimensional subspace Particular choice of K(u) = G(u)−1 could be K(u) = (C−1 + H(u))−1 , H(u) = EY|u DΦ(u), DΦ(u) (14) Computationally infeasible to update Fisher metric H(u) in each u Given eigenbasis {φi(x)} we define projection operator Pr as follows: Pr : X → Xr, u → ur := r i=1 φi φi, u (15) Truncate H(u) on the r-dimensional subspace Xr ⊂ X (X = Xr + X⊥) Hr(u)(v, w) = Prv, EY|u[DrΦ(u)DrΦ(u) T ]Prw , ∀v, w ∈ X (16) K(u)−1 can be approximated K(u)−1 ≈ Hr(u) + C−1 (17) S.Lan | ∞-Dimensional Geometric MCMC
- 22. 21 Prior-Based Dimension Reduction With the eigen-pair {λi, ui(x)}, the prior covariance operator C can be written and approximated as C = UΛU∗ ≈ UrΛrU∗ r (18) Then we can approximate the posterior covariance K(u) = (C−1 + H(u))−1 ≈ C + UrΛ 1 2 r (Dr − Ir)Λ 1 2 r U∗ r (19) where Dr := (ˆHr(u) + Ir)−1 and ˆHr(u) := Λ 1 2 r U∗ r H(u)UrΛ 1 2 r . By applying U∗ r and U∗ ⊥ to Langevin SDE (3) respectively we get dur = − 1 2 Drurdt − γr 2 Dr ur Φ(u; y)dt + √ DrdWr (20a) du⊥ = − 1 2 u⊥dt − γ⊥ 2 u⊥ Φ(u; y)dt + dW⊥ (20b) where ur = Λ − 1 2 r U∗ r u and u⊥ = Λ − 1 2 ⊥ U∗ ⊥u. S.Lan | ∞-Dimensional Geometric MCMC
- 23. 22 Karhunen-Loève Expansion Let X be a Hilbert space H = L2 (D; R) on bounded open D ⊂ Rd with Lipschitz boundary, and with inner-product ·, · and norm · Consider the following covariance operator C C := σ2 (αI − ∆)−s (21) Let {λ2 i } and {φi(x)} denote eigenvalues and eigenfunctions of C. If s > d/2 and λi i− s d , C defines a Gaussian measure N(0, C) that each draw u(·) ∼ N(0, C) admits Karhunen-Loève (K-L) expansion (Adler, 1981; Bogachev, 1998; Dashti and Stuart, 2015): u(x) = +∞ i=0 uiλiφi(x), ui iid ∼ N(0, 1) (22) S.Lan | ∞-Dimensional Geometric MCMC
- 24. 23 Laminar Jet formulation of the inverse problem We consider the following 2d Navier-Stokes equation: Momentum equation: − div ν( u + uT ) + u · u + p = 0, (23) where u = (u, v) is the velocity field and p the pressure; Continuity equation: div u = 0. (24) Denote σn = −pn + ν( u + uT ) · n as the boundary traction. u · n = −θ(y), σn × n = 0 on I := x = 0, y ∈ (− 1 2 Ly, 1 2 Ly) σn + β(u · n)− u = 0 on O := x = Lx, y ∈ (− 1 2 Ly, 1 2 Ly) u · n = 0, σn × n = 0 on B := x ∈ (0, Lx), y = ± 1 2 Ly where (u · n)− = u·n−|u·n| 2 , and β ∈ (0, 1] is the backflow stabilization parameter S.Lan | ∞-Dimensional Geometric MCMC
- 25. 24 Laminar Jet results Laminar Jet problem: the location of observations (left) and the forward PDE solutions with true unknown (right). Fluid viscosity ν = 3 × 10−2 . S.Lan | ∞-Dimensional Geometric MCMC
- 26. 25 Laminar Jet results Laminar Jet test problem: true inflow velocity profiles with different number of modes (left) and posterior estimates given by different algorithms (right). Shaded region shows the 95% credible band estimated with samples by ∞-mHMC. S.Lan | ∞-Dimensional Geometric MCMC
- 27. 26 Laminar Jet results Laminar Jet problem: the trace plots of data-misfit function evaluated with each sample (left, values have been offset to be better compared with) and the auto-correlation of data-misfits as a function of lag (right). S.Lan | ∞-Dimensional Geometric MCMC
