Generalized pCN Metropolis algorithms
for Bayesian inference in Hilbert spaces
Oliver Ernst, Daniel Rudolf, Björn Sprungk
Data Assimilation and Inverse Problems (U Warwick)
23rd February, 2016
FAKULTÄT FÜR MATHEMATIK
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Outline
1. Problem setting and Metropolis-Hastings algorithms in Hilbert spaces
2. The gpCN-Metropolis algorithm
3. Convergence
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Problem setting and Metropolis-Hastings algorithms in Hilbert spaces
Next
1. Problem setting and Metropolis-Hastings algorithms in Hilbert spaces
2. The gpCN-Metropolis algorithm
3. Convergence
Warwick, Feb 2016 · B. Sprungk
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Problem setting and Metropolis-Hastings algorithms in Hilbert spaces
Motivation: uncertainty quantification in groundwater flow modelling
Groundwater flow modelling:
I
PDE model for groundwater pressure head p, e.g.,
−∇ · (eκ(x) ∇p(x)) = 0
I
Noisy observations of κ and p at locations xj , j = 1, . . . , J
I
Interested in functional f of flux u(x) = −eκ(x) ∇p(x)
(e.g., breakthrough time of pollutants)
UQ approach:
I
Model unknown coefficient κ as (Gaussian) random function
I
Employ observational data to fit stochastic model for κ
I
Compute expectations or probabilities for resulting random f
Warwick, Feb 2016 · B. Sprungk
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Problem setting and Metropolis-Hastings algorithms in Hilbert spaces
Motivation: uncertainty quantification in groundwater flow modelling
Groundwater flow modelling:
I
PDE model for groundwater pressure head p, e.g.,
−∇ · (eκ(x) ∇p(x)) = 0
I
Noisy observations of κ and p at locations xj , j = 1, . . . , J
I
Interested in functional f of flux u(x) = −eκ(x) ∇p(x)
(e.g., breakthrough time of pollutants)
UQ approach:
I
Model unknown coefficient κ as (Gaussian) random function
I
Employ observational data to fit stochastic model for κ
I
Compute expectations or probabilities for resulting random f
Warwick, Feb 2016 · B. Sprungk
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Problem setting and Metropolis-Hastings algorithms in Hilbert spaces
Obtain stochastic model for κ via Bayes
I
Random function in Hilbert space H with CONS {φm }m∈N
X
κ(x, ω) =
ξm (ω) φm (x),
m≥1
where ξ = (ξm )m∈N random vector in `2
I
Employ measurements of κ to fit Gaussian prior µ0 for κ resp. ξ:
ξ ∼ µ0 = N (0, C0 )
I
on `2
Conditioning prior µ0 on noisy observations p ∈ RJ of p
p = G(ξ) + ε,
(ξ, ε) ∼ µ0 ⊗ N (0, Σ)
results in posterior via Bayes’ rule [Stuart, 2010]
µ(dξ) ∝ exp (−Φ(ξ)) µ0 (dξ),
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1
Φ(ξ) = |p − G(ξ)|2Σ−1 .
2
Problem setting and Metropolis-Hastings algorithms in Hilbert spaces
Obtain stochastic model for κ via Bayes
I
Random function in Hilbert space H with CONS {φm }m∈N
X
κ(x, ω) =
ξm (ω) φm (x),
m≥1
where ξ = (ξm )m∈N random vector in `2
I
Employ measurements of κ to fit Gaussian prior µ0 for κ resp. ξ:
ξ ∼ µ0 = N (0, C0 )
I
on `2
Conditioning prior µ0 on noisy observations p ∈ RJ of p
p = G(ξ) + ε,
(ξ, ε) ∼ µ0 ⊗ N (0, Σ)
results in posterior via Bayes’ rule [Stuart, 2010]
µ(dξ) ∝ exp (−Φ(ξ)) µ0 (dξ),
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1
Φ(ξ) = |p − G(ξ)|2Σ−1 .
