Department of Statistics

University of California

Berkeley, CA 94720-3860

stark@stat.berkeley.edu

National Science Foundation

Arlington, VA

11-12 June 1998

Observe data

**d = Kx + e + f**

Data vector * d*
is in

* x* is in

**K: X --> R**^{n}
is a linear operator.

* K* might be known only
approximately

* e* is an

i) the joint distribution of the components of

ii) a set

* f* is an

Seek to learn about * x*.

For example, we might wish to estimate

itself**x**a functional

of**L**, which might be linear or nonlinear (**x**:**L**)**M --> R; x |--> Lx**

Most approaches are based on approximating * X*
by a finite-dimensional subspace

Very few people worry about the
finite-dimensional approximation, but it is * absolutely
crucial*: ignoring it typically grossly understates
the uncertainty in the result by arbitrarily large multiples. The
apparent uncertainty in finite-dimensional approximations tends
to be driven by the approximation, rather than the data.

It has been quite difficult to get across to many physical scientists the message that the uncertainty within the finite parametrization tends to be far too optimistic.

Examples I have worked on include

seismic velocity in Earth's core

topography of the core-mantle boundary

seismic tomography

estimating the geomagnetic field at the core-mantle boundary

earthquake prediction and earthquake aftershocks

helioseismology

microwave cosmology

FTIR Spectroscopy

sparse signal recovery and deconvolution

In some problems, * Kx*
can be computed exactly for all

Similarly, sometimes the operator * K*
is not known exactly, for example the domain of integration can
itself depend on the data, which are known only with error.

In these approximate cases, it is sometimes
possible to bound the error in the forward computation using the
fact that * x* is in

Stark (1992) gives a fairly general treatment and
an example in helioseismology; Backus (1989) treats the case * C*
is an ellipsoid in a Hilbert space

Basic idea: seek the estimator **Â **(in
some class *A* of estimators) that does the best, in the
worst case (for the worst * x* in

**inf**_{Â in
A }**sup**_{x in
C, f in F
}**{ E****l****(Â*** d*,

for some loss functional * l*(
. , . ). (Typically, the loss is monotone increasing in |

Donoho (1994) establishes a close connection
between minimax statistical estimation and optimal deterministic
recovery of a linear functional * L *of a
model

* l*(

is of the form

**Â*** d*
= a +

with a in * R*,

This arises from two facts: (i) the difficulty of the full estimation problem can be shown to be that of the hardest one-dimensional subproblem, whose difficulty is intimately related to the modulus of continuity:

w(r; * L*,

(ii) Once the problem is reduced to one dimension, it is a parametric optimization problem: estimating the mean of a normal distribution with unit variance, and mean known to lie in an interval [-t, t]. The near-minimaxity of affine estimators follows from properties of that problem. A current topic of research is optimal variable-length confidence intervals for a bounded normal mean.

Donoho's work was extended to quadratic functionals by Donoho and Nussbaum (1990--yes, the extension appeared before the original result, thanks in part to speedy publishing in statistics).

It does not generally extend to recovery of the
entire object * x*, but Donoho (199?)
establishes such a result for sup-norm loss.

See Stark (1992a) for an application of Donoho's approach to a problem in geomagnetism.

The basic idea of "strict bounds" is to
find the range of values of some collection of functionals **{****L**_{q}**}**_{q
in Q}* *over the intersection of * C*
and the set

Agreement with the data is typically measured in
an **l**_{p}*
*norm. In that case, one often sees

**D = ****{ *** y*
in

The tolerance * t* is
chosen such that

**D = ****{ *** y*
in

with * t* given as before.
In the event the distribution of

**D = ****{ *** y*
in

The set * D* is a 1-

Because * x* is certainly
in

One then seeks for each *q *in* Q*

**L**_{q}^{-}*
= ***inf { ****L**_{q}* y
*:

By construction,

**P{ [****L**_{q}^{-},**
L**_{q}^{+}**]
contains ****L**_{q}**x****
for all ***q* in *Q*** } > **1-* ß*.

That is, the coverage probability of any
collection of intervals derived this way is simultaneous, because
all the intervals cover whenever * CD *contains

