Title: Statistical Approaches to Inverse Problems
1Statistical Approaches to Inverse Problems
- DIIG seminars on inverse problems Insight and
Algorithms - Niels Bohr InstituteCopenhagen, Denmark27-29
May 2002(Revised 13 May 2003) - P.B. Stark
- Department of Statistics
- University of California
- Berkeley, CA 94720-3860
- www.stat.berkeley.edu/stark
2Abstract
- It is useful to distinguish between the intrinsic
uncertainty of an inverse problem and the
uncertainty of applying any particular technique
to solve the inverse problem. The intrinsic
uncertainty depends crucially on the prior
constraints on the unknown (including prior
probability distributions in the case of Bayesian
analyses), on the forward operator, on the
statistics of the observational errors, and on
the nature of the properties of the unknown one
wishes to estimate. I will try to convey some
geometrical intuition for uncertainty, and the
relationship between the intrinsic uncertainty of
linear inverse problems and the uncertainty of
some common techniques applied to them.
3References Acknowledgements
- Donoho, D.L., 1994. Statistical Estimation and
Optimal Recovery, Ann. Stat., 22, 238-270. - Donoho, D.L., 1995. Nonlinear solution of linear
inverse problems by wavelet-vaguelette
decomposition, Appl. Comput. Harm. Anal.,2,
101-126. - Evans, S.N. and Stark, P.B., 2002. Inverse
Problems as Statistics, Inverse Problems, 18,
R1-R43 (in press). - Stark, P.B., 1992. Inference in
infinite-dimensional inverse problems
Discretization and duality, J. Geophys. Res., 97,
14,055-14,082. - Stark, P.B., 1992. Minimax confidence intervals
in geomagnetism, Geophys. J. Intl., 108, 329-338. - Created using TexPoint by G. Necula,
http//raw.cs.berkeley.edu/texpoint
4Outline
- Inverse Problems as Statistics
- Ingredients Models
- Forward and Inverse Problemsapplied perspective
- Statistical point of view
- Some connections
- Notation linear problems illustration
- Example geomagnetism from satellite observations
- Example seismic velocity from t(p) and x(p)
- Example differential rotation of the Sun from
normal mode splitting - Identifiability and uniqueness
- Sketch of identifiablity and extremal modeling
- Backus-Gilbert theory
- Example solar differential rotation
- Example seismic velocity in Earths core
5Outline, contd.
- Decision Theory
- Decision rules and estimators
- Comparing decision rules Loss and Risk
- Strategies Bayes/Minimax duality
- Mean distance error and bias
- Illustration Regularization
- Illustration Minimax estimation of linear
functionals - Example Gauss coefficients of the magnetic field
- Distinguishing Models metrics and consistency
6Inverse Problems as Statistics
- Measurable space X of possible data.
- Set ? of possible descriptions of the
worldmodels. - Family P Pq q 2 Q of probability
distributions on X, indexed by models ?. - Forward operator q a Pq maps model ? into a
probability measure on X . - Data X are a sample from Pq.
- Pq is whole story stochastic variability in the
truth, contamination by measurement error,
systematic error, censoring, etc.
7Models
- Set ? usually has special structure.
- ? could be a convex subset of a separable Banach
space T. (geomag, seismo, grav, MT, ) - Physical significance of ? generally gives qaPq
reasonable analytic properties, e.g., continuity.
8Forward Problems in Geophysics
- Composition of steps
- transform idealized description of Earth into
perfect, noise-free, infinite-dimensional data
(approximate physics) - censor perfect data to retain only a finite list
of numbers, because can only measure, record, and
compute with such lists - possibly corrupt the list with measurement error.
- Equivalent to single-step procedure with
corruption on par with physics, and mapping
incorporating the censoring.
9Inverse Problems
- Observe data X drawn from distribution P? for
some unknown ???. (Assume ? contains at least two
points otherwise, data superfluous.) - Use data X and the knowledge that ??? to learn
about ? for example, to estimate a parameter
g(?) (the value g(?) at ? of a continuous
G-valued function g defined on ?).
10Example Geomagnetism
11Geomagetic model parametrization
12Geomagnetic inverse problem
13Geophysical Inverse Problems
- Inverse problems in geophysics often solved
using applied math methods for Ill-posed problems
(e.g., Tichonov regularization, analytic
inversions) - Those methods are designed to answer different
questions can behave poorly with data (e.g., bad
bias variance) - Inference ? construction statistical viewpoint
more appropriate for interpreting geophysical
data.
