Download Introduction to Bayesian methods in inverse problems

Introduction
Bayesian inversion
Computation of integration based estimates
Numerical examples
Introduction to Bayesian methods in inverse
problems
Ville Kolehmainen1
1 Department of Applied Physics,
University of Eastern Finland, Kuopio, Finland
March 4 2013 Manchester, UK.
Introduction
Bayesian inversion
Computation of integration based estimates
Contents
Introduction
Bayesian inversion
Computation of integration based estimates
Numerical examples
Numerical examples
Introduction
Bayesian inversion
Computation of integration based estimates
Numerical examples
What are inverse problems?
• Consider measurement problems; estimate the quantity of
interest x ∈ Rn from (noisy) measurement of
A(x) ∈ Rd ,
where A is known mapping.
• Inverse problems are characterized as those measurement
problems which are ill-posed:
1. The problem is non-unique (e.g., less measurements than
unknowns)
2. Solution is unstable w.r.t measurement noise and modelling
errors (e.g., model reduction, inaccurately known nuisance
parameters in the model A(x)).
Introduction
Bayesian inversion
Computation of integration based estimates
Numerical examples
An introductory example:
• Consider 2D deconvolution (image deblurring); Given noisy
and blurred image
m ∈ Rn
m = Ax + e,
the objective is to reconstruct the original image x ∈ Rn .
• Forward model x 7→ Ax implements discrete convolution
(here the convolution kernel is Gaussian blurring kernel
with std of 6 pixels).
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Left; Original image x. Right; Observed image m = Ax + e.
Introduction
Bayesian inversion
Computation of integration based estimates
Numerical examples
Example (cont.)
• Matrix A has trivial null-space null(A) = {0} ⇒ solution is
unique (i.e. ∃(AT A)−1 ).
• However, the problem is unstable (A has "unclear"
∑
null-space, i.e., ∥Aw∥ = ∥ k σk ⟨w, vk ⟩uk ∥ ≃ 0 for certain
∥w∥ = 1)
• Figure shows the least squares (LS) solution
⇒ xLS = (AT A)−1 AT m
xLS = arg min ∥m − Ax∥2
x
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Left; true image x. Right; LS solution xLS
Introduction
Bayesian inversion
Computation of integration based estimates
Numerical examples
Solutions; Regularization.
• The ill posed problem is replaced by a well posed
approximation. Solution “hopefully” close to the true
solution.
• Typically modifications of the associated LS problem
∥m − Ax∥2 .
• Examples of methods; Tikhonov regularization, truncated
iterations.
• Consider the LS problem
xLS = arg min{∥m − Ax∥22 }
x
⇒ (AT A) xLS = AT m
| {z }
B
Uniqueness;
solution?
∃B −1
if null(A) = {0}. Stability of the
Introduction
Bayesian inversion
Computation of integration based estimates
Numerical examples
Regularization
• Example (cont.); In Tikhonov regularization, the LS
problem is replaced with
xTIK = arg min{∥m−Ax∥22 +α∥x∥22 }
⇒ (AT A + αI) xTIK = AT m
{z
}
|
x
B̃
∃B̃ −1 .
Uniqueness; α > 0 ⇒ null(B̃) = {0} ⇒
Stability of
the solution guaranteed by choosing α “large enough”.
• Regularization poses (implicit) prior about the solution.
These assumptions are sometimes well hidden.
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Left; true image x. Right; Tikhonov solution xTIK .
Introduction
Bayesian inversion
Computation of integration based estimates
Numerical examples
Solutions; Statistical (Bayesian) inversion.
• The inverse problem is recast as a problem of Bayesian
inference. The key idea is to extract estimates and assess
uncertainty about the unknowns based on
• Measured data
• Model of measurement process
• Model of a priori information
• Ill-posedness removed by explicit use of prior models!
• Systematic handling of model uncertainties and reduction
(⇒ approximation error theory)
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Left; true image x. Right; Bayesian estimate xCM .
Introduction
Bayesian inversion
Computation of integration based estimates
Numerical examples
Bayesian inversion.
• All variables in the model are considered as random
variables. The randomness reflects our uncertainty of their
actual values.
• The degree of uncertainty is coded in the probability
distribution models of these variables.
• The complete model of the inverse problem is the posterior
probability distribution
π(x | m) =
πpr (x)π(m | x)
,
π(m)
m = mobserved
(1)
where
• π(m | x) is the likelihood density; model of the measurement
process.
• πpr (x) is the prior density; model for a priori information.
• π(m) is a normalization constant
Introduction
Bayesian inversion
Computation of integration based estimates
Numerical examples
• The posterior density model is a function on n dimensional
space;
π(x | m) : Rn 7→ R+
where n is usually large.
