Harris recurrence of Metropolis-within-Gibbs and trans-dimensional Markov chains

A $\phi$-irreducible and aperiodic Markov chain with stationary probability distribution will converge to its stationary distribution from almost all starting points. The property of Harris recurrence allows us to replace ``almost all'' by ``all,'' which is potentially important when running Markov chain Monte Carlo algorithms. Full-dimensional Metropolis--Hastings algorithms are known to be Harris recurrent. In this paper, we consider conditions under which Metropolis-within-Gibbs and trans-dimensional Markov chains are or are not Harris recurrent. We present a simple but natural two-dimensional counter-example showing how Harris recurrence can fail, and also a variety of positive results which guarantee Harris recurrence. We also present some open problems. We close with a discussion of the practical implications for MCMC algorithms.Comment: Published at http://dx.doi.org/10.1214/105051606000000510 in the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

Output Feedback Invariants

The paper is concerned with the problem of determining a complete set of invariants for output feedback. Using tools from geometric invariant theory it is shown that there exists a quasi-projective variety whose points parameterize the output feedback orbits in a unique way. If the McMillan degree $n\geq mp$, the product of number of inputs and number of outputs, then it is shown that in the closure of every feedback orbit there is exactly one nondegenerate system.Comment: 15 page

Adaptive Gibbs samplers

We consider various versions of adaptive Gibbs and Metropolis- within-Gibbs samplers, which update their selection probabilities (and perhaps also their proposal distributions) on the fly during a run, by learning as they go in an attempt to optimise the algorithm. We present a cautionary example of how even a simple-seeming adaptive Gibbs sampler may fail to converge. We then present various positive results guaranteeing convergence of adaptive Gibbs samplers under certain conditions

Quantitative bounds on convergence of time-inhomogeneous Markov chains

Convergence rates of Markov chains have been widely studied in recent years. In particular, quantitative bounds on convergence rates have been studied in various forms by Meyn and Tweedie [Ann. Appl. Probab. 4 (1994) 981-1101], Rosenthal [J. Amer. Statist. Assoc. 90 (1995) 558-566], Roberts and Tweedie [Stochastic Process. Appl. 80 (1999) 211-229], Jones and Hobert [Statist. Sci. 16 (2001) 312-334] and Fort [Ph.D. thesis (2001) Univ. Paris VI]. In this paper, we extend a result of Rosenthal [J. Amer. Statist. Assoc. 90 (1995) 558-566] that concerns quantitative convergence rates for time-homogeneous Markov chains. Our extension allows us to consider f-total variation distance (instead of total variation) and time-inhomogeneous Markov chains. We apply our results to simulated annealing.Comment: Published at http://dx.doi.org/10.1214/105051604000000620 in the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

Degree of the generalized Pl\"ucker embedding of a Quot scheme and Quantum cohomology

We compute the degree of the generalized Pl\"ucker embedding $\kappa$ of a Quot scheme $X$ over \PP^1. The space $X$ can also be considered as a compactification of the space of algebraic maps of a fixed degree from \PP^1 to the Grassmanian $\rm{Grass}(m,n)$. Then the degree of the embedded variety $\kappa (X)$ can be interpreted as an intersection product of pullbacks of cohomology classes from $\rm{Grass}(m,n)$ through the map $\psi$ that evaluates a map from \PP^1 at a point x\in \PP^1. We show that our formula for the degree verifies the formula for these intersection products predicted by physicists through Quantum cohomology~\cite{va92}~\cite{in91}~\cite{wi94}. We arrive at the degree by proving a version of the classical Pieri's formula on the variety $X$, using a cell decomposition of a space that lies in between $X$ and $\kappa (X)$.Comment: 18 pages, Latex documen