14Elements of the Statistical View
- Distinguish between characteristics of the
problem, and characteristics of methods used to
draw inferences. - One fundamental property of a parameter
- g is identifiable if for all ?, z ? T,
- g(?) ? g(z) ? Ph ? Pz.
- In most inverse problems, g(?) ? not
identifiable, and few linear functionals of ? are
identifiable.
15Deterministic and Statistical Connections
- Identifiabilitydistinct parameter values yield
distinct probability distributions for the
observablessimilar to uniquenessforward
operator maps at most one model into the observed
data. - Consistencyparameter can be estimated with
arbitrary accuracy as the number of data
growsrelated to stability of a recovery
algorithmsmall changes in the data produce small
changes in the recovered model. - ? quantitative connections too.
16More Notation
- Let T be a separable Banach space, T its
normed dual. - Write the pairing between T and T
- lt, gt T x T ? R.
17Linear Forward Problems
- A forward problem is linear if
- T is a subset of a separable Banach space T
- X Rn, X (Xj)j1n
- For some fixed sequence (?j)j1n of elements of
T, - Xj h kj, q i ej, q 2 Q,
- where e (ej)j1n is a vector of stochastic
errors whose distribution does not depend on ?.
18Linear Forward Problems, contd.
- Linear functionals ?j are the representers
- Distribution P? is the probability distribution
of X. Typically, dim(T) ? at least, n lt
dim(T), so estimating ? is an underdetermined
problem. - Define
- K T ? Rn
- q ? (lt?j, ?gt)j1n .
- Abbreviate forward problem by X K? e, ? ? T.
19Linear Inverse Problems
- Use X K? e, and the constraint ? ? T to
estimate or draw inferences about g(?). - Probability distribution of X depends on ? only
through K?, so if there are two points - ?1, ?2 ? T such that K?1 K?2 but
- g(?1)?g(?2),
- then g(?) is not identifiable.
20Ex Sampling w/ systematic and random error
- Observe
- Xj f(tj) rj ej, j 1, 2, , n,
- f 2 C, a set of smooth of functions on 0, 1
- tj 2 0, 1
- rj? 1, j1, 2, , n
- ?j iid N(0, 1)
- Take Q C -1, 1n, X Rn, and q (f, r1, ,
rn). - Then Pq has density
- (2p)-n/2 exp-åj1n (xj f(tj)-rj)2.
21Sketch Identifiability
Pz Ph h z, so q not identifiable
g cannot be estimated with bounded bias
Pz Ph g(h) g(z), so g not identifiable
22Backus-Gilbert Theory
Let Q T be a Hilbert space. Let g 2 T T be
a linear parameter. Let kjj1n µ T. Then g(q)
is identifiable iff g L K for some 1 n
matrix L. If also Ee 0, then L X is
unbiased for g. If also e has covariance matrix S
EeeT, then the MSE of L X is L S LT.
23Sketch Backus-Gilbert
24Example Differential solar rotation
- Stellar oscillations known since late 1700s.
- Sun's oscillation observed in 1960 by Leighton,
Noyes, Simon. Explained as trapped acoustic waves
by Ulrich, Leibacher, Stein, 1970-1.
Source SOHO-SOI/MDI website
- gt107 modes predicted. gt250,000 identified 106
soon
Formal error bars inflated by 200. Hill et al.,
1996. Science 272, 1292-1295
25Pattern is Superposition of Modes
- Like vibrations of a spherical guitar string
- 3 quantum numbers l, m, n
- l and m are spherical surface wavenumbers
- n is radial wavenumber
Source GONG website
26Waves Trapped in Waveguide
- Low l modes sample more deeply
- p-modes do not sample core well
- Sun essentially opaque to EM transparent to
sound to neutrinos
Source forgotten!
27Spectrum is very Regular
- Explanation as modes, plus stellar evolutionary
theory, predict details of spectrum - Details confirmed in data by Deubner, 1975
Source GONG
28Oscillations Taste Solar Interior
- Frequencies sensitive to material properties
- Frequencies sensitive to differential rotation
- If Sun were spherically symmetric and did not
rotate, frequencies of the 2l1 modes with the
same l and n would be equal - Asphericity and rotation break the degeneracy
(Scheiner measured 27d equatorial rotation from
sunspots by 1630. Polar 33d.) - Like ultrasound for the Sun
29Linear forward problem for differential rotation
Dnlmn s klmnW(r,q)r dr dq Language change q
is latitude, W is rotation model. Relationship
assumes eigenfunctions and radial structure
known. Observational errors usually assumed to
be zero-mean independent normal random variables
with known variances.