2
Problem setting and Metropolis-Hastings algorithms in Hilbert spaces
Obtain stochastic model for κ via Bayes
I
Random function in Hilbert space H with CONS {φm }m∈N
X
κ(x, ω) =
ξm (ω) φm (x),
m≥1
where ξ = (ξm )m∈N random vector in `2
I
Employ measurements of κ to fit Gaussian prior µ0 for κ resp. ξ:
ξ ∼ µ0 = N (0, C0 )
I
on `2
Conditioning prior µ0 on noisy observations p ∈ RJ of p
p = G(ξ) + ε,
(ξ, ε) ∼ µ0 ⊗ N (0, Σ)
results in posterior via Bayes’ rule [Stuart, 2010]
µ(dξ) ∝ exp (−Φ(ξ)) µ0 (dξ),
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1
Φ(ξ) = |p − G(ξ)|2Σ−1 .
2
Problem setting and Metropolis-Hastings algorithms in Hilbert spaces
Markov chain Monte Carlo (MCMC) sampling
Eµ [f ] =
R
I
Employ MCMC sampling to compute
I
Construct Markov chain (ξ k )k∈N in H = `2 with transition kernel
H
f (ξ) µ(dξ).
M (η, A) := P(ξ k+1 ∈ A|ξ k = η)
which is µ-reversible:
M (ξ, dη) µ(dξ) = M (η, dξ) µ(dη)
I
Then µ stationary measure of chain and (under suitable conditions)
n
1X
f (ξ k )
n
n→∞
−−−→
k=1
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Eµ [f ]
a.s.
Problem setting and Metropolis-Hastings algorithms in Hilbert spaces
Efficiency of Markov chain Monte Carlo
I
µ-reversible Markov chain geometrically ergodic and ξ 1 ∼ µ, then
!
n
√
1X
d
n
f (ξ k ) − Eµ [f ]
−−→ N (0, σf2 ),
n
k=1
see [Roberts & Rosenthal, 1997], where
σf2
∞
X
=
γf (k),
γf (k) = Cov f (ξ 1 ), f (ξ 1+k )
k=−∞
I
Rapid decay of autocovariance function γf yields high statistical
efficiency.
I
Common measure of efficiency is effective sample size:
ESSf (n) = n
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σf2
Varµ (f )
.
Problem setting and Metropolis-Hastings algorithms in Hilbert spaces
The Metropolis-Hastings (MH) algorithm
Metropolis-Hastings algorithm for generating state ξ k+1 of Markov chain
1. Draw a new state η from proposal kernel P (ξ k , dη):
η ∼ P (ξ k ).
2. Accept proposal η with acceptance probability α(ξ k , η), i.e., draw
a ∼ Uni[0, 1] and set
(
η, a ≤ α(ξ k , η),
ξ k+1 =
ξ k , otherwise.
Transition kernel of resulting chain is called Metropolis kernel and reads
Z
M (ξ, dη) = α(ξ, η)P (ξ, dη) + 1 − α(ξ, ζ) P (ξ, dζ) δξ (dη).
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Problem setting and Metropolis-Hastings algorithms in Hilbert spaces
The Metropolis-Hastings (MH) algorithm
Metropolis-Hastings algorithm for generating state ξ k+1 of Markov chain
1. Draw a new state η from proposal kernel P (ξ k , dη):
η ∼ P (ξ k ).
2. Accept proposal η with acceptance probability α(ξ k , η), i.e., draw
a ∼ Uni[0, 1] and set
(
η, a ≤ α(ξ k , η),
ξ k+1 =
ξ k , otherwise.
Transition kernel of resulting chain is called Metropolis kernel and reads
Z
M (ξ, dη) = α(ξ, η)P (ξ, dη) + 1 − α(ξ, ζ) P (ξ, dζ) δξ (dη).
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Problem setting and Metropolis-Hastings algorithms in Hilbert spaces
MH algorithms in Hilbert space
To ensure µ-reversibility of Metropolis kernel we need to choose
dν >
α(ξ k , η) = min 1,
(ξ , η) ,
dν k
where ν(dξ, dη) := P (ξ, dη) µ(dξ) and ν > (dξ, dη) := ν(dη, dξ).
Since µ and µ0 are equivalent (Bayes’ rule), a µ0 -reversible proposal P ,
P (ξ, dη) µ0 (dξ) = P (η, dξ) µ0 (dη),
will yield
dµ
dν >
(ξ, η) =
(η)
dν
dµ0
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dµ
(ξ)
= exp Φ(ξ) − Φ(η) .
dµ0
9 / 28
Problem setting and Metropolis-Hastings algorithms in Hilbert spaces
MH algorithms in Hilbert space
To ensure µ-reversibility of Metropolis kernel we need to choose
dν >
α(ξ k , η) = min 1,
(ξ , η) ,
dν k
where ν(dξ, dη) := P (ξ, dη) µ(dξ) and ν > (dξ, dη) := ν(dη, dξ).
