393 research outputs found
Markovian stochastic approximation with expanding projections
Stochastic approximation is a framework unifying many random iterative
algorithms occurring in a diverse range of applications. The stability of the
process is often difficult to verify in practical applications and the process
may even be unstable without additional stabilisation techniques. We study a
stochastic approximation procedure with expanding projections similar to
Andrad\'{o}ttir [Oper. Res. 43 (1995) 1037-1048]. We focus on Markovian noise
and show the stability and convergence under general conditions. Our framework
also incorporates the possibility to use a random step size sequence, which
allows us to consider settings with a non-smooth family of Markov kernels. We
apply the theory to stochastic approximation expectation maximisation with
particle independent Metropolis-Hastings sampling.Comment: Published in at http://dx.doi.org/10.3150/12-BEJ497 the Bernoulli
(http://isi.cbs.nl/bernoulli/) by the International Statistical
Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm
On the ergodicity properties of some adaptive MCMC algorithms
In this paper we study the ergodicity properties of some adaptive Markov
chain Monte Carlo algorithms (MCMC) that have been recently proposed in the
literature. We prove that under a set of verifiable conditions, ergodic
averages calculated from the output of a so-called adaptive MCMC sampler
converge to the required value and can even, under more stringent assumptions,
satisfy a central limit theorem. We prove that the conditions required are
satisfied for the independent Metropolis--Hastings algorithm and the random
walk Metropolis algorithm with symmetric increments. Finally, we propose an
application of these results to the case where the proposal distribution of the
Metropolis--Hastings update is a mixture of distributions from a curved
exponential family.Comment: Published at http://dx.doi.org/10.1214/105051606000000286 in the
Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute
of Mathematical Statistics (http://www.imstat.org
Quantitative convergence rates for sub-geometric Markov chains
We provide explicit expressions for the constants involved in the
characterisation of ergodicity of sub-geometric Markov chains. The constants
are determined in terms of those appearing in the assumed drift and one-step
minorisation conditions. The result is fundamental for the study of some
algorithms where uniform bounds for these constants are needed for a family of
Markov kernels. Our result accommodates also some classes of inhomogeneous
chains.Comment: 14 page
A Region-Dependent Gain Condition for Asymptotic Stability
A sufficient condition for the stability of a system resulting from the
interconnection of dynamical systems is given by the small gain theorem.
Roughly speaking, to apply this theorem, it is required that the gains
composition is continuous, increasing and upper bounded by the identity
function. In this work, an alternative sufficient condition is presented for
the case in which this criterion fails due to either lack of continuity or the
bound of the composed gain is larger than the identity function. More
precisely, the local (resp. non-local) asymptotic stability of the origin
(resp. global attractivity of a compact set) is ensured by a region-dependent
small gain condition. Under an additional condition that implies convergence of
solutions for almost all initial conditions in a suitable domain, the almost
global asymptotic stability of the origin is ensured. Two examples illustrate
and motivate this approach
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