30Different Modes sample Sun differently
Left raypath for l100, n8 and l2, n8
p-modesRight raypath for l5, n10 g-mode.
g-modes have not been observed
l20 modes. Left m20. Middle m16. (Doppler
velocities) Right section through eigenfunction
of l20, m16, n 14.
Gough et al., 1996. Science 272, 1281-1283
31Linear Combinations of Splitting Kernels
Cuts through kernels for rotationA l15, m8.
B l28, m14. C l28, m24.D two targeted
combinations 0.7R, 60o 0.82R, 30oThompson et
al., 1996. Science 272, 1300-1305.
Estimated rotation rate as a function of depth at
three latitudes.Source SOHO-SOI/MDI website
32Backus-Gilbert Necessary conditions
- Let g be an identifiable real-valued parameter.
Suppose ? ?0?T, a symmetric convex set T ? T,
c?R, and g T ? R such that - ?0 T ? T
- For t ?T, g(?0 t) c g(t), and g(-t)
-g(t) - g(a1t1 a2t2) a1g(t1) a2g(t2), t1, t2 ? T,
a1, a2 ? 0, a1a2 1, and - supt ? T g(t) lt?.
- Then ? 1n matrix ? s.t. the restriction of g to
T is the restriction of ?.K to T.
33Backus-Gilbert Sufficient Conditions
- Suppose g (gi)i1m is an Rm-valued parameter
that can be written as the restriction to T of
?.K for some mn matrix ?. - Then
- g is identifiable.
- If Ee 0, ?.X is an unbiased estimator of g.
- If, in addition, e has covariance matrix S
EeeT, the covariance matrix of ?.X is ?.S.?T
whatever be P?.
34Decision Rules
- A (randomized) decision rule
- d X ? M1(A)
- x ? dx(.),
- is a measurable mapping from the space X of
possible data to the collection M1(A) of
probability distributions on a separable metric
space A of actions. - A non-randomized decision rule is a randomized
decision rule that, to each x ?X, assigns a unit
point mass at some value - a a(x) ? A.
35Why randomized rules?
- In some problems, have better behavior.
- Allowing randomized rules can make the set of
decisions convex (by allowing mixtures of
different decisions), which makes the math
easier. - If the risk is convex, Rao-Blackwell theorem says
that the optimal decision is not randomized.
(More on this later.)
36Example randomization natural
- Coin has chance 1/3 of landing with one side
showing chance 2/3 of the other showing. Dont
know which side is which. - Want to decide whether P(heads) 1/3 or 2/3.
- Toss coin 10 times. X heads.
- Toss fair coin once. U heads.
Use data to pick the more likely scenario, but if
data dont help, decide by tossing a fair coin.
37Estimators
- An estimator of a parameter g(?) is a decision
rule d for which the space A of possible actions
is the space G of possible parameter values. - gg(X) is common notation for an estimator of
g(?). - Usually write non-randomized estimator as a
G-valued function of x instead of a M1(G)-valued
function.
38Comparing Decision Rules
- Infinitely many decision rules and estimators.
- Which one to use?
- The best one!
- But what does best mean?
39Loss and Risk
- 2-player game Nature v. Statistician.
- Nature picks ? from T. ? is secret, but
statistician knows T. - Statistician picks d from a set D of rules. d is
secret. - Generate data X from P?, apply d.
- Statistician pays loss L(?, d(X)). L should be
dictated by scientific context, but - Risk is expected loss r(?, d) EqL(?, d(X))
- Good rule d has small risk, but what does small
mean?
40Strategy
- Rare that one d has smallest risk 8q?Q.
- d is admissible if not dominated (if no
estimator does at least as well for every q, and
better for some q). - Minimax decision minimizes rQ(d) supq?Qr(?, d)
over d?D - Minimax risk is rQ infd 2 D rQ(d)
- Bayes decision minimizes
- rp(d) sQr(q,d)p(dq) over d?D
- for a given prior probability distribution p on
Q. - Bayes risk is rp infd 2 D rp(d).
41Minimax is Bayes for least favorable prior
Pretty generally for convex ?, D,
concave-convexlike r,
- If minimax risk gtgt Bayes risk, prior p controls
the apparent uncertainty of the Bayes estimate.
42Common Risk Mean Distance Error (MDE)
- Let dG denote the metric on G, and let g be an
estimator of g. - MDE at ? of g is
- MDE?(g, g) Eq d(g(X), g(?)).
