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Isotropic probability measures in infinite-dimensional spaces.

G Backus1

  • 1Institute of Geophysics and Planetary Physics, A-025, University of California-San Diego, La Jolla, CA 92093.

Proceedings of the National Academy of Sciences of the United States of America
|December 1, 1987
PubMed
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Every isotropic probability measure on real sequences is a combination of a point mass at zero and a specific type of measure. These measures are uniquely defined by their finite-dimensional distributions, offering a new approach to inverse problems.

Area of Science:

  • Probability Theory
  • Stochastic Processes
  • Functional Analysis

Background:

  • Isotropic probability measures on infinite-dimensional spaces are fundamental in probability theory.
  • Understanding their properties is crucial for modeling complex systems and statistical inference.
  • Existing methods for linear inverse problems face challenges with certain types of prior information.

Purpose of the Study:

  • To characterize all isotropic probability measures on the space of real sequences R(infinity).
  • To establish a unique relationship between these measures and their finite-dimensional marginal distributions.
  • To explore the implications for linear inverse problems and Bayesian inference techniques.

Main Methods:

  • Decomposition of isotropic measures into a convex combination involving a measure concentrated at zero and a set I(0)(R(infinity)).

Related Experiment Videos

  • Characterization of measures in I(0)(R(infinity)) through their finite-dimensional marginal distributions and density functions.
  • Utilizing the property of complete monotonicity of density functions and Laplace transforms for establishing bijections.

    • Main Results:

    • Every isotropic probability measure on R(infinity) is a convex combination of the zero measure and a measure in I(0)(R(infinity)).
    • Measures in I(0)(R(infinity)) are uniquely determined by their finite-dimensional marginal distributions, which possess completely monotone density functions.
    • A bijection is established between I(0)(R(infinity)) and the set of probability measures on [0, infinity) via a Laplace transform relationship.
    • Isotropic measures in I(0)(R(infinity)) assign zero probability to the space of square-summable sequences.

    Conclusions:

    • The study provides a comprehensive characterization of isotropic probability measures on R(infinity).
    • The established bijection offers a novel way to represent and analyze these measures.
    • Linear inverse problems with prior information solely based on the inequality Sigma x(i)^2 <= 1 are unsuitable for stochastic inversion and Bayesian inference.