Prior probability and uncertainty

Abstract

Consider a stochastic system with output <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Y</tex> whose probability density <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">(pY \mid \Lambda)</tex> is a known function of the parameter <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\Lambda</tex> whose true value is unknown. Our aim is to assign a prior probability density <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> b(\Lambda)</tex> for <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\Lambda</tex> using all the available knowledge so that we can assess the probabilistic behavior of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Y</tex> from the corresponding marginal density <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">q(Y)</tex> . Usually the range of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\Lambda</tex> is known. In addition, by considering the distribution of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\Lambda</tex> in similar systems, we can define the density <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">f(\Lambda)</tex> , about which we may have some knowledge such as <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">E(\Lambda _ 1 ^ {2}) \geq 1/2</tex> , etc. We derive an expression for the uncertainty functional <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\phi(\cdot)</tex> involving <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">f(\Lambda) </tex> and <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">b(\Lambda)</tex> to quantify the discrepancy between the actual behavior of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Y</tex> and our assessment of its behavior. We pose a two-person zero-sum game with <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\phi</tex> as the payoff function so that <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">b(\Lambda)</tex> is chosen by us to minimize <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\phi</tex> , whereas <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">f(\Lambda)</tex> is chosen by nature to maximize <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\phi</tex> . We derive an asymptotic expression for the prior density, work out a few examples, and discuss the advantages of this method over others.

Cite this paper

@article{kashyap1971prior,
  title = {{Prior probability and uncertainty}},
  author = {Kashyap, R.L.},
  journal = {IEEE Transactions on Information Theory},
  volume = {17},
  number = {6},
  pages = {641--650},
  year = {1971},
  doi = {10.1109/tit.1971.1054725}
}

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