- When metric derives from norm, MDE is called mean
norm error (MNE). - When the norm is Hilbertian, MNE2 is called mean
squared error (MSE).
43Shrinkage
- Suppose X N(q, I) with dim(q) d 3.
- X not admissible for q for squared-error loss
(Stein, 1956). - Dominated by dS(X) X(1 a/(b X2)) for
small a and big b. - James-Stein better dJS(X) X(1-a/X2), for 0
lt a 2(d-2). - Better if take positive part of shrinkage factor
dJS(X) X(1-a/X2), for 0 lt a 2(d-2).
Not minimax, but close. - Implications for Backus-Gilbert estimates of d 3
linear functionals. - 9 extensions to other distributions see Evans
Stark (1996).
44Bias
- When G is a Banach space, can define bias at ? of
g - bias?(g, g) Eq g - g(?)
- (when the expectation is well-defined).
- If bias?(g, g) 0, say g is unbiased at ? (for
g). - If g is unbiased at ? for g for every ???, say g
is unbiased for g. If such g exists, say g is
unbiasedly estimable. - If g is unbiasedly estimable then g is
identifiable.
45Example Bounded Normal Mean
- Observe X N (q, 1). Know a priori q 2 -t, t.
- Want to estimate g(q) q.
- Let f() be the standard normal density.Let F()
be the standard normal cumulative distribution
function. - Suppose we elect to use squared-error loss
- L(q, d) (q - d)2
- r(q, d) Eq L(q, d(X)) Eq (q - d(X))2
- rQ(d) supq 2 Q r(q, d) supq 2 Q Eq (q -
d(X))2 - rQ infd 2 D supq 2 Q Eq (q - d(X))2
46Risk of X for bounded normal mean
- Consider simple estimator d(X) X.
- EX q, so X is unbiased for q, and q is
unbiasedly estimable. - r(q, X) Eq (q X)2 Var(X) 1.
- Consider Bayesian prior to capture the constraint
q 2 -t, t - p U-t, t, the uniform distribution on the
interval -t, t. - rp(X) s-tt r(q, X) p(dq) s-tt 1 (2t)-1 dq
1. - In this example, frequentist risk of X equals
Bayes risk of X for uniform prior p.
47X is not the best Truncation
- Easy to find an estimator better than X from both
frequentist and Bayes perspectives. - Truncation estimate dT
dT is biased, but has smaller MSE than X,
whatever be q 2 Q. (dT is the constrained maximum
likelihood estimate.)
48Risk of dT
x
Pq(X lt -t)
dT
f(xq)
0
t
-t
q
-t
0
q
t
49Minimax Estimation of BNM
- Truncation estimate better than X, but not
minimax. - Clear that r min(1, t2) MSE(X) 1, and rQ(0)
t2. - Minimax MSE estimator is a nonlinear shrinkage
estimator. - Minimax MSE risk is t2/(1t2).
50Bayes estimation of BNM
- Posterior density of q given x is
51Posterior Mean
- The mean of the posterior density minimizes the
Bayes risk, when the loss is squared error
52Bayes Estimator is Nonlinear Shrinkage
Philip B. Stark function f bayesUnif(x, tau) f
x - (normpdf(tau - x) - normpdf(-tau-x))./(normc
df(tau-x) - normcdf(-tau-x)) return
6
X
dT
4
dp
2
0
Bayes estimator dp, t3
-2
-4
-6
-6
-4
-2
0
2
4
6
For t 3, Bayes risk rp ¼ 0.7 (by simulation)
.Minimax risk rQ 0.75.
53Bayes/Minimax Risks
Philip B. Stark nsim 20000 risk 0 tau
5 for theta -tau.01tau pred
bayesunif(theta randn(nsim,1),tau) risk risk
(pred - theta)'(pred - theta)/nsim end risk/le
ngth(-tau.01tau)
Philip B. Stark
Philip B. Stark
Difference between knowing q 2 -t, t, and q
U-t, t.
54Sketch Regularization
55Consistency of Occams Inversion
- Common approach minimize norm (or other
regularization functional) subject to mean data
misfit 1. - Sometimes called Occams Inversion (Constable,
Parker and Constable) simplest hypothesis
consistent with the data. - In many circumstances, this estimator is
inconsistent as number of data grows, greater
and greater chance that the estimator is 0.
Allowable misfit grows faster than norm of
noise-free data image. - In common situations, consistency of the general
approach requires data redundancy and averaging.