- 28. 27 Laminar Jet results Method AP s/iter ESS(min,med,max) minESS/s spdup PDEsolns pCN 0.61 1.29 ( 5.24, 6.66, 13.33) 4.05E-04 1.00 22004 ∞-MALA 0.66 1.68 ( 5.38, 6.62, 19.53) 3.21E-04 0.79 33005 ∞-HMC 0.72 3.81 ( 5.41, 7.43, 16.44) 1.42E-04 0.35 82466 ∞-mMALA 0.68 5.97 ( 1075.24, 2851.22, 3867.08) 1.80E-02 44.47 2233205 ∞-mHMC 0.58 13.33 ( 2058.42, 3394.17, 4560.03) 1.54E-02 38.13 5575696 split ∞-MMALA 0.57 3.66 ( 1079.55, 1805.89, 2395.13) 2.95E-02 72.82 693065 split ∞-mHMC 0.60 6.88 ( 2749.63, 3974.36, 5498.03) 4.00E-02 98.67 1721694 Sampling Efficiency in Laminar Jet problem. AP: acceptance probability, s/iter: seconds per iteration, ESS(min,med,max): Effective Sample Size (minimum, median, maximum), and MinESS/s: minimum ESS per second. Dimension =100. HMC algorithms do leap frog for steps randomly chosen between 1 and 4. Split∗ truncates K-L at r=30. S.Lan | ∞-Dimensional Geometric MCMC
- 29. 28 Likelihood-Informed Dimension Reduction By whitening coordinates vi(x) = C− 1 2 ui(x), generalized eigenproblem H(u)ui(x) = λiC−1 ui(x) (25) can be shown equivalent to the eigen-problem for ppGNH H(v) H(v)vi(x) = C 1 2 H(u)C 1 2 vi(x) = λivi(x) (26) The local posterior covariance is approximated (Dr := (Ir + Λr)−1 ) K(v) = (I + H(v))−1 ≈ I + Vr(Dr − Ir)V∗ r (27) By applying V∗ r and V∗ ⊥ to whitened Langevin SDE respectively dvr = − 1 2 Drvrdt − γr 2 Dr vr Φ(v; y)dt + √ DrdWr (28a) dv⊥ = − 1 2 v⊥dt − γ⊥ 2 v⊥ Φ(v; y)dt + dW⊥ (28b) where vr = V∗ r v and v⊥ = V∗ ⊥v. S.Lan | ∞-Dimensional Geometric MCMC
- 30. 29 Connection to DILI (Cui et al., 2016) By considering the approximation H(v) ≈ VrΛrV∗ r , we have the posterior covariance projected to r-dimensional subspace: Kr = Covµ[V∗ r v] = V∗ r (I + H(v))−1 Vr ≈ Dr := (Ir + Λr)−1 (29) While DILI computes the posterior covariance in the subspace Kr := Covµ[V∗ r v] empirically. They both have the following approximate posterior covariance Kv = Covµ[v] ≈ VrKrV∗ r + I − VrV∗ r (30) Since we directly work with (29), empirical calculation of Kr is avoided; The approximation (29) is already in diagonal form thus the rotation Ψr = VrWr in DILI is not needed. They capture the similar geometric feature of subspace. S.Lan | ∞-Dimensional Geometric MCMC
- 31. 30 Elliptic Inverse Problem formulation of the inverse problem Consider the elliptic inverse problem as in (DILI Cui et al., 2016) defined on the unit square domain Ω = [0, 1]2 : − · (k(s) p(s)) = f (s), s ∈ Ω k(s) p(s), n(s) = 0, s ∈ ∂Ω ∂Ω p(s)dl(s) = 0 (31) where k(s) is the transmissivity field, p(s) is the potential function, f (s) is the forcing term, and n(s) is the outward normal to the boundary. The inverse problem is to infer u = log k from observations {yn}. 25 observations arise from the solutions (solved on a 80 × 80 mesh) contaminated by additive Gaussian error: yn = p(xn) + εn, εn ∼ N(0, σ2 y), SNR := max s {u(s)}/σy = 100 S.Lan | ∞-Dimensional Geometric MCMC
- 32. 31 Elliptic Inverse Problem results S.Lan | ∞-Dimensional Geometric MCMC
- 33. 32 Elliptic Inverse Problem results Elliptic inverse problem (SNR = 100): Bayesian posterior mean estimates of the log-transmissivity field u(s) based on 2000 samples by various MCMC algorithms; the upper-left corner shows the MAP estimate. S.Lan | ∞-Dimensional Geometric MCMC