• To obtain "practical solution", the posterior model is
summarized by estimates that answer to questions such
as;
• “What is the most probable value of x over π(x | m)? ”
• “What is the mean of x over π(x | m)? ”
• “In what interval are the values of x with 90% (posterior)
probability? ”
Introduction
Bayesian inversion
Computation of integration based estimates
Typical point estimates
• Maximum a posteriori (MAP) estimate:
π(xMAP | m) = arg max π(x | m).
x
• Conditional mean (CM) estimate:
∫
xCM =
• Covariance:
Γx|m =
Rn
xπ(x | m)dx.
∫
Rn
(x − xCM )(x − xCM )T π(x | m) dx
Numerical examples
Introduction
Bayesian inversion
Computation of integration based estimates
Numerical examples
• Confidence intervals; Given 0 < τ < 100, compute ak and
bk s.t.
∫
∫
ak
−∞
∞
πk (xk ) dxk =
πk (xk ) dxk =
bk
100 − τ
200
where πk (xk ) is the marginal density
∫
πk (xk ) =
π(x | m) dx1 · · · dxk −1 dxk +1 · · · dxn .
Rn−1
The interval Ik (τ ) = [ak bk ] contains τ % of the mass of the
marginal density.
Introduction
Bayesian inversion
Computation of integration based estimates
Numerical examples
Computation of the integration based estimates
• Many estimators are of the form
∫
f̄ (x) =
Rn
f (x)π(x | m)dx
• Examples:
• f (x) = x
xCM
• f (x) = (x − xCM )(x − xCM )T
Γx|m
• etc ...
• Analytical evaluation in most cases impossible
• Traditional numerical quadratures not applicable when n is
large (number of points needed unreasonably large,
support of π(x | m) may not be well known)
• ⇒ Monte Carlo integration.
Introduction
Bayesian inversion
Computation of integration based estimates
Numerical examples
Monte Carlo integration
• Monte Carlo integration
1. Draw an ensemble {x (k ) , k = 1, 2, . . . , N} of i.i.d samples
from π(x).
2. Estimate
∫
N
1 ∑
f (x)π(x)dx ≈
f (x (k) )
N
Rn
k =1
• Convergence (law of large numbers)
∫
N
1 ∑
(k )
f (x)π(x)dx (a.c)
limN→∞
f (x ) →
N
Rn
k =1
∑N
(k ) ) reduces
• Variance of the estimator f̄ = N1
k =1 f (x
∝
1
N
Introduction
Bayesian inversion
Computation of integration based estimates
Numerical examples
Simple example of Monte Carlo integration
• Let x ∈ R2 and
{
G = [−1, 1] × [−1, 1]
x∈
/ G.
∫
The task is to compute integral g(x) = R2 χD π(x)dx where
D is the unit disk on R2 .
• Using N = 5000 samples, we get estimate g(x) = 0.7814
(true value π/4 = 0.7854)
π(x) =
x ∈ G,
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Introduction
Bayesian inversion
Computation of integration based estimates
Numerical examples
MCMC
• Direct sampling of the posterior usually not possible ⇒
Markov chain Monte Carlo (MCMC).
• MCMC is Monte Carlo integration using Markov chains
(dependent samples);
1. Draw {x (k ) , k = 1, 2, . . . , N} ∼ π(x) by simulating a Markov
chain (with equilibrium distribution π(x)).
2. Estimate
∫
N
1 ∑
f (x)π(x)dx ≈
f (x (k) )
N
Rn
k =1
• Variance of the estimator reduces as ∝ τ /N where τ is the
integrated autocorrelation time of the chain.
• Algorithms for MCMC;
1. Metropolis Hastings algorithm
2. Gibbs sampler
Introduction
Bayesian inversion
Computation of integration based estimates
Numerical examples
Metropolis-Hastings algorihtm
• Generation of the ensemble {x (k ) , k = 1, . . . , N} ∼ π(x)
using Metropolis Hastings algorithm;
1. Pick an initial value x (1) ∈ Rn and set ℓ = 1
2. Set x = x (ℓ) .
3. Draw a candidate sample x ′ from proposal density
x ′ ∼ q(x, x ′ )
and compute the acceptance factor
(
)
π(x ′ )q(x ′ , x)
α(x, x ′ ) = min 1,
.
π(x)q(x, x ′ )
4. Draw t ∈ [0, 1] from uniform probability density
(t ∼ uni(0, 1)).
5. If α(x, x ′ ) ≥ t, set x (ℓ+1) = x ′ , else x (ℓ+1) = x. Increment
ℓ → ℓ + 1.
