Gibbs Sampling, Exponential Families and Orthogonal Polynomials

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.

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)

References in corpus (8)

• Mixing times of lozenge tiling and card shuffling Markov chains
• Renewal theory and computable convergence rates for geometrically ergodic Markov chains
• Sufficient burn-in for Gibbs samplers for a hierarchical random effects model
• Separation cut-offs for birth and death chains
• A Probablistic Origin for a New Class of Bivariate Polynomials
• Markov chain comparison
• A Note on Wetting Transition for Gradient Fields
• Characterization of the cubic exponential families by orthogonality of polynomials

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