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The empirical KL-measure of MCMC convergence
Department of Mathematical Statistics, Chalmers University of Technology.
Linköping University, Department of Mathematics, Mathematical Statistics . Linköping University, The Institute of Technology.
(English)Manuscript (preprint) (Other academic)
Abstract [en]

A new measure based on comparison of empirical distributions for sub sequences or parallel runs and the full sequence of Markov chain Monte Carlo simulation, is proposed as a criterion of stability or convergence. The measure is also put forward as a loss function when the design, including the proposal function, of a Markov chain is optimised. The comparison of empirical distributions is based on a Kullback-Leibler (KL) type distance over value sets defined by the output data. The singularity problem for such a measure is removed in a simple way.

The leading term in a series expansion of the measure gives an interpretation in terms of the relative uncertainty of cell frequencies measured by their average coefficient of variation. The validity of the leading term is studied by simulation in two analytically tractable cases with Markov dependency and selected acceptance rates. The agreement between the leading term and the KL-measure is close, in particular when the simulations are extensive enough for stable results. Comparisons with established criteria turn out favourably in examples studied.

Keyword [en]
Convergence diagnostics, Kullback-Leibler distance, proposal distribution, parallel chains, single chain
National Category
Natural Sciences
URN: urn:nbn:se:liu:diva-86859OAI: diva2:582901
Available from: 2013-01-07 Created: 2013-01-07 Last updated: 2013-01-07
In thesis
1. Computer based statistical treatment in models with incidental parameters: inspired by car crash data
Open this publication in new window or tab >>Computer based statistical treatment in models with incidental parameters: inspired by car crash data
2003 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

Bootstrap and Markov chain Monte Carlo methods have received much attention in recent years. We study computer intensive methods that can be used in complex situations where it is not possible to express the likelihood estimates or the posterior analytically. The work is inspired by a set of car crash data from real traffic.

We formulate and develop a model for car crash data that aims to estimate and compare the relative collision safety among different car models. This model works sufficiently well, although complications arise due to a growing vector of incidental parameters. The bootstrap is shown to be a useful tool for studying uncertainties of the estimates of the structural parameters. This model is further extended to include driver characteristics. In a Poisson model with similar, but simpler structure, estimates of the structural parameter in the presence of incidental parameters are studied. The profile likelihood, bootstrap and the delta method are compared for deterministic and random incidental parameters. The same asymptotic properties, up to first order, are seen for deterministic as well as random incidental parameters.

The search for suitable methods that work in complex model structures leads us to consider Markov chain Monte Carlo (MCMC) methods. In the area of MCMC, we consider particularly the question of how and when to claim convergence of the MCMC run in situations where it is only possible to analyse the output values of the run and also how to compare different MCMC modellings. In Metropolis-Hastings algorithm, different proposal functions lead to different realisations. We develop a new convergence diagnostic, based on the Kullback-Leibler distance, which is shown to be particularly useful when comparing different runs. Comparisons with established methods turn out favourably for the KL.

In both models, a Bayesian analysis is made where the posterior distribution is obtained by MCMC methods. The credible intervals are compared to the corresponding confidence intervals from the bootstrap analysis and are shown to give the same qualitative conclusions.

Place, publisher, year, edition, pages
Linköping: Linköpings universitet, 2003. 34 p.
Linköping Studies in Science and Technology. Dissertations, ISSN 0345-7524 ; 814
National Category
urn:nbn:se:liu:diva-22219 (URN)1377 (Local ID)91-7373-625-2 (ISBN)1377 (Archive number)1377 (OAI)
Public defence
2003-05-09, Sal Visionen, Hus B, Linköpings Universitet, Linköping, 13:15 (Swedish)
Available from: 2009-10-07 Created: 2009-10-07 Last updated: 2013-01-07

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Vadeby, Anna
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