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12
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 19 (6 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.
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 5 (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
Separation cutoffs for random walk on irreducible representations, arXiv: math.PR/0703291
, 2007
"... Abstract. Random walk on the irreducible representations of the symmetric and general linear groups is studied. A separation distance cutoff is proved and the exact separation distance asymptotics are determined. A key tool is a method for writing the multiplicities in the Kronecker tensor powers of ..."
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Cited by 3 (3 self)
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Abstract. Random walk on the irreducible representations of the symmetric and general linear groups is studied. A separation distance cutoff is proved and the exact separation distance asymptotics are determined. A key tool is a method for writing the multiplicities in the Kronecker tensor powers of a fixed representation as a sum of nonnegative terms. Connections are made with the LagrangeSylvester interpolation approach to Markov chains. 1.
Commutation relations and Markov chains
"... Abstract. It is shown that the combinatorics of commutation relations is well suited for analyzing the convergence rate of certain Markov chains. Examples studied include random walk on irreducible representations, a local random walk on partitions whose stationary distribution is the Ewens distribu ..."
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Cited by 2 (0 self)
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Abstract. It is shown that the combinatorics of commutation relations is well suited for analyzing the convergence rate of certain Markov chains. Examples studied include random walk on irreducible representations, a local random walk on partitions whose stationary distribution is the Ewens distribution, and some birthdeath chains. 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 2 (1 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.
unknown title
, 2009
"... The passage time distribution for a birthanddeath chain: Strong stationary duality gives a first stochastic proof ..."
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The passage time distribution for a birthanddeath chain: Strong stationary duality gives a first stochastic proof
A SHARP ANALYSIS OF THE MIXING TIME FOR RANDOM WALK ON ROOTED TREES
, 908
"... Abstract. We define an analog of Plancherel measure for the set of rooted unlabeled trees on n vertices, and a Markov chain which has this measure as its stationary distribution. Using the combinatorics of commutation relations, we show that order n 2 steps are necessary and suffice for convergence ..."
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Abstract. We define an analog of Plancherel measure for the set of rooted unlabeled trees on n vertices, and a Markov chain which has this measure as its stationary distribution. Using the combinatorics of commutation relations, we show that order n 2 steps are necessary and suffice for convergence to the stationary distribution. 1.
DUALITY AND INTERTWINING FOR DISCRETE MARKOV KERNELS: A RELATION AND EXAMPLES
, 907
"... Abstract. 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 Sie ..."
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Abstract. 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.