Explicit error bounds for lazy reversible Markov chain Monte Carlo

Rudolf, Daniel

doi:10.1016/j.jco.2008.05.005

Cited by 23 publications

(28 citation statements)

References 15 publications

(30 reference statements)

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Indeed, the existing papers [22,23,24,25,26] reported several difficulties to evaluate the variance of the sample mean in the continuous probability space even with the discrete time Markov chain. So, it is remained to extend the obtained results to the continuous case.…”

Section: Resultsmentioning

confidence: 99%

Information geometry approach to parameter estimation in Markov chains

Hayashi¹,

Watanabe²

2016

Ann. Statist.

View full text Add to dashboard Cite

Abstract: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. Primary 62M05.

show abstract

Section: Resultsmentioning

confidence: 99%

Information geometry approach to parameter estimation in Markov chains

Hayashi¹,

Watanabe²

2016

Ann. Statist.

View full text Add to dashboard Cite

show abstract

“…Results of this or related type have been obtained for finite or compact state space X and bounded target function f in [2,20,48]. Niemiro and Pokarowski in [39] give results for relative precision estimation.…”

Section: Introductionmentioning

confidence: 97%

Rigorous confidence bounds for MCMC under a geometric drift condition

Łatuszyński

Niemiro

2011

Journal of Complexity

View full text Add to dashboard Cite

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 $f:\stany \to R,$ let $I=\int_{\stany} f(x) \pi(x) dx$ be the value of interest and $\hat{I}_{t,n}=(1/n)\sum_{i=t}^{t+n-1}f(X_i)$ its MCMC estimate. Precisely, we derive lower bounds for the length of the trajectory $n$ and burn-in time $t$ which ensure that $$P(|\hat{I}_{t,n}-I|\leq \varepsilon)\geq 1-\alpha.$$ 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 $\varepsilon, \alpha.$ Next we analyse 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 Bayesian estimators in practically relevant models. We illustrate our bounds numerically in a simple example

show abstract

“…There are still no useful error bounds on the accuracy for MCMC methods; a recent authoritative review of the latest research is given in [40] and [28]- [29]. In particular, the error bounds are of the deceptively simple form: (14) ) ( d N c e Ω ≤ in which N denotes the number of particles and d is dimension of the state vector, and c is a so-called "constant."…”

Section: Open Problems In Particle Flowmentioning

confidence: 99%

“…In contrast, Roberts and Rosenthal [43] show very interesting numerical results for adaptive MCMC methods, from which they conclude that "practical implementation raises many important and largely unstudied problems." In other words, there are no useful error bounds for MCMC methods; as shown in [29], the simple bounds in the literature are many orders of magnitude from being tight. As explained above, simple bounds cannot be tight and tight bounds cannot be simple.…”

Section: Open Problems In Particle Flowmentioning

confidence: 99%

A plethora of open problems in particle flow research for nonlinear filters, Bayesian decisions, Bayesian learning, and transport

Daum

Huang

2016

Signal Processing, Sensor/Information Fusion, and Target Recognition XXV

View full text Add to dashboard Cite

We describe many open problems for research in particle flows to compute Bayes' rule for nonlinear filters, Bayesian decisions and Bayesian learning as well as transport. Particle flow mitigates particle degeneracy, which is the main cause of the curse of dimensionality for particle filters. Particle flow filters are many orders of magnitude faster to compute in real time compared with standard particle filters for the same accuracy for difficult high dimensional problems.

show abstract

Explicit error bounds for lazy reversible Markov chain Monte Carlo

Cited by 23 publications

References 15 publications

Information geometry approach to parameter estimation in Markov chains

Information geometry approach to parameter estimation in Markov chains

Rigorous confidence bounds for MCMC under a geometric drift condition

A plethora of open problems in particle flow research for nonlinear filters, Bayesian decisions, Bayesian learning, and transport

Contact Info

Product

Resources

About