235 research outputs found
Explicit error bounds for lazy reversible Markov Chain Monte Carlo
We prove explicit, i.e., non-asymptotic, error bounds for Markov Chain Monte
Carlo methods, such as the Metropolis algorithm. The problem is to compute the
expectation (or integral) of f with respect to a measure which can be given by
a density with respect to another measure. A straight simulation of the desired
distribution by a random number generator is in general not possible. Thus it
is reasonable to use Markov chain sampling with a burn-in. We study such an
algorithm and extend the analysis of Lovasz and Simonovits (1993) to obtain an
explicit error bound
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
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
Nonasymptotic bounds on the mean square error for MCMC estimates via renewal techniques
The Nummellinâs 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
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
Error bounds for computing the expectation by Markov chain Monte Carlo
We study the error of reversible Markov chain Monte Carlo methods for
approximating the expectation of a function. Explicit error bounds with respect
to different norms of the function are proven. By the estimation the well known
asymptotical limit of the error is attained, i.e. there is no gap between the
estimate and the asymptotical behavior. We discuss the dependence of the error
on a burn-in of the Markov chain. Furthermore we suggest and justify a specific
burn-in for optimizing the algorithm
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