432 research outputs found
Computational limits to nonparametric estimation for ergodic processes
A new negative result for nonparametric estimation of binary ergodic
processes is shown. I The problem of estimation of distribution with any degree
of accuracy is studied. Then it is shown that for any countable class of
estimators there is a zero-entropy binary ergodic process that is inconsistent
with the class of estimators. Our result is different from other negative
results for universal forecasting scheme of ergodic processes.Comment: submitted to IEEE trans I
Empirical processes, typical sequences and coordinated actions in standard Borel spaces
This paper proposes a new notion of typical sequences on a wide class of
abstract alphabets (so-called standard Borel spaces), which is based on
approximations of memoryless sources by empirical distributions uniformly over
a class of measurable "test functions." In the finite-alphabet case, we can
take all uniformly bounded functions and recover the usual notion of strong
typicality (or typicality under the total variation distance). For a general
alphabet, however, this function class turns out to be too large, and must be
restricted. With this in mind, we define typicality with respect to any
Glivenko-Cantelli function class (i.e., a function class that admits a Uniform
Law of Large Numbers) and demonstrate its power by giving simple derivations of
the fundamental limits on the achievable rates in several source coding
scenarios, in which the relevant operational criteria pertain to reproducing
empirical averages of a general-alphabet stationary memoryless source with
respect to a suitable function class.Comment: 14 pages, 3 pdf figures; accepted to IEEE Transactions on Information
Theor
Fundamental Limitations of Disturbance Attenuation in the Presence of Side Information
In this paper, we study fundamental limitations of disturbance attenuation of feedback systems, under the assumption that the controller has a finite horizon preview of the disturbance. In contrast with prior work, we extend Bode's integral equation for the case where the preview is made available to the controller via a general, finite capacity, communication system. Under asymptotic stationarity assumptions, our results show that the new fundamental limitation differs from Bode's only by a constant, which quantifies the information rate through the communication system. In the absence of asymptotic stationarity, we derive a universal lower bound which uses Shannon's entropy rate as a measure of performance. By means of a case-study, we show that our main bounds may be achieved
Universal Coding and Prediction on Martin-L\"of Random Points
We perform an effectivization of classical results concerning universal
coding and prediction for stationary ergodic processes over an arbitrary finite
alphabet. That is, we lift the well-known almost sure statements to statements
about Martin-L\"of random sequences. Most of this work is quite mechanical but,
by the way, we complete a result of Ryabko from 2008 by showing that each
universal probability measure in the sense of universal coding induces a
universal predictor in the prequential sense. Surprisingly, the effectivization
of this implication holds true provided the universal measure does not ascribe
too low conditional probabilities to individual symbols. As an example, we show
that the Prediction by Partial Matching (PPM) measure satisfies this
requirement. In the almost sure setting, the requirement is superfluous.Comment: 12 page
Bayesian Entropy Estimation for Countable Discrete Distributions
We consider the problem of estimating Shannon's entropy from discrete
data, in cases where the number of possible symbols is unknown or even
countably infinite. The Pitman-Yor process, a generalization of Dirichlet
process, provides a tractable prior distribution over the space of countably
infinite discrete distributions, and has found major applications in Bayesian
non-parametric statistics and machine learning. Here we show that it also
provides a natural family of priors for Bayesian entropy estimation, due to the
fact that moments of the induced posterior distribution over can be
computed analytically. We derive formulas for the posterior mean (Bayes' least
squares estimate) and variance under Dirichlet and Pitman-Yor process priors.
Moreover, we show that a fixed Dirichlet or Pitman-Yor process prior implies a
narrow prior distribution over , meaning the prior strongly determines the
entropy estimate in the under-sampled regime. We derive a family of continuous
mixing measures such that the resulting mixture of Pitman-Yor processes
produces an approximately flat prior over . We show that the resulting
Pitman-Yor Mixture (PYM) entropy estimator is consistent for a large class of
distributions. We explore the theoretical properties of the resulting
estimator, and show that it performs well both in simulation and in application
to real data.Comment: 38 pages LaTeX. Revised and resubmitted to JML
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