Gibbs Sampling, Exponential Families and Orthogonal Polynomials

2008/08/28 by Persi Diaconis, Kshitij Khare, Laurent Saloff-Coste
Mathematics · #stat.ME #math.CA

paper · pdf · doi:10.1214/07-sts252

published as Statistical Science 2008, Vol. 23, No. 2, 151-178 · This paper commented in: [arXiv:0808.3855], [arXiv:0808.3856], [arXiv:0808.3859], [arXiv:0808.3861]. Rejoinder in [arXiv:0808.3864]. Published in at http://dx.doi.org/10.1214/07-STS252 the Statistical Science (http://www.imstat.org/sts/) by the Institute of Mathematical Statistics (http://www.imstat.org)

Abstract

We give families of examples where sharp rates of convergence to stationarity of the widely used Gibbs sampler are available. The examples involve standard exponential families and their conjugate priors. In each case, the transition operator is explicitly diagonalizable with classical orthogonal polynomials as eigenfunctions.

Citations

• Comment: Gibbs Sampling, Exponential Families and Orthogonal Polynomials
• Comment: Gibbs Sampling, Exponential Families, and Orthogonal Polynomials
• Comment: Lancaster Probabilities and Gibbs Sampling
• Comment: On Random Scan Gibbs Samplers
• Rejoinder: Gibbs Sampling, Exponential Families and Orthogonal Polynomials

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• Extensions of the solidarity principle of the spectral gap for Gibbs samplers to their blocked and collapsed variants