58 research outputs found
Robust adaptive Metropolis algorithm with coerced acceptance rate
The adaptive Metropolis (AM) algorithm of Haario, Saksman and Tamminen
[Bernoulli 7 (2001) 223-242] uses the estimated covariance of the target
distribution in the proposal distribution. This paper introduces a new robust
adaptive Metropolis algorithm estimating the shape of the target distribution
and simultaneously coercing the acceptance rate. The adaptation rule is
computationally simple adding no extra cost compared with the AM algorithm. The
adaptation strategy can be seen as a multidimensional extension of the
previously proposed method adapting the scale of the proposal distribution in
order to attain a given acceptance rate. The empirical results show promising
behaviour of the new algorithm in an example with Student target distribution
having no finite second moment, where the AM covariance estimate is unstable.
In the examples with finite second moments, the performance of the new approach
seems to be competitive with the AM algorithm combined with scale adaptation.Comment: 21 pages, 3 figure
Conditional convex orders and measurable martingale couplings
Strassen's classical martingale coupling theorem states that two real-valued
random variables are ordered in the convex (resp.\ increasing convex)
stochastic order if and only if they admit a martingale (resp.\ submartingale)
coupling. By analyzing topological properties of spaces of probability measures
equipped with a Wasserstein metric and applying a measurable selection theorem,
we prove a conditional version of this result for real-valued random variables
conditioned on a random element taking values in a general measurable space. We
also provide an analogue of the conditional martingale coupling theorem in the
language of probability kernels and illustrate how this result can be applied
in the analysis of pseudo-marginal Markov chain Monte Carlo algorithms. We also
illustrate how our results imply the existence of a measurable minimiser in the
context of martingale optimal transport.Comment: 21 page
Markovian stochastic approximation with expanding projections
Stochastic approximation is a framework unifying many random iterative
algorithms occurring in a diverse range of applications. The stability of the
process is often difficult to verify in practical applications and the process
may even be unstable without additional stabilisation techniques. We study a
stochastic approximation procedure with expanding projections similar to
Andrad\'{o}ttir [Oper. Res. 43 (1995) 1037-1048]. We focus on Markovian noise
and show the stability and convergence under general conditions. Our framework
also incorporates the possibility to use a random step size sequence, which
allows us to consider settings with a non-smooth family of Markov kernels. We
apply the theory to stochastic approximation expectation maximisation with
particle independent Metropolis-Hastings sampling.Comment: Published in at http://dx.doi.org/10.3150/12-BEJ497 the Bernoulli
(http://isi.cbs.nl/bernoulli/) by the International Statistical
Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm
On the ergodicity of the adaptive Metropolis algorithm on unbounded domains
This paper describes sufficient conditions to ensure the correct ergodicity
of the Adaptive Metropolis (AM) algorithm of Haario, Saksman and Tamminen
[Bernoulli 7 (2001) 223--242] for target distributions with a noncompact
support. The conditions ensuring a strong law of large numbers require that the
tails of the target density decay super-exponentially and have regular
contours. The result is based on the ergodicity of an auxiliary process that is
sequentially constrained to feasible adaptation sets, independent estimates of
the growth rate of the AM chain and the corresponding geometric drift
constants. The ergodicity result of the constrained process is obtained through
a modification of the approach due to Andrieu and Moulines [Ann. Appl. Probab.
16 (2006) 1462--1505].Comment: Published in at http://dx.doi.org/10.1214/10-AAP682 the Annals of
Applied Probability (http://www.imstat.org/aap/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Quantitative convergence rates for sub-geometric Markov chains
We provide explicit expressions for the constants involved in the
characterisation of ergodicity of sub-geometric Markov chains. The constants
are determined in terms of those appearing in the assumed drift and one-step
minorisation conditions. The result is fundamental for the study of some
algorithms where uniform bounds for these constants are needed for a family of
Markov kernels. Our result accommodates also some classes of inhomogeneous
chains.Comment: 14 page
Convergence properties of pseudo-marginal markov chain monte carlo algorithms
We study convergence properties of pseudo-marginal Markov chain Monte Carlo
algorithms (Andrieu and Roberts [Ann. Statist. 37 (2009) 697-725]). We find
that the asymptotic variance of the pseudo-marginal algorithm is always at
least as large as that of the marginal algorithm. We show that if the marginal
chain admits a (right) spectral gap and the weights (normalised estimates of
the target density) are uniformly bounded, then the pseudo-marginal chain has a
spectral gap. In many cases, a similar result holds for the absolute spectral
gap, which is equivalent to geometric ergodicity. We consider also unbounded
weight distributions and recover polynomial convergence rates in more specific
cases, when the marginal algorithm is uniformly ergodic or an independent
Metropolis-Hastings or a random-walk Metropolis targeting a super-exponential
density with regular contours. Our results on geometric and polynomial
convergence rates imply central limit theorems. We also prove that under
general conditions, the asymptotic variance of the pseudo-marginal algorithm
converges to the asymptotic variance of the marginal algorithm if the accuracy
of the estimators is increased.Comment: Published at http://dx.doi.org/10.1214/14-AAP1022 in the Annals of
Applied Probability (http://www.imstat.org/aap/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Conditional particle filters with diffuse initial distributions
Conditional particle filters (CPFs) are powerful smoothing algorithms for
general nonlinear/non-Gaussian hidden Markov models. However, CPFs can be
inefficient or difficult to apply with diffuse initial distributions, which are
common in statistical applications. We propose a simple but generally
applicable auxiliary variable method, which can be used together with the CPF
in order to perform efficient inference with diffuse initial distributions. The
method only requires simulatable Markov transitions that are reversible with
respect to the initial distribution, which can be improper. We focus in
particular on random-walk type transitions which are reversible with respect to
a uniform initial distribution (on some domain), and autoregressive kernels for
Gaussian initial distributions. We propose to use on-line adaptations within
the methods. In the case of random-walk transition, our adaptations use the
estimated covariance and acceptance rate adaptation, and we detail their
theoretical validity. We tested our methods with a linear-Gaussian random-walk
model, a stochastic volatility model, and a stochastic epidemic compartment
model with time-varying transmission rate. The experimental findings
demonstrate that our method works reliably with little user specification, and
can be substantially better mixing than a direct particle Gibbs algorithm that
treats initial states as parameters.Comment: 21 pages, 17 figure
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