6,615 research outputs found
Optimum estimation via gradients of partition functions and information measures: a statistical-mechanical perspective
In continuation to a recent work on the statistical--mechanical analysis of
minimum mean square error (MMSE) estimation in Gaussian noise via its relation
to the mutual information (the I-MMSE relation), here we propose a simple and
more direct relationship between optimum estimation and certain information
measures (e.g., the information density and the Fisher information), which can
be viewed as partition functions and hence are amenable to analysis using
statistical--mechanical techniques. The proposed approach has several
advantages, most notably, its applicability to general sources and channels, as
opposed to the I-MMSE relation and its variants which hold only for certain
classes of channels (e.g., additive white Gaussian noise channels). We then
demonstrate the derivation of the conditional mean estimator and the MMSE in a
few examples. Two of these examples turn out to be generalizable to a fairly
wide class of sources and channels. For this class, the proposed approach is
shown to yield an approximate conditional mean estimator and an MMSE formula
that has the flavor of a single-letter expression. We also show how our
approach can easily be generalized to situations of mismatched estimation.Comment: 21 pages; submitted to the IEEE Transactions on Information Theor
Mutual Information and Minimum Mean-square Error in Gaussian Channels
This paper deals with arbitrarily distributed finite-power input signals
observed through an additive Gaussian noise channel. It shows a new formula
that connects the input-output mutual information and the minimum mean-square
error (MMSE) achievable by optimal estimation of the input given the output.
That is, the derivative of the mutual information (nats) with respect to the
signal-to-noise ratio (SNR) is equal to half the MMSE, regardless of the input
statistics. This relationship holds for both scalar and vector signals, as well
as for discrete-time and continuous-time noncausal MMSE estimation. This
fundamental information-theoretic result has an unexpected consequence in
continuous-time nonlinear estimation: For any input signal with finite power,
the causal filtering MMSE achieved at SNR is equal to the average value of the
noncausal smoothing MMSE achieved with a channel whose signal-to-noise ratio is
chosen uniformly distributed between 0 and SNR
On mutual information, likelihood-ratios and estimation error for the additive Gaussian channel
This paper considers the model of an arbitrary distributed signal x observed
through an added independent white Gaussian noise w, y=x+w. New relations
between the minimal mean square error of the non-causal estimator and the
likelihood ratio between y and \omega are derived. This is followed by an
extended version of a recently derived relation between the mutual information
I(x;y) and the minimal mean square error. These results are applied to derive
infinite dimensional versions of the Fisher information and the de Bruijn
identity. The derivation of the results is based on the Malliavin calculus.Comment: 21 pages, to appear in the IEEE Transactions on Information Theor
Information Theoretic Proofs of Entropy Power Inequalities
While most useful information theoretic inequalities can be deduced from the
basic properties of entropy or mutual information, up to now Shannon's entropy
power inequality (EPI) is an exception: Existing information theoretic proofs
of the EPI hinge on representations of differential entropy using either Fisher
information or minimum mean-square error (MMSE), which are derived from de
Bruijn's identity. In this paper, we first present an unified view of these
proofs, showing that they share two essential ingredients: 1) a data processing
argument applied to a covariance-preserving linear transformation; 2) an
integration over a path of a continuous Gaussian perturbation. Using these
ingredients, we develop a new and brief proof of the EPI through a mutual
information inequality, which replaces Stam and Blachman's Fisher information
inequality (FII) and an inequality for MMSE by Guo, Shamai and Verd\'u used in
earlier proofs. The result has the advantage of being very simple in that it
relies only on the basic properties of mutual information. These ideas are then
generalized to various extended versions of the EPI: Zamir and Feder's
generalized EPI for linear transformations of the random variables, Takano and
Johnson's EPI for dependent variables, Liu and Viswanath's
covariance-constrained EPI, and Costa's concavity inequality for the entropy
power.Comment: submitted for publication in the IEEE Transactions on Information
Theory, revised versio
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