Since µ and µ0 are equivalent (Bayes’ rule), a µ0 -reversible proposal P ,
P (ξ, dη) µ0 (dξ) = P (η, dξ) µ0 (dη),
will yield
dν >
dµ
(ξ, η) =
(η)
dν
dµ0
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dµ
(ξ)
= exp Φ(ξ) − Φ(η) .
dµ0
9 / 28
Problem setting and Metropolis-Hastings algorithms in Hilbert spaces
Examples
Gaussian random walk-proposal:
P (ξ) = N (ξ, s2 C0 )
I
s ∈ R+ is a stepsize parameter typically tuned such that acceptance
rate
Z Z
ᾱ :=
α(ξ, η) P (ξ, dη) µ(dξ) ≈ 0.234.
I
This proposal P is not µ0 reversible.
H
H
√
Preconditioned Crank-Nicolson-proposal: P (ξ) = N ( 1 − s2 ξ, s2 C0 )
I
Derived from a CN-scheme for (the drift of) the SDE
√
dξ t = (0 − ξ t ) dt + 2 dBt
where Bt is C0 -Brownian motion in H , see [Cotter et al., 2013].
I
This proposal is µ0 reversible.
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Problem setting and Metropolis-Hastings algorithms in Hilbert spaces
Examples
Gaussian random walk-proposal:
P (ξ) = N (ξ, s2 C0 )
I
s ∈ R+ is a stepsize parameter typically tuned such that acceptance
rate
Z Z
ᾱ :=
α(ξ, η) P (ξ, dη) µ(dξ) ≈ 0.234.
I
This proposal P is not µ0 reversible.
H
H
√
Preconditioned Crank-Nicolson-proposal: P (ξ) = N ( 1 − s2 ξ, s2 C0 )
I
Derived from a CN-scheme for (the drift of) the SDE
√
dξ t = (0 − ξ t ) dt + 2 dBt
where Bt is C0 -Brownian motion in H , see [Cotter et al., 2013].
I
This proposal is µ0 reversible.
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Problem setting and Metropolis-Hastings algorithms in Hilbert spaces
Implications for MCMC algorithm performance
Problem: Bayesian inference in 2D groundwater flow model.
Acceptance rate vs. stepsize s for different dimensions M of ξ ∈ RM .
Random walk-proposal P (ξ) = N (ξ, s2 C0 )
1
0.9
Average acceptance rate
0.8
0.7
0.6
0.5
M = 10
0.4
M = 40
0.3
M = 160
M = 640
0.2
0.1
0 −3
10
−2
−1
10
10
stepsize s
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0
10
Problem setting and Metropolis-Hastings algorithms in Hilbert spaces
Implications for MCMC algorithm performance
Problem: Bayesian inference in 2D groundwater flow model.
Acceptance rate vs. stepsize s for different dimensions M of ξ ∈ RM .
√
pCN-proposal P (ξ) = N ( 1 − s2 ξ, s2 C0 )
1
0.9
Average acceptance rate
0.8
0.7
0.6
0.5
0.4
M = 10
M = 40
0.3
M = 160
0.2
M = 640
0.1
0 −3
10
−2
−1
10
10
stepsize s
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0
10
The gpCN-Metropolis algorithm
Next
1. Problem setting and Metropolis-Hastings algorithms in Hilbert spaces
2. The gpCN-Metropolis algorithm
3. Convergence
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The gpCN-Metropolis algorithm
Motivation
Observation:
Higher statistical efficiency when proposal employs same covariance as
target measure µ, see [Tierney, 1994], [Roberts & Rosenthal, 2001], . . .
Example: µ = N (0, C ) in 2D, MH with different proposal covariances
1
1
0.5
0.5
0
0
−0.5
−0.5
−1
−1.5
−1
−0.5
0
0.5
1
−1
−1.5
1.5
P (ξ) = N (ξ, s2 I )
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−1
−0.5
0
0.5
P (ξ) = N (ξ, s2 C )
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1
1.5
The gpCN-Metropolis algorithm
Motivation
Observation:
Higher statistical efficiency when proposal employs same covariance as
target measure µ, see [Tierney, 1994], [Roberts & Rosenthal, 2001], . . .