56Singular Value Decomposition, Linear Problems
- Assume Q ½ T , separable Hilbert space ejj1n
iid N(0, s2)kjj1n linearly independent - K is compact infinite-dimensional null
space.Let K ltn ! T be the adjoint operator to
K. - 9 n triples (nj, xj, lj)j1n, nj 2 T, xj 2 X
and lj 2 lt, such that - Knj lj xj,
- K xj lj nj.
- njj1n can be chosen to be orthonormal in T
xjj1n can be chosen to be orthonormal in X. - lj gt 0, 8 j. Order s.t. l1 l2 L gt 0.
- (nj, xj, lj)j1n are singular value
decomposition of K.
57Singular Value Weighting
- Can write minimum norm model that fits data
exactly as - dMN(X) åj1n lj-1 (xj X) nj.
- Write q q q? (span of nj and its
orthocomplement) - Biasq(dMN) Eq dMN(X) - q q?.
- Varq dMN Eq åj1n lj-1 (xj e) nj 2 s2
åj1n lj-2. - Components associated with small lj make variance
big noise components multiplied by lj-1. - Singular value truncation reconstruct q using
nj with lj t - dSVT åj1m lj-1 (xj X) nj, where m max k
lk gt t. - Mollifies the noise magnification but increases
bias.
58Bias of SVT
- Bias of SVT bigger than of MNE by projection of q
onto span njjm1n. - Variance of SVT smaller by s2 åj m 1n lj-2.
- With adequate prior information about q (to
control bias) can exploit bias-variance tradeoff
to reduce MSE. - SVT in family of estimators that re-weight the
singular functions in the reconstruction - dw åj1n w(lj) (xj X) nj.
- Regularization using norm penalty, with
regularization parameter l, corresponds to - w(u) u/(u2 l).
- These tend to have smaller norm smaller than
maximum likelihood estimate can be viewed as
shrinkage.
59Examples of Singular Functions
- Linear, time-invariant filters complex sinusoids
- Circular convolution sinusoids
- Band and time-limiting prolate spheroidal
wavefunctions - Main-field geomagnetism spherical harmonics,
times radial polynomials in r-1 - Abel transform Jacobi polynomials and Chebychev
polynomials - See Donoho (1995) for more examples and
references.
60Minimax Estimation of Linear parameters
- Observe X Kq e 2 Rn, with
- q 2 Q µ T, T a separable Hilbert space
- Q convex
- eii1n iid N(0,s2).
- Seek to learn about g(q) Q ! R, linear, bounded
on Q - For variety of risks (MSE, MAD, length of
fixed-length confidence interval), minimax risk
is controlled by modulus of continuity of g,
calibrated to the noise level. - Full problem no harder than hardest 1-dimensional
subproblem reduces to BNM (Donoho, 1994).
61Example Geomagnetism
- Q q 2 l2(w) ål11 wl åm-ll qlm 2 q .
- Estimate g(q) qlm.
- Symmetry of Q and linearity of K, g, let us
characterize the modulus
The problem is to maximize a linear functional of
a vector in the intersection of two ellipsoids.
In the main-field geomagnetism problem, as the
data sampling becomes more uniform over the
spherical idealization of a satellite orbit, both
the norm (prior information) and the operator K
are diagonalized by spherical harmonics.
62Modulus of Continuity
63Distinguishing two models
- Data tell the difference between two models z and
h if the L1 distance between Pz and Ph is large
64L1 and Hellinger distances
65Consistency in Linear Inverse Problems
- Xi ?i? ?i, i1, 2, 3, ???, subset of
separable Banach space?i? ? linear, bounded
on ? ?i iid ? - ? consistently estimable w.r.t. weak topology iff
?Tk, Tk Borel function of X1, . . . , Xk, s.t.
????, ??gt0, ?? ??, - limk Pq?Tk - ??gt? 0
66Importance of the Error Distribution
- µ a prob. measure on ? µa(B) µ(B-a), a? ?
- Pseudo-metric on ?
- If restriction to ? converges to metric
compatible with weak topology, can estimate ?
consistently in weak topology. - For given sequence of functionals ki, µ rougher
? consistent estimation easier.
67Summary
- Solving inverse problem means different things
to different audiences. - Statistical viewpoint is useful abstraction.
Physics in mapping ? ? P?Prior information in
constraint ???. - There is more information in the assertion q p,
with p supported on Q, than there is in the
constraint q 2 Q. - Separating model q from parameters g(q) of
interest is useful Sabatiers well posed
questions. Many interesting questions can be
answered without knowing the entire model. - Thinking about measures of performance is useful.
- Difficulty of problem ? performance of specific
method.