Preface
Published online by Cambridge University Press: 06 July 2010
Summary
Traditional courses for engineers in filtering and signal processing have been based on elementary linear algebra, Hilbert space theory and calculus. However, the key objective underlying such procedures is the (recursive) estimation of indirectly observed states given observed data. This means that one is discussing conditional expected values, given the observations. The correct setting for conditional expected value is in the context of measurable spaces equipped with a probability measure, and the initial object of this book is to provide an overview of required measure theory. Secondly, conditional expectation, as an inverse operation, is best formulated as a form of Bayes’ Theorem. A mathematically pleasing presentation of Bayes’ theorem is to consider processes as being initially defined under a “reference probability.” This is an idealized probability under which all the observations are independent and identically distributed. The reference probability is a much nicer measure under which to work. A suitably defined change of measure then transforms the distribution of the observations to their real world form. This setting for the derivation of the estimation and filtering results enables more general results to be obtained in a transparent way.
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- Measure Theory and FilteringIntroduction and Applications, pp. ix - xPublisher: Cambridge University PressPrint publication year: 2004