Example: µ = N (0, C ) in 2D, MH with different proposal covariances
1
1
0.5
0.5
0
0
−0.5
−0.5
−1
−1.5
−1
−0.5
0
0.5
1
−1
−1.5
1.5
P (ξ) = N (ξ, s2 I )
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−1
−0.5
0
0.5
P (ξ) = N (ξ, s2 C )
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1
1.5
The gpCN-Metropolis algorithm
Motivation
Observation:
Higher statistical efficiency when proposal employs same covariance as
target measure µ, see [Tierney, 1994], [Roberts & Rosenthal, 2001], . . .
Example: µ = N (0, C ) in 2D, MH with different proposal covariances
ACRF
ACRF
1
1
0.9
X
0.9
X1
0.8
X2
0.8
X
1
0.7
0.7
0.6
0.6
0.5
0.5
0.4
0.4
0.3
0.3
0.2
0.2
0.1
0
0
2
0.1
50
100
0
0
150
lag
100
2
P (ξ) = N (ξ, s I )
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50
lag
2
P (ξ) = N (ξ, s C )
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150
The gpCN-Metropolis algorithm
How to approximate posterior covariance in advance
I
If forward map G were linear,then
µ = N (m, C) ,
I
C = (C0−1 + G∗ Σ−1 G)−1 .
Idea: Linearization of nonlinear G at ξ 0
e
G(ξ) ≈ G(ξ)
:= G(ξ 0 ) + Jξ,
J = ∇G(ξ 0 )
yields approximation to posterior covariance
e = (C −1 + J ∗ Σ−1 J)−1 .
C≈C
0
I
Good choice for ξ 0 might be maximum a posteriori estimate:
−1/2
ξ MAP = argmin |p − G(ξ)|2Σ−1 + kC0 ξk2 .
ξ
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The gpCN-Metropolis algorithm
How to approximate posterior covariance in advance
I
If forward map G were linear,then
µ = N (m, C) ,
I
C = (C0−1 + G∗ Σ−1 G)−1 .
Idea: Linearization of nonlinear G at ξ 0
e
G(ξ) ≈ G(ξ)
:= G(ξ 0 ) + Jξ,
J = ∇G(ξ 0 )
yields approximation to posterior covariance
e = (C −1 + J ∗ Σ−1 J)−1 .
C≈C
0
I
Good choice for ξ 0 might be maximum a posteriori estimate:
−1/2
ξ MAP = argmin |p − G(ξ)|2Σ−1 + kC0 ξk2 .
ξ
Warwick, Feb 2016 · B. Sprungk
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The gpCN-Metropolis algorithm
How to approximate posterior covariance in advance
I
If forward map G were linear,then
µ = N (m, C) ,
I
C = (C0−1 + G∗ Σ−1 G)−1 .
Idea: Linearization of nonlinear G at ξ 0
e
G(ξ) ≈ G(ξ)
:= G(ξ 0 ) + Jξ,
J = ∇G(ξ 0 )
yields approximation to posterior covariance
e = (C −1 + J ∗ Σ−1 J)−1 .
C≈C
0
I
Good choice for ξ 0 might be maximum a posteriori estimate:
−1/2
ξ MAP = argmin |p − G(ξ)|2Σ−1 + kC0 ξk2 .
ξ
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The gpCN-Metropolis algorithm
Generalized pCN-proposals
I
Class of admissible proposal covariances:
CΓ = (C0−1 + Γ)−1 ,
I
Γ positive, self-adjoint and bounded
Define generalized pCN-proposal kernel as
PΓ (ξ) = N (AΓ ξ, s2 CΓ ),
cf. operator weighted proposals [Law, 2013] and [Cui et al., 2014]
I
Enforcing µ0 -reversibility of PΓ – as for pCN-proposal P0 – yields
1/2
AΓ = C 0
I
p
−1/2
I − s2 (I + HΓ )−1 C0 ,
Boundedness of AΓ on H not obvious
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1/2
1/2
HΓ := C0 ΓC0 .
