8,256 research outputs found
Control Variates for Reversible MCMC Samplers
A general methodology is introduced for the construction and effective
application of control variates to estimation problems involving data from
reversible MCMC samplers. We propose the use of a specific class of functions
as control variates, and we introduce a new, consistent estimator for the
values of the coefficients of the optimal linear combination of these
functions. The form and proposed construction of the control variates is
derived from our solution of the Poisson equation associated with a specific
MCMC scenario. The new estimator, which can be applied to the same MCMC sample,
is derived from a novel, finite-dimensional, explicit representation for the
optimal coefficients. The resulting variance-reduction methodology is primarily
applicable when the simulated data are generated by a conjugate random-scan
Gibbs sampler. MCMC examples of Bayesian inference problems demonstrate that
the corresponding reduction in the estimation variance is significant, and that
in some cases it can be quite dramatic. Extensions of this methodology in
several directions are given, including certain families of Metropolis-Hastings
samplers and hybrid Metropolis-within-Gibbs algorithms. Corresponding
simulation examples are presented illustrating the utility of the proposed
methods. All methodological and asymptotic arguments are rigorously justified
under easily verifiable and essentially minimal conditions.Comment: 44 pages; 6 figures; 5 table
Nudging the particle filter
We investigate a new sampling scheme aimed at improving the performance of
particle filters whenever (a) there is a significant mismatch between the
assumed model dynamics and the actual system, or (b) the posterior probability
tends to concentrate in relatively small regions of the state space. The
proposed scheme pushes some particles towards specific regions where the
likelihood is expected to be high, an operation known as nudging in the
geophysics literature. We re-interpret nudging in a form applicable to any
particle filtering scheme, as it does not involve any changes in the rest of
the algorithm. Since the particles are modified, but the importance weights do
not account for this modification, the use of nudging leads to additional bias
in the resulting estimators. However, we prove analytically that nudged
particle filters can still attain asymptotic convergence with the same error
rates as conventional particle methods. Simple analysis also yields an
alternative interpretation of the nudging operation that explains its
robustness to model errors. Finally, we show numerical results that illustrate
the improvements that can be attained using the proposed scheme. In particular,
we present nonlinear tracking examples with synthetic data and a model
inference example using real-world financial data
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