Typically, those optimization problems cannot be
solved exactly, but for a large class of problems, their values
can be bounded above and below by pairs of finite-dimensional
optimization problems (exploiting conjugate duality). The results
can be purely algebraic (they need not require a topology on * X*),
which is an advantage if the prior physical information does not
endow

finite-d approx

lower bound * |-------*|

true range of uncertainty

conservative finite-dimensional approximation

Somewhat surprisingly, there are problems in
which the exact bounds can be found by solving finite-dimensional
problems. For example, the problem of finding upper and lower
bounds (a confidence envelope) for a probability density function
from independent, indentically distributed samples, subject to
the prior information that the density has at most *k*
modes, can be solved by a finite number of finite-dimensional
linear programs, as can the problem of finding a lower confidence
bound for the number of modes of a density. See Hengartner and
Stark (1995) for details.

Directions I think hold promise for progress in the next few years include

finding optimal variable-length confidence intervals for linear functionals in linear inverse problems by studying the bounded normal mean problem, then using Donoho's results to apply them to inverse problems in separable Hilbert spaces

exploiting the geometry of the prior information, the measure of misfit, the forward mapping, and the functionals of interest, to sharpen inferences about collections of functionals

treating certain non-convex constraint sets

, such as that arising from sparsity, which is a star-shaped, combinatorial constraint**C**using the augmented Lagrangian to study problems in which the data mapping

is nonlinear**K**making one-sided inferences about convex, lower-semicontinuous functionals

(I think this is fairly straightforward)**L**using interval arithmetic to account for roundoff and numerical precision in the calculation of confidence intervals

Backus, G.E., 1989. Confidence set
inference with a prior quadratic bound, *Geophys. J., 97*,
119-150.

Backus, G.E., 1989. Trimming and
procrastination as inversion techniques, *Phys. Earth Planet.
Inter., 98*, 101-142.

Donoho, 1994. Statistical Estimation and Optimal
Recovery, *Ann. Stat., 22*, 238-270.

Donoho, D.L., 199?. Exact Asymptotic Minimax
Risk for Sup Norm Loss via Optimal Recovery, *Th. Prob. and
Rel. Fields*

Donoho, D.L. and M. Nussbaum, 1990. Minimax
Quadratic Estimation of a Quadratic Functional, *J.
Complexity, 6*, 290-323.

Genovese, C.R., P.B. Stark and M.J. Thompson, 1995.
Uncertainties for Two-Dimensional Models of Solar Rotation from
Helioseismic Eigenfrequency Splitting, *Ap. J., 443*,
843-854.

Genovese, C.R. and P.B. Stark, 1996. Data
Reduction and Statistical Consistency in Linear Inverse Problems,
*Phys. Earth Planet. Inter., 98*, 143-162.

Hengartner, N.W. and P.B. Stark, 1995.
Finite-sample confidence envelopes for shape-restricted
densities, *Ann. Stat., 23*, 525-550.

Pulliam, R.J. and P.B. Stark, 1993. Bumps on the core-mantle
boundary: Are they facts or artifacts?, *J. Geophys. Res., 98*,
1943-1956.

Pulliam, R.J. and P.B. Stark, 1994. Confidence regions for
mantle heterogeneity, *J. Geophys. Res., 99*, 6931-6943.

Stark, P.B., 1992a. Minimax Confidence
Intervals in Geomagnetism, *Geophys. J. Intl., 108*,
329-338.

Stark, P.B., 1992b. Inference in
Infinite-Dimensional Inverse Problems: Discretization and
Duality, *J. Geophys. Res., 97*, 14,055-14,082.

Stark, P.B. and N.W. Hengartner, 1993. Reproducing Earth's
kernel: Uncertainty of the shape of the core-mantle boundary from
PKP and PcP travel-times, *J. Geophys. Res., 98*,1957-1972.

Stark, P.B., 1993. Uncertainty of the COBE quadrupole
detection, *Ap. J. Lett., 408*, L73-L76.

Stark, P.B., 1995. Reply to Comment by Morelli and Dziewonski,
*J. Geophys. Res., 100*, 15,399-15,402.

©1998, P.B. Stark. All rights reserved.