Advertisement

Bayesian Density Estimation

  • Sibusiso Sibisi
  • John Skilling
Conference paper
Part of the Fundamental Theories of Physics book series (FTPH, volume 70)

Abstract

We develop a fully Bayesian solution to the density estimation problem. Smoothness of the estimates f is incorporated through the integral formulation f(x) = ∫ dx′ф(x′) K(x,x′) involving an appropriately smooth kernel function K. The analysis involves integration over the underlying space of densities ф. The key to this approach lies in properly setting up a measure on this space consistent with passage to the continuum limit of continuous x. With this done, a flat prior suffices to complete a well-posed definition of the problem.

Keywords

Continuum Limit Dirichlet Form Hypothesis Space Optimal Width Bayesian Solution 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    Ferguson, T. S. (1973). A Bayesian analysis of some nonparametric problems. Ann. Statist.1, 209– 230.MathSciNetzbMATHCrossRefGoogle Scholar
  2. [2]
    Ferguson, T. S. (1974). Prior distributions on spaces of probability measures. Ann. Statist.2, 615– 629.MathSciNetzbMATHCrossRefGoogle Scholar
  3. [3]
    Good, I. J. and Gaskins, R. A. (1971). Nonparametric roughness penalties for probability densities. Biometrika, 58, 255–277.MathSciNetzbMATHCrossRefGoogle Scholar
  4. [4]
    Gu, C. (1993). Smoothing spline density estimation: a dimensionless automatic algorithm. J. Amer. Statist. Ass., 88, 495 – 504.zbMATHCrossRefGoogle Scholar
  5. [5]
    Izenman, A. J. (1991). Recent developments in nonparametric density estimation. J. Amer. Statist. Ass., 86, 205 – 224.MathSciNetzbMATHCrossRefGoogle Scholar
  6. [6]
    Neal, R. M. (1992). Bayesian mixture modelling. In Maximum Entropy and Bayesian Methods (eds C. R. Smith, G. J. Erickson and P. O. Neudorfer ), 197 – 211, Dordrecht: Kluwer.Google Scholar
  7. [7]
    Powell, M. J. D. (1987). Radial basis functions for multivariable interpolation: a review. In Algorithms for Approximation(eds J. C. Mason and M. G. Cox ), 143 – 167, Oxford: Clarendon.Google Scholar
  8. [8]
    Scott, D. W. (1992). Multivariate Density Estimation. New York: Wiley.zbMATHCrossRefGoogle Scholar
  9. [9]
    Silverman, B. W. (1985). Some aspects of the spline smoothing approach to nonparametric regression curve fitting. J. R. Statist. Soc. B, 47, 1–52.zbMATHGoogle Scholar
  10. [10]
    Silverman, B. W. (1986). Density Estimation. London: Chapman and Hall.zbMATHGoogle Scholar
  11. [11]
    Smith, A. F. M. and Makov, U. E. (1978). A quasi-Bayes sequential procedure for mixtures. J. R. Statist. Soc. B, 40, 106–112.MathSciNetzbMATHGoogle Scholar
  12. [12]
    Thompson, J. R. and Tapia, R. A. (1990). Nonparametric Function Estimation, Modeling and Simulation. Philadelphia: SIAMzbMATHGoogle Scholar
  13. [13]
    West, M. (1992). Modelling with mixtures. In Bayesian Statistics 4(eds J. M. Bernardo, J. O. Berger, A. P. Dawid and A. F. M. Smith ), 503–524, Oxford: Clarendon.Google Scholar
  14. [14]
    Whittle, P. (1958). On the smoothing of probability density functions. J. R. Statist. Soc. B, 20, 334 – 343.MathSciNetzbMATHGoogle Scholar

Copyright information

© Kluwer Academic Publishers 1996

Authors and Affiliations

  • Sibusiso Sibisi
    • 1
  • John Skilling
    • 1
  1. 1.Cavendish LaboratoryUniversity of CambridgeEngland

Personalised recommendations