39 research outputs found
Optimal scaling for the transient phase of the random walk Metropolis algorithm: The mean-field limit
We consider the random walk Metropolis algorithm on with
Gaussian proposals, and when the target probability measure is the -fold
product of a one-dimensional law. In the limit , it is well known
(see [Ann. Appl. Probab. 7 (1997) 110-120]) that, when the variance of the
proposal scales inversely proportional to the dimension whereas time is
accelerated by the factor , a diffusive limit is obtained for each component
of the Markov chain if this chain starts at equilibrium. This paper extends
this result when the initial distribution is not the target probability
measure. Remarking that the interaction between the components of the chain due
to the common acceptance/rejection of the proposed moves is of mean-field type,
we obtain a propagation of chaos result under the same scaling as in the
stationary case. This proves that, in terms of the dimension , the same
scaling holds for the transient phase of the Metropolis-Hastings algorithm as
near stationarity. The diffusive and mean-field limit of each component is a
diffusion process nonlinear in the sense of McKean. This opens the route to new
investigations of the optimal choice for the variance of the proposal
distribution in order to accelerate convergence to equilibrium (see [Optimal
scaling for the transient phase of Metropolis-Hastings algorithms: The longtime
behavior Bernoulli (2014) To appear]).Comment: Published at http://dx.doi.org/10.1214/14-AAP1048 in the Annals of
Applied Probability (http://www.imstat.org/aap/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Nonasymptotic bounds on the estimation error of MCMC algorithms
We address the problem of upper bounding the mean square error of MCMC
estimators. Our analysis is nonasymptotic. We first establish a general result
valid for essentially all ergodic Markov chains encountered in Bayesian
computation and a possibly unbounded target function . The bound is sharp in
the sense that the leading term is exactly ,
where is the CLT asymptotic variance. Next, we
proceed to specific additional assumptions and give explicit computable bounds
for geometrically and polynomially ergodic Markov chains under quantitative
drift conditions. As a corollary, we provide results on confidence estimation.Comment: Published in at http://dx.doi.org/10.3150/12-BEJ442 the Bernoulli
(http://isi.cbs.nl/bernoulli/) by the International Statistical
Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm). arXiv admin
note: text overlap with arXiv:0907.491
Analysis of Langevin Monte Carlo via convex optimization
In this paper, we provide new insights on the Unadjusted Langevin Algorithm.
We show that this method can be formulated as a first order optimization
algorithm of an objective functional defined on the Wasserstein space of order
. Using this interpretation and techniques borrowed from convex
optimization, we give a non-asymptotic analysis of this method to sample from
logconcave smooth target distribution on . Based on this
interpretation, we propose two new methods for sampling from a non-smooth
target distribution, which we analyze as well. Besides, these new algorithms
are natural extensions of the Stochastic Gradient Langevin Dynamics (SGLD)
algorithm, which is a popular extension of the Unadjusted Langevin Algorithm.
Similar to SGLD, they only rely on approximations of the gradient of the target
log density and can be used for large-scale Bayesian inference
Nonasymptotic bounds on the mean square error for MCMC estimates via renewal techniques
The Nummellin’s split chain construction allows to decompose a Markov
chain Monte Carlo (MCMC) trajectory into i.i.d. "excursions". Regenerative MCMC
algorithms based on this technique use a random number of samples. They have
been proposed as a promising alternative to usual fixed length simulation [25, 33,
14]. In this note we derive nonasymptotic bounds on the mean square error (MSE)
of regenerative MCMC estimates via techniques of renewal theory and sequential
statistics. These results are applied to costruct confidence intervals. We then focus
on two cases of particular interest: chains satisfying the Doeblin condition and a geometric
drift condition. Available explicit nonasymptotic results are compared for
different schemes of MCMC simulation