6. When ℓ = N stop, else repeat from step 2.
Introduction
Bayesian inversion
Computation of integration based estimates
Numerical examples
• Normalization constant of π do not need to be known.
• Great flexibility in choosing the proposal density q(x, x ′ );
almost any density would do the job (eventually).
• However, the choice of q(x, x ′ ) is a crucial part of
successful MCMC; it determines the efficiency
(autocorrelation time τ ) of the algorithm ⇒ q(x, x ′ ) should
be s.t. τ as small as possible
• No systematic methods for choosing q(x, x ′ ). More based
on “intuition & art”.
Introduction
Bayesian inversion
Computation of integration based estimates
Numerical examples
• The updating may proceed
• All unknowns at a time
• Single component xj at a time
• A block of unknowns at a time
• The order of updating (in single component and blockwise
updating) may be
• Update elements chosen in random (random scan)
• Systematic order
• The proposal can be mixture of several densities
q(x, x ′ ) =
t
∑
i=1
pi qi (x, x ′ ),
t
∑
i=1
pi = 1
Introduction
Bayesian inversion
Computation of integration based estimates
Numerical examples
• Proposal parameteres are usually calibrated by pilot runs;
aim at finding parameters giving best efficiency (minimal
τ ).
• Determining the "burn-in"; Trial runs (e.g, running several
chains & checking that they are consistent). Often
determined from the plot of “posterior trace”
{π(x (k ) ), k = 1, . . . , N} (see Figure).
Introduction
Bayesian inversion
Computation of integration based estimates
Numerical examples
Numerical examples
• In summary, Bayesian solution of an inverse problem
consist of the following steps:
1. Construct the likelihood model π(m|x). This step includes
the development of the model x 7→ A(x) and modeling the
measurement noise and modelling errors.
2. Construction of the prior model πpr (x).
3. Develop computation methods for the summary statistics.
• We consider the the following numerical examples:
1. 2D deconvolution (intro example continued)
2. EIT with experimental data
Introduction
Bayesian inversion
Computation of integration based estimates
Numerical examples
Example 1; 2D deconvolution
• We are given a blurred and noisy image
m ∈ Rn
m = Ax + e,
• Assume that we know a priori that the true solution is
binary image representing some text.
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Left; true image x. Right; Noisy, blurred image m = Ax + e.
Introduction
Bayesian inversion
Computation of integration based estimates
Numerical examples
Likelihood model π(m | x):
• Joint density
π(m, x, e) = π(m | x, e)π(e | x)π(x) = π(m, e | x)π(x)
• In case of m = Ax + e, we have
π(m | x, e) = δ(m − Ax − e), and
∫
π(m | x) =
π(m, e | x) de
∫
=
δ(m − Ax − e)π(e | x) de
= πe | x (m − Ax | x)
Introduction
Bayesian inversion
Computation of integration based estimates
Numerical examples
• In the (usual) case of mutually independent x and e, we
have πe | x (e | x) = πe (e) and
π(m|x) = πe (m − Ax)
• The model of the noise: π(e) = N (0, Γe ) →,
(
))
1(
2
π(m | x) ∝ exp −
∥Le (m − Ax)∥
,
2
where LTe Le = Γ−1
e .
• Remark; we have assumed that the (numerical) model Ax
is exact (i.e., no modelling errors).
Introduction
Bayesian inversion
Computation of integration based estimates
Numerical examples
Prior model πpr (x)
• The prior knowledge:
• x can obtain the values xj ∈ {0, 1}
• Case xj = 0 (black, background), case xj = 1 (white, text).
• The pixels with value xj = 1 are known to be clustered in
"blocky structures" which have short boundaries.
• We model the prior knowledge with Ising prior

πpr (x) ∝ exp α
n ∑
∑

δ(xi , xj )
i=1 j∈Ni
where δ is the Kronecker delta
{
1, xi = xj
δ(xi , xj ) =
0, xi =
̸ xj
Introduction
Bayesian inversion
Computation of integration based estimates
Numerical examples
Posterior model π(x | m):
• Posterior model for the deblurring problem;


n ∑
∑
1
π(x|m) ∝ exp − ∥Le (m − Ax)∥22 + α
δ(xi , xj )
2
i=1 j∈Ni
• Posterior explored with the Metropolis Hastings algorithm.
Introduction
Bayesian inversion
Computation of integration based estimates
Numerical examples
Metropolis Hastings proposal
∑
• The proposal is a mixture q(x, x ′ ) =
2
′
i=1 ξi qi (x, x )
of two
move types.