- 34. 33 Elliptic Inverse Problem results Method h AP s/iter ESS(min,med,max) minESS/s spdup PDEsolns pCN 0.01 0.57 0.99 (2.67,6.95,37.79) 0.0013 1.00 2501 ∞-MALA 0.04 0.61 1.62 (4.32,15.34,51.45) 0.0013 0.99 5002 ∞-HMC 0.04 0.59 3.52 (24.36,92.13,184.84) 0.0035 2.57 12342 DR-∞-mMALA 0.52 0.67 8.85 (127.25,210.84,460.07) 0.0072 5.34 80032 DR-∞-mHMC 0.25 0.56 22.97 (190.2,322.29,687.11) 0.0041 3.08 198176 DILI (0.1, 0.2) 0.69 1.59 (30.52,133.67,221.97) 0.0096 7.13 6612 aDR-∞-mMALA 0.25 0.71 1.61 (12.09,89.17,174.36) 0.0037 2.79 6612 aDR-∞-mHMC 0.10 0.69 3.63 (70.99,234.42,364.31) 0.0098 7.26 14056 Sampling efficiency in elliptic inverse problem (SNR=100). Column labels are as follows. h: step size(s) used for making MCMC proposal; AP: average acceptance probability; s/iter: average seconds per iteration; ESS(min,med,max): minimum, median, maximum of Effective Sample Size across all posterior coordinates; min(ESS)/s: minimum ESS per second; spdup: speed-up relative to base pCN algorithm; PDEsolns: number of PDE solutions during execution. S.Lan | ∞-Dimensional Geometric MCMC
- 35. 34 ∞-dimensional Spherical Hamiltonian Monte Carlo qqqq qqq qqqqq qq qq qq qq qqqq qqqq qqqq qqqq qqq qqq qqqq qq qq qqqq qqqq qqq qqq qq qqqq qq qq qqqq qq qq qq qq qq qqqq qqq qqq qqqq qqqq qq qq qq qqqq qq qq qq qq qqqq qq qq qq qq qqqq qq qqqq qq qqqq qq qq qq qq qq qq qq qq qqq q qqqq qq qq qqqq qqqq qq qq qq qq qq qq qqqq qq qq qqqq qqqq qq qq qq qq qqqq qq qqqq qqqq qq qq qqqq qq qq qq qq qqqq qqqq qq qq qq qq qq qq qqqq q qqqqq qq qq qq qq qqqq qq qqqq qqqq qq qq qq qqqq qq qq qq qq qq qq qq qq qqqq qq qq qq qqqq qqqq qq qq qq qq qqqq qqqq qq qq qq qq qq qq qq qq qq qqqq qqqq qq qq qq qqqq qq qq qq qqqq qq qq qq qq qq qqqq qq qqqq qqqq qq qq qq qq qq qq qqqq qq qq qq qq qq qq qq qq qqqq qq qq qq qqqq qq qq qqqq qq qq qqqq qq qqqq qqqq qqqq qq qq qq qq qq qq qq qq qq qq qq qqqq qq qq qq qqq q qqqq qqqq qqqq qq qq qqqq qq qq qqqq qq qq qqqq qqqq qq qq qq qq qq qq qq qq qqqq qq qqqq qq qq qqqq qqqq qq qq qq qq qq qqqq qqqq qq qq qq qqqq qqqq qq qqqq qq qq qq qq qq qq qq qq 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q q qqq q q q q qq qq q q q q q q q q q qq q q q qqq q q q q q q q q q qq q q q q q q q qq q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q qq q q q q q q q q q q q qq q qq q qq qq q q q q q q q q q q q q q q q qq q q q q q qq q q q q q q qq q qq q q q q q q q q q q q q q q q q q q q q qq q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q qq q q qqq q q q qq q q q q q qq q q q q q q q q q q q q q q q q q q q q qq q q q q qq q q q q q q q q qq q q q q q q q qq q q q q q q q q q q qq q q qqq q q q q q q q q q q q q q q q q qqq q q q q q q q q q qqqq q q q q qqq q q q q q q qq q q q q qq q q q q q q q q q q q q q q qq q q q q q q q q q q q q qq qq qq q q q q q q q q q q q q q q q qq q q q q q q q q qq q q q q q q q q q q q q qq q q q q q q q q q q q qq q q q q q q q qqq q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q qqq q q q q q q q q q q q q q qq q q q qq q q q q q q q q q q qq q qq q q q q q q q q q q q q q qq q qq q q q q q q q q q qq q qq qq q q q qq q q q q q q q q q q q q q q qqqq q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q qq q q q q q q