The gpCN-Metropolis algorithm
Generalized pCN-proposals
I
Class of admissible proposal covariances:
CΓ = (C0−1 + Γ)−1 ,
I
Γ positive, self-adjoint and bounded
Define generalized pCN-proposal kernel as
PΓ (ξ) = N (AΓ ξ, s2 CΓ ),
cf. operator weighted proposals [Law, 2013] and [Cui et al., 2014]
I
Enforcing µ0 -reversibility of PΓ – as for pCN-proposal P0 – yields
1/2
AΓ = C 0
I
p
−1/2
I − s2 (I + HΓ )−1 C0 ,
Boundedness of AΓ on H not obvious
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1/2
1/2
HΓ := C0 ΓC0 .
The gpCN-Metropolis algorithm
Generalized pCN-Metropolis
gpCN Metropolis
The Metropolis algorihtm with the gpCN-proposal kernel
PΓ (ξ) = N (AΓ ξ, s2 CΓ ),
where s ∈ (0, 1), and the acceptance probability
α(ξ, η) = min {1, exp(Φ(ξ) − Φ(η))}
is well-defined and yields a µ-reversible Markov chain in H .
It is called gpCN Metropolis and its Metropolis kernel denoted by MΓ .
Note, for Γ = 0 we recover the pCN Metropolis algorithm with kernel M0 .
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The gpCN-Metropolis algorithm
Numerical experiment
I
Setting
dp
dx (x)
I
1D model:
d
dx
I
Observations:
4
p = p(0.2j) j=1 + ε,
I
Prior:
κ Brownian bridge on [0, 1], i.e.,
eκ(x)
√
= 0,
p(0) = 0, p(1) = 2
ε ∼ N (0, σε2 I)
M
2 X
ξm (ω) sin(mπx),
ξm ∼ N (0, m−2 )
π m=1
R1
Quantity of interest: f (ξ) = 0 exp(κ(x, ξ)) dx
κ(x, ξ(ω)) ≈
I
I
Proposals for MH-MCMC
I
Results
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The gpCN-Metropolis algorithm
Numerical experiment
I
Setting
I
Proposals for MH-MCMC
pCN:
P1 (ξ) = N (ξ, s2 C0 )
√
P2 (ξ) = N ( 1 − s2 ξ, s2 C0 )
Gauss-Newton RW (GN-RW):
P3 (ξ) = N (ξ, s2 CΓ )
gpCN:
P4 (ξ) = N (AΓ ξ, s2 CΓ )
Gaussian random walk (RW):
Γ = σε−2 JJ > ,
I
Results
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J = ∇G(ξ MAP )
The gpCN-Metropolis algorithm
Numerical experiment
I
Setting
I
Proposals for MH-MCMC
I
Results
Autocorrelation for M = 50, σε = 0.1
1
RW
pCN
GN-RW
gpCN
0.8
0.6
0.4
0.2
0
100
102
lag
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The gpCN-Metropolis algorithm
Numerical experiment
I
Setting
I
Proposals for MH-MCMC
I
Results
Autocorrelation for M = 400, σε = 0.01
1
RW
pCN
GN-RW
gpCN
0.8
0.6
0.4
0.2
0
100
102
lag
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The gpCN-Metropolis algorithm
Numerical experiment
I
Setting
I
Proposals for MH-MCMC
I
Results
Effective sample size vs. dimension
ESS
105
104
103
RW
pCN
GN-RW
gpCN
102
103
dimension M
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The gpCN-Metropolis algorithm
Numerical experiment
I
Setting
I
Proposals for MH-MCMC
I
Results
Effective sample size vs. noise variance
ESS
105
104
103
10-4
RW
pCN
GN-RW
gpCN
10-3
noise variance
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10-2
σ2ǫ
Convergence
Next
1. Problem setting and Metropolis-Hastings algorithms in Hilbert spaces
2. The gpCN-Metropolis algorithm
3. Convergence
Warwick, Feb 2016 · B. Sprungk
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Convergence
Geometric ergodicity and spectral gaps
Consider Markov chain {ξ k }k∈N0 with µ-reversible transition kernel M .
I
Kernel M is L2µ -geometrically ergodic if r > 0 exists such that
kµ − νM k kTV ≤ Cν exp(−r k)
I
∀ν :
dν
∈ L2µ (H ).
dµ
Define associated Markov operator M : L2µ → L2µ by
Z
Mf (ξ) :=
f (η) M (ξ, dη).