Move 1: choose update element xi ∈ R by drawing
index i with uniform probability n1 and change
the value of xi .
Move 2: Let N ∗ (x) denote the set of active edges in
image x (edge lij connects pixels xi and xj in
the lattice; it is active if xi ̸= xj ). Pick an active
update edge w.p.
1
∗
|N (x)|
Pick one of the two pixels related to the
chosen edge w.p 12 and change the value of
the chosen pixel.
Introduction
Bayesian inversion
Computation of integration based estimates
Numerical examples
Results
• Left image; True image x
• Middle; Tikhonov regularized solution
xTIK = arg min{∥m − Ax∥22 + α∥x∥22 }
x
∫
• Right image; CM estimate xCM = Rn xπ(x|m)dx.
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Left; true image x, Middle; xTIK . Right; xCM .
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Introduction
Bayesian inversion
Computation of integration based estimates
Numerical examples
Results
• Left image; xCM
• Posterior variances diag(Γx|m )
• Right; A sample from π(x | m)
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Left; xCM , Middle; posterior variances diag(Γx|m ). Right; A
sample x from π(x | m).
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Introduction
Bayesian inversion
Computation of integration based estimates
Numerical examples
Example 2: "tank EIT"
• Joint work with Geoff Nicholls (Oxford) & Colin Fox (U.
Otago, NZ)
• Data measured with the Kuopio EIT device.
• 16 electrodes, adjacent current patterns.
• Target: plastic cylinders in salt water.
Measured voltages
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Introduction
Bayesian inversion
Computation of integration based estimates
Numerical examples
• Measurement model
V = U(σ) + e,
e ∼ N (e∗ , Γe )
where V denotes the measured voltages and U(σ)
FEM-based forward map.
• e and σ modelled mutually independent.
• e∗ and Γe estimated by repeated measurements.
• Posterior model
{
}
1
T −1
π(σ | V ) ∼ exp − (V − U(σ)) Γe (V − U(σ)) πpr (σ)
2
• We compute results with three different prior models πpr (σ).
• Sampling carried out with the Metropolis-Hasting sampling
algorithm.
Introduction
Bayesian inversion
Computation of integration based estimates
Numerical examples
Prior models πpr (σ)
1 S MOOTHNESS
PRIOR
(
πpr (σ) ∝ π+ (σ) exp −α
n
∑
i=1
)
Hi (σ) ,
Hi (σ) =
∑
|σi −σj |2 .
j∈Ni
2 M ATERIAL
TYPE PRIOR (F OX & N ICHOLLS ,1998)
• The possible material types {1, 2, . . . , C} inside the body
are known but the segmentation to these materials is
unknown. The material distibution is represented by an
(auxliary) dicrete valued random field τ (pixel values
τj ∈ {1, 2, . . . , C}).
• The possible values of conductivity σ(τ ) for different
materials are known only approximately. Gaussian prior for
the material conductivities.
• The different material types are assumed to be clustered in
“blocky structures” (e.g., organs, etc)
Introduction
Bayesian inversion
Computation of integration based estimates
Numerical examples
• Prior model πpr (σ) = πpr (σ|τ )πpr (τ ), where
(
πpr (σ|τ ) ∝ π+ (σ) exp −α
n
∑
)
Gi (σ)
i=1
n
∏
π̃(σi |τi )
i=1
where π̃(σi |τi ) = N (η(τi ), ξ(τi )2 ) and
∑
∑
(σi − σj )2
Gi (σ) =
i
• πpr (τ ) is an Ising prior:
(
πpr (τ ) ∝ exp β
n
∑
i=1
j∈{k |k ∈Ni andτk =τi }
)
Ti (τ ) ,
Ti (τ ) =
∑
δ(τi , τj ),
j∈Ni
where δ(τi , τj ) is the Kronecker delta function.
(2)
Introduction
Bayesian inversion
Computation of integration based estimates
Numerical examples
3 C IRCLE
PRIOR
• Domain Ω with known (constant) background conductivity
σbg is assumed to contain unknown number of circular
inclusions (i.e., dimension of state not fixed)
• The inlcusions have known contrast
• Sizes and locations of inclusions unknown
• For the number N of the circular inclusions we write a point
process prior:
πpr (σ) = β N exp(−βA)δ(σ ∈ A),
where
• β = 1/A (A is the area of the domain Ω)
• A is set of feasible circle configurations (the circle
inclusions are disjoint) and δ is the indicator function.
Introduction
Bayesian inversion
Computation of integration based estimates
Numerical examples
CM estimates with the different priors
• Smoothness prior (Top right), material type prior (Bottom
left) and circle prior (Bottom right)