Statistical Approaches to Inverse Problems

1
Statistical 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

2
Abstract

• 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.

3
References 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

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Outline

• 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

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Outline, 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

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Inverse 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.

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Models

• 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.

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Forward 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.

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Inverse 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 ?).

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Example Geomagnetism
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Geomagetic model parametrization
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Geomagnetic inverse problem
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Geophysical 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.

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Elements 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.

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Deterministic 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.

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More 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.

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Linear 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 ?.

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Linear 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.

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Linear 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.

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Ex 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.

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Sketch 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
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Backus-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.
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Sketch Backus-Gilbert
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Example 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
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Pattern 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
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Waves 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!
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Spectrum is very Regular

• Explanation as modes, plus stellar evolutionary
  theory, predict details of spectrum
• Details confirmed in data by Deubner, 1975

Source GONG
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Oscillations 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

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Linear 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.
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Different 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
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Linear 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
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Backus-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.

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Backus-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?.

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Decision 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.

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Why 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.)

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Example 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.
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Estimators

• 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.

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Comparing Decision Rules

• Infinitely many decision rules and estimators.
• Which one to use?
• The best one!
• But what does best mean?

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Loss 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?

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Strategy

• 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).

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Minimax 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.

42
Common 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).

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Shrinkage

• 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).

44
Bias

• 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.

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Example 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

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Risk 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.

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X 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.)
48
Risk of dT
x
Pq(X lt -t)
dT
f(xq)
0
t
-t
q
-t
0
q
t
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Minimax 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).

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Bayes estimation of BNM

• Posterior density of q given x is

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Posterior Mean

• The mean of the posterior density minimizes the
  Bayes risk, when the loss is squared error

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Bayes 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.
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Bayes/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
t rp(simulation) rQ
0.5 0.08 0.33
1 0.25 0.50
2 0.55 0.67
3 0.70 0.75
4 0.77 0.80
5 0.82 0.83
t ! 1 1 1
Difference between knowing q 2 -t, t, and q
U-t, t.
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Sketch Regularization
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Consistency 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.

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Singular 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.

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Singular 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.

58
Bias 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.

59
Examples 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.

60
Minimax 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).

61
Example 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.
62
Modulus of Continuity
63
Distinguishing two models

• Data tell the difference between two models z and
  h if the L1 distance between Pz and Ph is large

64
L1 and Hellinger distances
65
Consistency 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

66
Importance 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.

67
Summary

• 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.