13,459 research outputs found
Estimating the spectral gap of a trace-class Markov operator
The utility of a Markov chain Monte Carlo algorithm is, in large part,
determined by the size of the spectral gap of the corresponding Markov
operator. However, calculating (and even approximating) the spectral gaps of
practical Monte Carlo Markov chains in statistics has proven to be an extremely
difficult and often insurmountable task, especially when these chains move on
continuous state spaces. In this paper, a method for accurate estimation of the
spectral gap is developed for general state space Markov chains whose operators
are non-negative and trace-class. The method is based on the fact that the
second largest eigenvalue (and hence the spectral gap) of such operators can be
bounded above and below by simple functions of the power sums of the
eigenvalues. These power sums often have nice integral representations. A
classical Monte Carlo method is proposed to estimate these integrals, and a
simple sufficient condition for finite variance is provided. This leads to
asymptotically valid confidence intervals for the second largest eigenvalue
(and the spectral gap) of the Markov operator. In contrast with previously
existing techniques, our method is not based on a near-stationary version of
the Markov chain, which, paradoxically, cannot be obtained in a principled
manner without bounds on the spectral gap. On the other hand, it can be quite
expensive from a computational standpoint. The efficiency of the method is
studied both theoretically and empirically
Geometric ergodicity of trans-dimensional Markov chain Monte Carlo algorithms
This article studies the convergence properties of trans-dimensional MCMC
algorithms when the total number of models is finite. It is shown that, for
reversible and some non-reversible trans-dimensional Markov chains, under mild
conditions, geometric convergence is guaranteed if the Markov chains associated
with the within-model moves are geometrically ergodic. This result is proved in
an framework using the technique of Markov chain decomposition. While the
technique was previously developed for reversible chains, this work extends it
to the point that it can be applied to some commonly used non-reversible
chains. Under geometric convergence, a central limit theorem holds for ergodic
averages, even in the absence of Harris ergodicity. This allows for the
construction of simultaneous confidence intervals for features of the target
distribution. This procedure is rigorously examined in a trans-dimensional
setting, and special attention is given to the case where the asymptotic
covariance matrix in the central limit theorem is singular. The theory and
methodology herein are applied to reversible jump algorithms for two Bayesian
models: an autoregression with Laplace errors and unknown model order, and a
probit regression with variable selection
Analysis of two-component Gibbs samplers using the theory of two projections
The theory of two projections is utilized to study two-component Gibbs
samplers. Through this theory, previously intractable problems regarding the
asymptotic variances of two-component Gibbs samplers are reduced to elementary
matrix algebra exercises. It is found that in terms of asymptotic variance, the
two-component random-scan Gibbs sampler is never much worse, and could be
considerably better than its deterministic-scan counterpart, provided that the
selection probability is appropriately chosen. This is especially the case when
there is a large discrepancy in computation cost between the two components.
The result contrasts with the known fact that the deterministic-scan version
has a faster convergence rate. A modified version of the deterministic-scan
sampler that accounts for computation cost behaves similarly to the random-scan
version. As a side product, some general formulas for characterizing the
convergence rate of a possibly non-reversible or time-inhomogeneous Markov
chain in an operator theoretic framework are developed
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