q q q q q qq q q q q qq q q q q qq q q q q q q q q q qqq q q q q q q q qq q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q qq q q q q q q q q q q q qq q q q q q q q q q q q q q q q qq q qq q q q qq q q qq q q q q q q q qq q qq q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q qq q q q qq q q q q q q q q qq q q q q qq q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q qq q q q qq q q qq q q q q q q q q qq q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q qq q q q q q q qq q q q q qq q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q qq q q q qq q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q qq q q q q q q q q q q q q q q q qq q q q q q q q q q q qq q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q qq q qq q q qq q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q qq q q q q q q q qq q q q q q q q q q q q q q q q q q q q qq q q qq q q q q q q q q q q q qq q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q qq q q qq q q q q q q q q q q q qq q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q qq qq q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q qq q q q q qq q q q q qq q q q q q q q q q q qq q q q q q q q q q q q q q q qq q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q qq qq q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q qq q q q q q q q q q qq q q q q q q q q qq q q qq q q q q q q q q q q q q q q q q q qq q q q q q qq q q q q q q qq q q q q q qq q q q q q qq q q q q q q q q q q qq q qq q q q q q q q q q q q q qq q qq q q q q q q q q q qq q qq qq q q q q q q q q q qq q q qqq qq q q qqq θ θD+1 q A B S.Lan | ∞-Dimensional Geometric MCMC
- 36. 35 Representation of Probability Densities Consider probability distributions over smooth manifolds D. Having fixed a background measure µ, let P := p : D → R | p ≥ 0, D p(x) µ(dx) = 1 (32) Define the following nonparametric Fisher metric on the tangent space TpP := φ ∈ C∞ (D) | D φ(x) µ(dx) = 0 : gF (φ, ψ)p := D φ(x)ψ(x) p(x) µ(dx). (33) The square-root mapping S : (P, gF ) → (Q, ·, · 2), S(p) = q = 2 √ p is a Riemannian isometry, where Q is ∞-dimensional sphere in L2 (D) Q := q : D → R | D q(x)2 µ(dx) = 1 , f , h 2 = D fhdµ(x) (34) S.Lan | ∞-Dimensional Geometric MCMC
- 37. 36 Nonparametric Density Modeling It is easier to work with root density q ∈ Q (e.g. clean geodesic flow). Restrict Gaussian process prior q(·) ∼ GP(0, K(·)) to Q, where the covariance operator K = σ2 (α − ∆)−s has eigen-pairs {λ2 i , φi(x)}∞ i=1. Then q(x) 2 = ∞ i=1 qiφi(x) 2 = 1 with qi ∼ N(0, λ2 i ) implies q 2 2 := ∞ i=1 q2 i = 1, i.e. q := (qi) ∈ S∞ (35) Given data x = {xn ∈ D}N n , we have the posterior density π(q|x) ∝ π(q) π(x|q) = ∞ i=1 exp − q2 i /(2λ2 i ) δ q 2 (1) N n=1 q2 (xn) (36) Sampling q = (qi) can be done by spherical HMC (Lan et al., 2014). S.Lan | ∞-Dimensional Geometric MCMC
- 38. 37 Spherical Hamiltonian Monte Carlo (Lan et al., 2014) Truncate q := (qi) at I, q := (qi)I i=1. Define the total energy E(q, v) := U(q) + K(v; q) = ˜U(q) + K0(v; q) (37) ˜U(q) := U(q) − 1 2 log |G(q−I)| = − log π(q|x) + log |qI| (38) K0(v; q) := 1 2 vT −IG(q−I)v−I = 1 2 vT v (39) Discretizing the Hamiltonian equation results in v− = v − h 2 P(q)g(q) q v+ = r 0 0 v− 2 cos( v− 2r−1 h) + sin( v− 2r−1 h) − sin( v− 2r−1 h) + cos( v− 2r−1 h) r−1 0 0 v− −1 2 q v− v = v+ − h 2 P(q )g(q ) (40) where g(q) := q ˜U(q), P(q) := ID − r−2 qqT . S.Lan | ∞-Dimensional Geometric MCMC
