EPFL
 Biomedical Imaging GroupSTI
EPFL
  Publications
English only   BIG > Publications > Bayesian Interpolators


 CONTENTS
 Home Page
 News & Events
 People
 Publications
 Tutorials and Reviews
 Research
 Demos
 Download Algorithms

 DOWNLOAD
 PDF
 Postscript
 All BibTeX References

On the Linearity of Bayesian Interpolators for Non-Gaussian Continuous-Time AR(1) Processes

A. Amini, P. Thévenaz, J.P. Ward, M. Unser

IEEE Transactions on Information Theory, vol. 59. no. 8, pp. 5063-5074, August 2013.



Bayesian estimation problems involving Gaussian distributions often result in linear estimation techniques. Nevertheless, there are no general statements as to whether the linearity of the Bayesian estimator is restricted to the Gaussian case. The two common strategies for non-Gaussian models are either finding the best linear estimator or numerically evaluating the Bayesian estimator by Monte Carlo methods. In this paper, we focus on Bayesian interpolation of non-Gaussian first-order autoregressive (AR) processes where the driving innovation can admit any symmetric infinitely divisible distribution characterized by the Lévy-Khintchine representation theorem. We redefine the Bayesian estimation problem in the Fourier domain with the help of characteristic forms. By providing analytic expressions, we show that the optimal interpolator is linear for all symmetric α-stable distributions. The Bayesian interpolator can be expressed in a convolutive form where the kernel is described in terms of exponential splines. We also show that the limiting case of Lévy-type AR(1) processes, the system of which has a pole at the origin, always corresponds to a linear Bayesian interpolator made of a piecewise linear spline, irrespective of the innovation distribution. Finally, we show the two mentioned cases to be the only ones within the family for which the Bayesian interpolator is linear.


@ARTICLE(http://bigwww.epfl.ch/publications/amini1303.html,
AUTHOR="Amini, A. and Th{\'{e}}venaz, P. and Ward, J.P. and Unser, M.",
TITLE="On the Linearity of {B}ayesian Interpolators for Non-{G}aussian
        Continuous-Time {AR(1)} Processes",
JOURNAL="{IEEE} Transactions on Information Theory",
YEAR="2013",
volume="59",
number="8",
pages="5063--5074",
month="August",
note="")

© 2013 IEEE. Personal use of this material is permitted. However, permission to reprint/republish this material for advertising or promotional purposes or for creating new collective works for resale or redistribution to servers or lists, or to reuse any copyrighted component of this work in other works must be obtained from IEEE.
This material is presented to ensure timely dissemination of scholarly and technical work. Copyright and all rights therein are retained by authors or by other copyright holders. All persons copying this information are expected to adhere to the terms and constraints invoked by each author's copyright. In most cases, these works may not be reposted without the explicit permission of the copyright holder.