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20
Gibbs sampling, exponential families and orthogonal polynomials
 Statistical Sciences
, 2008
"... 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 ort ..."
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Cited by 40 (10 self)
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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. Key words and phrases: Gibbs sampler, running time analyses, exponential families, conjugate priors, location families, orthogonal polynomials, singular value decomposition. 1.
On times to quasistationarity for birth and death processes
, 2007
"... The purpose of this paper is to present a probabilistic proof of the wellknown result stating that the time needed by a continuoustime finite birth and death process for going from the left end to the right end of its state space is a sum of independent exponential variables whose parameters are t ..."
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Cited by 21 (3 self)
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The purpose of this paper is to present a probabilistic proof of the wellknown result stating that the time needed by a continuoustime finite birth and death process for going from the left end to the right end of its state space is a sum of independent exponential variables whose parameters are the sign reversed eigenvalues of the underlying generator with a Dirichlet condition at the right end. The exponential variables appear as fastest strong quasistationary times for successive dual processes associated to the original absorbed process. As an aftermath, we get an interesting probabilistic representation of the time marginal laws of the process in terms of “local equilibria”.
The passage time distribution for a birthanddeath chain: Strong stationary duality gives a first stochastic proof
, 2009
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Fastest mixing Markov chain on graphs with symmetries
 SIAM J. Optim
"... We show how to exploit symmetries of a graph to efficiently compute the fastest mixing Markov chain on the graph (i.e., find the transition probabilities on the edges to minimize the secondlargest eigenvalue modulus of the transition probability matrix). Exploiting symmetry can lead to significant ..."
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Cited by 15 (1 self)
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We show how to exploit symmetries of a graph to efficiently compute the fastest mixing Markov chain on the graph (i.e., find the transition probabilities on the edges to minimize the secondlargest eigenvalue modulus of the transition probability matrix). Exploiting symmetry can lead to significant reduction in both the number of variables and the size of matrices in the corresponding semidefinite program, thus enable numerical solution of largescale instances that are otherwise computationally infeasible. We obtain analytic or semianalytic results for particular classes of graphs, such as edgetransitive and distancetransitive graphs. We describe two general approaches for symmetry exploitation, based on orbit theory and blockdiagonalization, respectively. We also establish the connection between these two approaches. Key words. Markov chains, eigenvalue optimization, semidefinite programming, graph automorphism, group representation. 1
ABRUPT CONVERGENCE AND ESCAPE BEHAVIOR FOR BIRTH AND DEATH CHAINS
, 2009
"... We link two phenomena concerning the asymptotical behavior of stochastic processes: (i) abrupt convergence or cutoff phenomenon, and (ii) the escape behavior usually associated to exit from metastability. The former is characterized by convergence at asymptotically deterministic times, while the c ..."
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Cited by 8 (2 self)
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We link two phenomena concerning the asymptotical behavior of stochastic processes: (i) abrupt convergence or cutoff phenomenon, and (ii) the escape behavior usually associated to exit from metastability. The former is characterized by convergence at asymptotically deterministic times, while the convergence times for the latter are exponentially distributed. We compare and study both phenomena for discretetime birthanddeath chains on Z with drift towards zero. In particular, this includes energydriven evolutions with energy functions in the form of a single well. Under suitable drift hypotheses, we show that there is both an abrupt convergence towards zero and escape behavior in the other direction. Furthermore, as the evolutions are reversible, the law of the final escape trajectory coincides with the time reverse of the law of cutoff paths. Thus, for evolutions defined by onedimensional energy wells with sufficiently steep walls, cutoff and escape behavior are related by time inversion.
Comparison of cutoffs between lazy walks and markovian semigroups
 In preparation
, 2012
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On absorbtion times and Dirichlet eigenvalues
"... This paper gives a stochastic representation in spectral terms for the absorbtion time T of a finite Markov chain which is irreducible and reversible outside the absorbing point. This yields quantitative informations on the parameters of a similar representation due to O’Cinneide for general chains ..."
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Cited by 5 (0 self)
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This paper gives a stochastic representation in spectral terms for the absorbtion time T of a finite Markov chain which is irreducible and reversible outside the absorbing point. This yields quantitative informations on the parameters of a similar representation due to O’Cinneide for general chains admitting real eigenvalues. In the discrete time setting, if the underlying Dirichlet eigenvalues (namely the eigenvalues of the Markov transition operator restricted to the functions vanishing on the absorbing point) are nonnegative, we show that T is distributed as a mixture of sums of independent geometric laws whose parameters are successive Dirichlet eigenvalues (starting from the smallest one). The mixture weights depend on the starting law. This result leads to a probabilistic interpretation of the spectrum, in terms of strong random times and local equilibria through a simple intertwining relation. Next this study is extended to the continuous time framework, where geometric laws have to be replaced by exponential distributions having the (opposite) Dirichlet eigenvalues of the generator as parameters. Returning to the discrete time setting we consider the influence of negative eigenvalues which are given another probabilistic meaning. These results generalize results of Karlin and McGregor and Keilson for birth and death chains.
DUALITY AND INTERTWINING FOR DISCRETE MARKOV KERNELS: A RELATION AND EXAMPLES
, 2009
"... We work out some relations between duality and intertwining in the context of discrete Markov chains, fixing up the background of previous relations first established for birth and death chains and their Siegmund duals. In view of the results, the monotone properties resulting from the Siegmund du ..."
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Cited by 5 (1 self)
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We work out some relations between duality and intertwining in the context of discrete Markov chains, fixing up the background of previous relations first established for birth and death chains and their Siegmund duals. In view of the results, the monotone properties resulting from the Siegmund dual of birth and death chains are revisited in some detail, with emphasis on the non neutral Moran model. We also introduce an ultrametric type dual extending the Siegmund kernel. Finally we discuss the sharp dual, following closely the DiaconisFill study.