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The following sections are included:

  • Formal Statement of the Problem

    • State and measurement vectors

    • The forward model

    • Weighting function matrix

    • Vector spaces

  • Linear Problems without Measurement Error

    • Subspaces of state space

    • Identifying the null space and the row space

  • Linear Problems with Measurement Error

    • Describing experimental error

    • The Bayesian approach to inverse problems

      • Bayes' theorem

      • Example: The Linear problem with Gaussian statistics

  • Degrees of Freedom

    • How many independent quantities can be measured?

    • Degrees of freedom for signal

  • Information Content of a Measurement

    • The Fisher information matrix

    • Shannon information content

      • Entropy of a probability density function

      • Entropy of a Gaussian distribution

      • Information content in the linear Gaussian case

  • The Standard Example: Information Content and Degrees of Freedom

  • Probability Density Functions and the Maximum Entropy Principle