- 39. 38 Nonparametric Density Modeling simulation result q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q qq q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q qq q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q qq q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q 0.00 0.25 0.50 0.75 0.0 0.2 0.4 0.6 0.8 Dimension2 q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q 0.00 0.25 0.50 0.75 1.00 0.00 0.25 0.50 0.75 1.00 q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q 0.00 0.25 0.50 0.75 1.00 0.00 0.25 0.50 0.75 1.00 Dimension 1 Dimension2 q q q q q qq q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q 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q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q 0.00 0.25 0.50 0.75 1.00 0.00 0.25 0.50 0.75 1.00 Dimension 1 q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q qq q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q qq q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q qq q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q 0.00 0.25 0.50 0.75 0.0 0.2 0.4 0.6 0.8 Dimension2 q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q 0.00 0.25 0.50 0.75 1.00 0.00 0.25 0.50 0.75 1.00 q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q 0.00 0.25 0.50 0.75 1.00 0.00 0.25 0.50 0.75 1.00 Dimension 1 Dimension2 q q q q q qq q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q 0.00 0.25 0.50 0.75 1.00 0.00 0.25 0.50 0.75 1.00 Dimension 1 Figure: The contours (black) of the posterior median from 1,000 draws of the χ2 -process density sampler. Each posterior is conditioned on 1,000 data points (red). S.Lan | ∞-Dimensional Geometric MCMC
- 40. 39 Nonparametric Density Modeling result of real data q qqq q q qq qqq qq qqqqq qq q q qqq q qqqq q qq q q q q qqq q q q q qqq qq q qq q q qq q q q q q qq qqqq q q qq q q qq qqq qq qq qq qqq qq qq q qq q q q qqqqq qqq qqqqq qqq q q q q qqq q q qqq q q q q qqqq q q qq qq q q q q qq qq qq q q q q qqq qq q qq q q q qq q qq q q qq q q qq q q qq qq q q qq q q q qqq q q q q q qq qq q q qq q q q q qqq q qqqq q q qq q q q q q q qqq qq qq q q qqq qqqqq qq q qq q qqqq q q q qq q q q q q q qq q q q q q q q q qq q q q q q q q q q q q q q q q qq q q q q q qq q q q qq q q q qq qq q q q q qq q q q qq qqqq q q q q q q qqqq q q q q q q q q qqqq q q q qqq q q q q q q q q q q q q qq q qqq qqq q q q q q q q qq q q q qq q q qq q qqqq qq q qq q qq q q q qq q q q q q q q q q q qq q qqq qq q qqq q q qqq q q qq qq qqqqq q q q qq q q q q q q q q q q q qq q qq q q q q q q q q q q q q q qq q q qq q q qq q q q q qq q q q q q q q q q q qq q qq q q qq qq q q q q q q q q q q q qqq q qq q q q q qq q qqq qq qq q qqqqq q q q qq q q qqqq qq q q q q q q q q q q q q q q qq qq qq q q q q q q qq qq q q q qq q q q q q qq q q qqq q q q qq q q qq qqq qq qq q qq qqq q q q q q q qq q q q q q q q qq q q q qqq q qq q q q q qq q q q q q q qq q qq q qq q qq q q q q q q q q q q q q qq qqqq q q q qqq qqq q qq q q q q q q q qq qqq q q qqq q q q q q q q q q q q q