11 research outputs found
Error bounds of MCMC for functions with unbounded stationary variance
We prove explicit error bounds for Markov chain Monte Carlo (MCMC) methods to
compute expectations of functions with unbounded stationary variance. We assume
that there is a so that the functions have finite -norm. For
uniformly ergodic Markov chains we obtain error bounds with the optimal order
of convergence and if there exists a spectral gap we almost get the
optimal order. Further, a burn-in period is taken into account and a recipe for
choosing the burn-in is provided.Comment: 13 page
Time series prediction via aggregation : an oracle bound including numerical cost
We address the problem of forecasting a time series meeting the Causal
Bernoulli Shift model, using a parametric set of predictors. The aggregation
technique provides a predictor with well established and quite satisfying
theoretical properties expressed by an oracle inequality for the prediction
risk. The numerical computation of the aggregated predictor usually relies on a
Markov chain Monte Carlo method whose convergence should be evaluated. In
particular, it is crucial to bound the number of simulations needed to achieve
a numerical precision of the same order as the prediction risk. In this
direction we present a fairly general result which can be seen as an oracle
inequality including the numerical cost of the predictor computation. The
numerical cost appears by letting the oracle inequality depend on the number of
simulations required in the Monte Carlo approximation. Some numerical
experiments are then carried out to support our findings
Dimension-Independent MCMC Sampling for Inverse Problems with Non-Gaussian Priors
The computational complexity of MCMC methods for the exploration of complex
probability measures is a challenging and important problem. A challenge of
particular importance arises in Bayesian inverse problems where the target
distribution may be supported on an infinite dimensional space. In practice
this involves the approximation of measures defined on sequences of spaces of
increasing dimension. Motivated by an elliptic inverse problem with
non-Gaussian prior, we study the design of proposal chains for the
Metropolis-Hastings algorithm with dimension independent performance.
Dimension-independent bounds on the Monte-Carlo error of MCMC sampling for
Gaussian prior measures have already been established. In this paper we provide
a simple recipe to obtain these bounds for non-Gaussian prior measures. To
illustrate the theory we consider an elliptic inverse problem arising in
groundwater flow. We explicitly construct an efficient Metropolis-Hastings
proposal based on local proposals, and we provide numerical evidence which
supports the theory.Comment: 26 pages, 7 figure
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
Information Geometry Approach to Parameter Estimation in Markov Chains
We consider the parameter estimation of Markov chain when the unknown
transition matrix belongs to an exponential family of transition matrices.
Then, we show that the sample mean of the generator of the exponential family
is an asymptotically efficient estimator. Further, we also define a curved
exponential family of transition matrices. Using a transition matrix version of
the Pythagorean theorem, we give an asymptotically efficient estimator for a
curved exponential family.Comment: Appendix D is adde
Spectral gaps for a Metropolis-Hastings algorithm in infinite dimensions
We study the problem of sampling high and infinite dimensional target
measures arising in applications such as conditioned diffusions and inverse
problems. We focus on those that arise from approximating measures on Hilbert
spaces defined via a density with respect to a Gaussian reference measure. We
consider the Metropolis-Hastings algorithm that adds an accept-reject mechanism
to a Markov chain proposal in order to make the chain reversible with respect
to the target measure. We focus on cases where the proposal is either a
Gaussian random walk (RWM) with covariance equal to that of the reference
measure or an Ornstein-Uhlenbeck proposal (pCN) for which the reference measure
is invariant. Previous results in terms of scaling and diffusion limits
suggested that the pCN has a convergence rate that is independent of the
dimension while the RWM method has undesirable dimension-dependent behaviour.
We confirm this claim by exhibiting a dimension-independent Wasserstein
spectral gap for pCN algorithm for a large class of target measures. In our
setting this Wasserstein spectral gap implies an -spectral gap. We use
both spectral gaps to show that the ergodic average satisfies a strong law of
large numbers, the central limit theorem and nonasymptotic bounds on the mean
square error, all dimension independent. In contrast we show that the spectral
gap of the RWM algorithm applied to the reference measures degenerates as the
dimension tends to infinity.Comment: Published in at http://dx.doi.org/10.1214/13-AAP982 the Annals of
Applied Probability (http://www.imstat.org/aap/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Rigorous confidence bounds for MCMC under a geometric drift condition
We assume a drift condition towards a small set and bound the mean square error of estimators obtained by taking averages along a single trajectory of a Markov chain Monte Carlo algorithm. We use these bounds to construct fixed-width nonasymptotic confidence intervals. For a possibly unbounded function f:XâR, let be the value of interest and its MCMC estimate. Precisely, we derive lower bounds for the length of the trajectory n and burn-in time t which ensure that
The bounds depend only and explicitly on drift parameters, on the V-norm of f, where V is the drift function and on precision and confidence parameters . Next we analyze an MCMC estimator based on the median of multiple shorter runs that allows for sharper bounds for the required total simulation cost. In particular the methodology can be applied for computing posterior quantities in practically relevant models. We illustrate our bounds numerically in a simple example