Comment: Gibbs Sampling, Exponential Families and Orthogonal Polynomials
نویسندگان
چکیده
Patrizia Berti is Professor, Dipartimento di Matematica Pura ed Applicata “G. Vitali”, Universita’ di Modena e Reggio-Emilia, via Campi 213/B, 41100 Modena, Italy e-mail: [email protected]. Guido Consonni is Professor, Dipartimento di Economia Politica e Metodi Quantitativi, Universita’ di Pavia, via S. Felice 5, 27100 Pavia, Italy e-mail: [email protected]. Luca Pratelli is Professor, Accademia Navale, viale Italia 72, 57100 Livorno, Italy e-mail: [email protected]. Pietro Rigo is Professor, Dipartimento di Economia Politica e Metodi Quantitativi, Universita’ di Pavia, via S. Felice 5, 27100 Pavia, Italy e-mail: [email protected].
منابع مشابه
Gibbs Sampling, Exponential Families and Orthogonal Polynomials
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.
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1. General remarks. Let K be a reversible Markov kernel on a measurable space (S, B) with stationary distribution P. Regard K as a linear operator, for some (real) eigenvalue β j. Under mild additional conditions, (1) 4 K (s, ·) − P 2 ≤ j>0 β 2 j ϕ 2 j (s) for all s ∈ S, where ·· is total variation norm and K the-th iterate of K. Using (1) is quite natural in MCMC where information on the conve...
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It is our pleasure to congratulate the authors (hereafter DKSC) on an interesting paper that was a delight to read. While DKSC provide a remarkable collection of connections between different representations of the Markov chains in their paper, we will focus on the “running time analysis” portion. This is a familiar problem to statisticians; given a target population, how can we obtain a repres...
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