q q q qq q qq q q qq q q q q q q q q q qqq q qq q q qq q q q q q q q q q q q q qq q q q q q q q q q q q q q qq qq qqq q qq q q q q q 0.00 0.25 0.50 0.75 1.00 0.00 0.25 0.50 0.75 1.00 Dimension 1 Dimension2 Pointwise posterior mean qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q qq q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q 0.00 0.25 0.50 0.75 1.00 0.00 0.25 0.50 0.75 1.00 Dimension 1 Posterior predictive sample q qqq q q qq qqq qq qqqqq qq q q qqq q qqqq q qq q q q q qqq q q q q qqq qq q qq q q qq q q q q q qq qqqq q q qq q q qq qqq qq qq qq qqq qq qq q qq q q q qqqqq qqq qqqqq qqq q q q q qqq q q qqq q q q q qqqq q q qq qq q q q q qq qq qq q q q q qqq qq q qq q q q qq q qq q q qq q q qq q q qq qq q q qq q q q qqq q q q q q qq qq q q qq q q q q qqq q qqqq q q qq q q q q q q qqq qq qq q q qqq qqqqq qq q qq q qqqq q q q qq q q q q q q qq q q q q q q q q qq q q q q q q q q q q q q q q q qq q q q q q qq q q q qq q q q qq qq q q q q qq q q q qq qqqq q q q q q q qqqq q q q q q q q q qqqq q q q qqq q q q q q q q q q q q q qq q qqq qqq q q q q q q q qq q q q qq q q qq q qqqq qq q qq q qq q q q qq q q q q q q q q q q qq q qqq qq q qqq q q qqq q q qq qq qqqqq q q q qq q q q q q q q q q q q qq q qq q q q q q q q q q q q q q qq q q qq q q qq q q q q qq q q q q q q q q q q qq q qq q q qq qq q q q q q q q q q q q qqq q qq q q q q qq q qqq qq qq q qqqqq q q q qq q q qqqq qq q q q q q q q q q q q q q q qq qq qq q q q q q q qq qq q q q qq q q q q q qq q q qqq q q q qq q q qq qqq qq qq q qq qqq q q q q q q qq q q q q q q q qq q q q qqq q qq q q q q qq q q q q q q qq q qq q qq q qq q q q q q q q q q q q q qq qqqq q q q qqq qqq q qq q q q q q q q qq qqq q q qqq q q q q q q q q q q q q q q q qq q qq q q qq q q q q q q q q q qqq q qq q q qq q q q q q q q q q q q q qq q q q q q q q q q q q q q qq qq qqq q qq q q q q q 0.00 0.25 0.50 0.75 1.00 0.00 0.25 0.50 0.75 1.00 Dimension 1 Dimension2 Pointwise posterior mean qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q qq q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q 0.00 0.25 0.50 0.75 1.00 0.00 0.25 0.50 0.75 1.00 Dimension 1 Posterior predictive sample Figure: Hutchings’ bramble canes data: the first figure depicts the 823 bramble canes (red), a heatmap of the pointwise posterior mean (black is low, white is high), and a single contour at density 0.3 (blue) including all but a few points. The second figure shows 823 draws from the χ2 -process density posterior predictive distribution, obtained using a rejection sampling scheme. S.Lan | ∞-Dimensional Geometric MCMC
- 41. 40 Conclusion Geometric information (gradient, metric) helps MCMC in mixing but comes at computational cost, alleviated by dimension reduction. Key: find intrinsic finite dimensional subspace where most information concentrates and manifold MCMC can be applied Light-computation algorithms, e.g. pCN, preconditioned MALA, can be applied in the less informative complementary subspace MCMC defined on manifold can help solve challenging statistical problems, e.g. density estimation, modeling covariance/correlation matrices. S.Lan | ∞-Dimensional Geometric MCMC
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- 44. Thank you !