H
I
L2µ -spectral gap of operator M:
I
M is L2µ -geometrically ergodic iff gapµ (M) > 0.
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gapµ (M) := 1 − kM − Eµ kL2µ →L2µ
Convergence
Proving geometric ergodicity of gpCN-Metropolis
I
For pCN-Metropolis kernel M0 an L2µ -spectral gap was proven in
dµ
[Hairer et al., 2014] under certain conditions on dµ
.
0
I
Our Strategy:
Take a comparative approach and relate gapµ (MΓ ) to gapµ (M0 ) in
order to prove
0 < gapµ (M0 )
I
=⇒
0 < gapµ (MΓ ).
Idea behind:
If kernels M0 and MΓ do not “differ” too much, then (maybe) they
inherit convergence properties from each other.
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Convergence
Comparison result
Theorem (Comparison of spectral gaps)
For i = 1, 2 let Mi be µ-reversible Metropolis kernels with proposals Pi
and acceptance probability α.
Assume that
1. the associated Markov operators Mi are positive,
2. the Radon-Nikodym derivative ρ(ξ; η) :=
3. and for a β > 1 we have
R R
sup
µ(A)∈(0, 21 ]
A Ac
dP1 (ξ)
dP2 (ξ) (η)
ρβ (ξ; η) P2 (ξ, dη) µ(dξ)
< ∞,
µ(A)
Then
gapµ (M1 )2β ≤ Kβ gapµ (M2 )β−1 .
Warwick, Feb 2016 · B. Sprungk
exists,
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Convergence
Application to pCN-/gpCN-Metropolis
Lemma (Positivity of Markov operators)
The Markov operator MΓ associated to gpCN-Metropolis is positive for
any admissible Γ.
Lemma (Absolute continuity, integrability of density)
There exists a density between the pCN- and gpCN-proposal
ρΓ (ξ; η) :=
dP0 (ξ)
(η)
dPΓ (ξ)
and constants b, K < ∞ such that for β < 1 +
Z
H
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ρβΓ (ξ; η) PΓ (ξ, dη) ≤ K exp
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1
2kHΓ k
b2
2
kξk2 .
Convergence
Restricted measures and Markov chains
I
For R > 0 set BR := {ξ ∈ H : kξk < R} and define
µR (dξ) :=
I
1
1B (ξ) µ(dξ).
µ(BR ) R
For Metropolis kernel M with proposal P and acceptance probability
α set
Z
MR (ξ, dη) := αR (ξ, η)P (ξ, dη)+ 1 −
αR (ξ, ζ)P (ξ, dζ) δξ (dη)
H
where αR (ξ, η) := 1BR (η)α(ξ, η).
Lemma
If M is self-adjoint and positive on L2µ (H ), then so is MR on L2µR (H )
and
c
gapµR (MR ) ≥ gapµ (M) − sup M (ξ, BR
).
ξ∈BR
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Convergence
Restricted measures and Markov chains
I
For R > 0 set BR := {ξ ∈ H : kξk < R} and define
µR (dξ) :=
I
1
1B (ξ) µ(dξ).
µ(BR ) R
For Metropolis kernel M with proposal P and acceptance probability
α set
Z
MR (ξ, dη) := αR (ξ, η)P (ξ, dη)+ 1 −
αR (ξ, ζ)P (ξ, dζ) δξ (dη)
H
where αR (ξ, η) := 1BR (η)α(ξ, η).
Lemma
If M is self-adjoint and positive on L2µ (H ), then so is MR on L2µR (H )
and
c
gapµR (MR ) ≥ gapµ (M) − sup M (ξ, BR
).
ξ∈BR
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Convergence
Main result
Theorem (Spectral gap of gpCN-Metropolis for restricted measure)
If
gapµ (M0 ) > 0,
then for any admissible Γ and any > 0 there exists a number R < ∞
such that
kµ − µR kTV < and gapµR (MΓ,R ) > 0.
Proof employs previous lemmas and comparison theorem to show
gapµ (M0 ) > 0
Warwick, Feb 2016 · B. Sprungk
=⇒
gapµR (M0,R ) > 0
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=⇒
gapµR (MΓ,R ) > 0.
