Bayesian Approaches to Non- and Semiparametric Density Estimation

This paper proposes and discusses several Bayesian attempts at nonparametric and semiparametric density estimation. The main categories of these ideas are as follows: ( 1) Build a non parametric prior around a given parametric model. We look at cases where the nonparametric part of the construction is a Dirichlet process or relatives thereof. (2) Express the density as an additive expansion of orthogonal basis functions, and place priors on the coefficients. Here attention is given to a certain robust Hermite expansion around the normal distribution. Multiplicative expansions are also considered. (3) Express the unknown density as locally being of a certain parametric form, then construct suitable local likelihood functions to express information content, and place local priors on the local parameters.

[1]  Jeffrey S. Simonoff,et al.  A Penalty Function Approach to Smoothing Large Sparse Contingency Tables , 1983 .

[2]  I. Good,et al.  Density Estimation and Bump-Hunting by the Penalized Likelihood Method Exemplified by Scattering and Meteorite Data , 1980 .

[3]  Jianqing Fan,et al.  Variable Bandwidth and Local Linear Regression Smoothers , 1992 .

[4]  Tom Leonard.  Density Estimation, Stochastic Processes and Prior Information , 1978 .

[5]  D. Freedman,et al.  On inconsistent Bayes estimates of location , 1986 .

[6]  L. Kuo.  Computations of mixtures of dirichlet processes , 1986 .

[7]  R. A. Gaskins,et al.  Nonparametric roughness penalties for probability densities , 2022 .

[8]  N. Hjort.  Nonparametric Bayes Estimators Based on Beta Processes in Models for Life History Data , 1990 .

[9]  Hani Doss.  Bayesian Nonparametric Estimation of the Median; Part I: Computation of the Estimates , 1985 .

[10]  M. C. Jones,et al.  Locally parametric nonparametric density estimation , 1996 .

[11]  C. D. Kemp,et al.  Density Estimation for Statistics and Data Analysis , 1987 .

[12]  W. Sudderth,et al.  Polya Trees and Random Distributions , 1992 .

[13]  M. West.  Kernel density estimation and marginalization consistency , 1991 .

[14]  P. Lenk.  A bayesian nonparametric density estimator , 1993 .

[15]  S. Yakowitz,et al.  PARAMETRIC/NONPARAMETRIC MIXTURE DENSITY ESTIMATION WITH APPLICATION TO FLOOD‐FREQUENCY ANALYSIS1 , 1985 .

[16]  Ingram Olkin,et al.  A Semiparametric Approach to Density Estimation , 1987 .

[17]  D. W. Scott,et al.  Multivariate Density Estimation, Theory, Practice and Visualization , 1992 .

[18]  S. Dalal.  Dirichlet invariant processes and applications to nonparametric estimation of symmetric distribution functions , 1979 .

[19]  Jorma Rissanen,et al.  Density estimation by stochastic complexity , 1992, IEEE Trans. Inf. Theory.

[20]  D. Freedman,et al.  On the consistency of Bayes estimates , 1986 .

[21]  Nils Lid Hjort,et al.  Dynamic Likelihood Hazard Rate Estimation , 1993 .

[22]  Matthew P. Wand,et al.  Kernel Smoothing , 1995 .

[23]  R. M. Korwar,et al.  Contributions to the Theory of Dirichlet Processes , 1973 .

[24]  N. Hjort,et al.  Nonparametric Density Estimation with a Parametric Start , 1995 .

[25]  Peter J. Lenk,et al.  Towards a practicable Bayesian nonparametric density estimator , 1991 .

[26]  T. Ferguson.  Prior Distributions on Spaces of Probability Measures , 1974 .

[27]  C. Antoniak.  Mixtures of Dirichlet Processes with Applications to Bayesian Nonparametric Problems , 1974 .

[28]  D. Pollard,et al.  Asymptotics for minimisers of convex processes , 2011, 1107.3806.

[29]  Nils Lid Hjort.  Local Bayesian Regression , 1994 .

[30]  G. Schwarz.  Estimating the Dimension of a Model , 1978 .

[31]  T. Ferguson.  BAYESIAN DENSITY ESTIMATION BY MIXTURES OF NORMAL DISTRIBUTIONS , 1983 .

[32]  N. Hjort.  Bayesian and Empirical Bayesian Bootstrapping , 1991 .

[33]  Hani Doss.  Bayesian Nonparametric Estimation of the Median; Part II: Asymptotic Properties of the Estimates , 1985 .

[34]  Thorburn Daniel,et al.  A Bayesian approach to density estimation , 1986 .

[35]  M. West,et al.  Semiparametric estimation of parametric hazard rates , 1992 .

[36]  M. C. Jones.  Kernel density estimation when the bandwidth is large , 1993 .

[37]  T. Ferguson.  A Bayesian Analysis of Some Nonparametric Problems , 1973 .

[38]  M. Escobar,et al.  Bayesian Density Estimation and Inference Using Mixtures , 1995 .

[39]  C. Loader.  Local Likelihood Density Estimation , 1996 .

[40]  Michel Mouchart,et al.  Bayesian Analysis of Mixtures : Some results on Exact Estimability and Identification , 1992 .

[41]  M. Lavine.  More Aspects of Polya Tree Distributions for Statistical Modelling , 1992 .