5,936 research outputs found
Lower Bounds on Exponential Moments of the Quadratic Error in Parameter Estimation
Considering the problem of risk-sensitive parameter estimation, we propose a
fairly wide family of lower bounds on the exponential moments of the quadratic
error, both in the Bayesian and the non--Bayesian regime. This family of
bounds, which is based on a change of measures, offers considerable freedom in
the choice of the reference measure, and our efforts are devoted to explore
this freedom to a certain extent. Our focus is mostly on signal models that are
relevant to communication problems, namely, models of a parameter-dependent
signal (modulated signal) corrupted by additive white Gaussian noise, but the
methodology proposed is also applicable to other types of parametric families,
such as models of linear systems driven by random input signals (white noise,
in most cases), and others. In addition to the well known motivations of the
risk-sensitive cost function (i.e., the exponential quadratic cost function),
which is most notably, the robustness to model uncertainty, we also view this
cost function as a tool for studying fundamental limits concerning the tail
behavior of the estimation error. Another interesting aspect, that we
demonstrate in a certain parametric model, is that the risk-sensitive cost
function may be subjected to phase transitions, owing to some analogies with
statistical mechanics.Comment: 28 pages; 4 figures; submitted for publicatio
Meta learning of bounds on the Bayes classifier error
Meta learning uses information from base learners (e.g. classifiers or
estimators) as well as information about the learning problem to improve upon
the performance of a single base learner. For example, the Bayes error rate of
a given feature space, if known, can be used to aid in choosing a classifier,
as well as in feature selection and model selection for the base classifiers
and the meta classifier. Recent work in the field of f-divergence functional
estimation has led to the development of simple and rapidly converging
estimators that can be used to estimate various bounds on the Bayes error. We
estimate multiple bounds on the Bayes error using an estimator that applies
meta learning to slowly converging plug-in estimators to obtain the parametric
convergence rate. We compare the estimated bounds empirically on simulated data
and then estimate the tighter bounds on features extracted from an image patch
analysis of sunspot continuum and magnetogram images.Comment: 6 pages, 3 figures, to appear in proceedings of 2015 IEEE Signal
Processing and SP Education Worksho
Conservative classical and quantum resolution limits for incoherent imaging
I propose classical and quantum limits to the statistical resolution of two
incoherent optical point sources from the perspective of minimax parameter
estimation. Unlike earlier results based on the Cram\'er-Rao bound, the limits
proposed here, based on the worst-case error criterion and a Bayesian version
of the Cram\'er-Rao bound, are valid for any biased or unbiased estimator and
obey photon-number scalings that are consistent with the behaviors of actual
estimators. These results prove that, from the minimax perspective, the
spatial-mode demultiplexing (SPADE) measurement scheme recently proposed by
Tsang, Nair, and Lu [Phys. Rev. X 6, 031033 (2016)] remains superior to direct
imaging for sufficiently high photon numbers.Comment: 12 pages, 2 figures. v2: focused on imaging, cleaned up the math,
added new analytic and numerical results. v3: restructured and submitte
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