Convergence
Outlook: gpCN-Metropolis with local proposal covariance
Density ρΓ (ξ) =
I
dP0 (ξ)
dPΓ (ξ)
allows for state-dependent proposal covariances:
Let ξ 7→ Γ(ξ) be measurable and define proposal kernel
Ploc (ξ) := N (AΓ(ξ) ξ, s2 CΓ(ξ) ).
I
Following an approach as in [Beskos et al., 2008] we get
Ploc (ξ, dη) µ0 (dξ) =
ρΓ(η) (η, ξ)
Ploc (η, dξ) µ0 (dη).
ρΓ(ξ) (ξ, η)
I
Obtain µ-reversible MH algorithm with proposal Ploc and
ρΓ(ξ) (ξ, η)
αloc (ξ, η) := min 1, exp(Φ(η) − Φ(ξ))
ρΓ(η) (η, ξ)
I
Same approach works for
Warwick, Feb 2016 · B. Sprungk
√
0 (ξ) := N ( 1 − s2 ξ, s2 C
Ploc
Γ(ξ) ).
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Convergence
Numerical experiment revisited
Same problem setting as before, but now also consider “local” proposals
√
local pCN:
P5 (ξ) = N ( 1 − s2 ξ, s2 CΓ(ξ) )
local gpCN:
P6 (ξ) = N (AΓ(ξ) ξ, s2 CΓ(ξ) )
with Γ(ξ) = ∇G(ξ)> Σ−1 ∇G(ξ).
ESS
Effective sample size vs. dimension
104
pCN
gpCN
local pCN
local gpCN
103
102
dimension M
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Convergence
Numerical experiment revisited
Same problem setting as before, but now also consider “local” proposals
√
local pCN:
P5 (ξ) = N ( 1 − s2 ξ, s2 CΓ(ξ) )
local gpCN:
P6 (ξ) = N (AΓ(ξ) ξ, s2 CΓ(ξ) )
with Γ(ξ) = ∇G(ξ)> Σ−1 ∇G(ξ).
Effective sample size vs. noise variance
ESS
104
pCN
gpCN
local pCN
local gpCN
102
10-4
10-3
noise variance σ2ǫ
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10-2
Conclusions
I
Generalized pCN-Metropolis allows to beneficially employ
approximation of posterior covariance for proposal kernel
I
gpCN seems to perform independent of dimension and noise variance
I
Geometric ergodicity of gpCN-Metropolis proven by general
framework for relating spectral gaps of Metropolis algorithms
I
gpCN-Metropolis with state-dependent proposal covariance possible
Some open issues:
I
Proof of higher efficiency of gpCN-Metropolis
I
Proof and theoretical understanding of variance-independence
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Conclusions
I
Generalized pCN-Metropolis allows to beneficially employ
approximation of posterior covariance for proposal kernel
I
gpCN seems to perform independent of dimension and noise variance
I
Geometric ergodicity of gpCN-Metropolis proven by general
framework for relating spectral gaps of Metropolis algorithms
I
gpCN-Metropolis with state-dependent proposal covariance possible
Some open issues:
I
Proof of higher efficiency of gpCN-Metropolis
I
Proof and theoretical understanding of variance-independence
Warwick, Feb 2016 · B. Sprungk
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References
A. Beskos, G. Roberts, A. Stuart, and J. Voss.
MCMC methods for diffusion bridges.
Stoch. Dynam., 8(3):319–350, 2008.
S. Cotter, G. Roberts, A. Stuart, and D. White.
MCMC methods for functions: Modifying old algorithms to make them faster.
Stat. Sci., 28(3):283–464, 2013.
T. Cui, K. Law, and Y. Marzouk.
Dimension-independent likelihood-informed MCMC.
arXiv:1411.3688, 2014.
M. Hairer, A. Stuart, and S. Vollmer.
Spectral gaps for a Metropolis-Hastings algorithm in infinite dimensions.
Ann. Appl. Probab., 24(6):2455–2490, 2014.
D. Rudolf, B. Sprungk.
On a generalization of the preconditioned Crank-Nicolson Metropolis algorithm.
arXiv:1504.03461, 2015.
A. Stuart.
Inverse problems: a Bayesian perspective.
Acta Numer., 19:451–559, 2010.
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