1,156 research outputs found
Adaptive Gibbs samplers
We consider various versions of adaptive Gibbs and Metropolis-
within-Gibbs samplers, which update their selection probabilities (and perhaps also their proposal distributions) on the
fly during a run, by learning
as they go in an attempt to optimise the algorithm. We present a cautionary
example of how even a simple-seeming adaptive Gibbs sampler may fail to
converge. We then present various positive results guaranteeing convergence
of adaptive Gibbs samplers under certain conditions
Complexity Results for MCMC derived from Quantitative Bounds
This paper considers how to obtain MCMC quantitative convergence bounds which
can be translated into tight complexity bounds in high-dimensional settings. We
propose a modified drift-and-minorization approach, which establishes a
generalized drift condition defined in a subset of the state space. The subset
is called the ``large set'' and is chosen to rule out some ``bad'' states which
have poor drift property when the dimension gets large. Using the ``large set''
together with a ``centered'' drift function, a quantitative bound can be
obtained which can be translated into a tight complexity bound. As a
demonstration, we analyze a certain realistic Gibbs sampler algorithm and
obtain a complexity upper bound for the mixing time, which shows that the
number of iterations required for the Gibbs sampler to converge is constant
under certain conditions on the observed data and the initial state. It is our
hope that this modified drift-and-minorization approach can be employed in many
other specific examples to obtain complexity bounds for high-dimensional Markov
chains.Comment: 42 page
Detecting multiple authorship of United States Supreme Court legal decisions using function words
This paper uses statistical analysis of function words used in legal
judgments written by United States Supreme Court justices, to determine which
justices have the most variable writing style (which may indicated greater
reliance on their law clerks when writing opinions), and also the extent to
which different justices' writing styles are distinguishable from each other.Comment: Published in at http://dx.doi.org/10.1214/10-AOAS378 the Annals of
Applied Statistics (http://www.imstat.org/aoas/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Harris recurrence of Metropolis-within-Gibbs and trans-dimensional Markov chains
A -irreducible and aperiodic Markov chain with stationary probability
distribution will converge to its stationary distribution from almost all
starting points. The property of Harris recurrence allows us to replace
``almost all'' by ``all,'' which is potentially important when running Markov
chain Monte Carlo algorithms. Full-dimensional Metropolis--Hastings algorithms
are known to be Harris recurrent. In this paper, we consider conditions under
which Metropolis-within-Gibbs and trans-dimensional Markov chains are or are
not Harris recurrent. We present a simple but natural two-dimensional
counter-example showing how Harris recurrence can fail, and also a variety of
positive results which guarantee Harris recurrence. We also present some open
problems. We close with a discussion of the practical implications for MCMC
algorithms.Comment: Published at http://dx.doi.org/10.1214/105051606000000510 in the
Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute
of Mathematical Statistics (http://www.imstat.org
Minimising MCMC variance via diffusion limits, with an application to simulated tempering
We derive new results comparing the asymptotic variance of diffusions by
writing them as appropriate limits of discrete-time birth-death chains which
themselves satisfy Peskun orderings. We then apply our results to simulated
tempering algorithms to establish which choice of inverse temperatures
minimises the asymptotic variance of all functionals and thus leads to the most
efficient MCMC algorithm.Comment: Published in at http://dx.doi.org/10.1214/12-AAP918 the Annals of
Applied Probability (http://www.imstat.org/aap/) by the Institute of
Mathematical Statistics (http://www.imstat.org
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