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Inference from inadequate and inaccurate data, I.

G Backus1

  • 1UNIVERSITY OF CALIFORNIA (LA JOLLA).

Proceedings of the National Academy of Sciences of the United States of America
|January 1, 1970
PubMed
Summary

This study presents a method for estimating unknown properties of a physical object using limited measurements. It leverages Hilbert space models and statistical likelihoods for accurate property prediction.

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Inference from Inadequate and Inaccurate Data, III.

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Area of Science:

  • Mathematical Physics
  • Statistical Inference
  • Functional Analysis

Background:

  • Physical objects often require numerous parameters for full specification.
  • Limited measurements (D) are taken for an object (E) needing more than D parameters.
  • The observer aims to estimate P additional properties of E.

Purpose of the Study:

  • To describe a method for estimating unknown properties of a physical object.
  • To handle cases where the object is modeled within a Hilbert space.
  • To incorporate prior beliefs about the model's norm (likely smaller than M).

Main Methods:

  • Modeling the physical object E using a member m(E) of a Hilbert space.
  • Assuming the Hilbert norm of m(E) is likely smaller than a known number M.
  • Treating observed and sought-after properties as continuous linear functionals on the Hilbert space (except in Section 6).

Main Results:

  • A procedure is outlined for estimating P properties based on D measurements.
  • The method is applicable when object properties are linear functionals.
  • Section 6 extends the approach to Frechet-differentiable non-linear functionals.

Conclusions:

  • The described method provides a framework for property estimation under specific modeling assumptions.
  • It offers a way to infer more information than directly measured.
  • Future work will address unbounded functionals and more general topological spaces.

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