120 research outputs found
Stability properties of some particle filters
Under multiplicative drift and other regularity conditions, it is established
that the asymptotic variance associated with a particle filter approximation of
the prediction filter is bounded uniformly in time, and the nonasymptotic,
relative variance associated with a particle approximation of the normalizing
constant is bounded linearly in time. The conditions are demonstrated to hold
for some hidden Markov models on noncompact state spaces. The particle
stability results are obtained by proving -norm multiplicative stability and
exponential moment results for the underlying Feynman-Kac formulas.Comment: Published in at http://dx.doi.org/10.1214/12-AAP909 the Annals of
Applied Probability (http://www.imstat.org/aap/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Dimension-free Wasserstein contraction of nonlinear filters
For a class of partially observed diffusions, sufficient conditions are given
for the map from initial condition of the signal to filtering distribution to
be contractive with respect to Wasserstein distances, with rate which has no
dependence on the dimension of the state-space and is stable under tensor
products of the model. The main assumptions are that the signal has affine
drift and constant diffusion coefficient, and that the likelihood functions are
log-concave. Contraction estimates are obtained from an -process
representation of the transition probabilities of the signal reweighted so as
to condition on the observations
Twisted particle filters
We investigate sampling laws for particle algorithms and the influence of
these laws on the efficiency of particle approximations of marginal likelihoods
in hidden Markov models. Among a broad class of candidates we characterize the
essentially unique family of particle system transition kernels which is
optimal with respect to an asymptotic-in-time variance growth rate criterion.
The sampling structure of the algorithm defined by these optimal transitions
turns out to be only subtly different from standard algorithms and yet the
fluctuation properties of the estimates it provides can be dramatically
different. The structure of the optimal transition suggests a new class of
algorithms, which we term "twisted" particle filters and which we validate with
asymptotic analysis of a more traditional nature, in the regime where the
number of particles tends to infinity.Comment: Published in at http://dx.doi.org/10.1214/13-AOS1167 the Annals of
Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical
Statistics (http://www.imstat.org
Calculating principal eigen-functions of non-negative integral kernels: particle approximations and applications
Often in applications such as rare events estimation or optimal control it is
required that one calculates the principal eigen-function and eigen-value of a
non-negative integral kernel. Except in the finite-dimensional case, usually
neither the principal eigen-function nor the eigen-value can be computed
exactly. In this paper, we develop numerical approximations for these
quantities. We show how a generic interacting particle algorithm can be used to
deliver numerical approximations of the eigen-quantities and the associated
so-called "twisted" Markov kernel as well as how these approximations are
relevant to the aforementioned applications. In addition, we study a collection
of random integral operators underlying the algorithm, address some of their
mean and path-wise properties, and obtain error estimates. Finally,
numerical examples are provided in the context of importance sampling for
computing tail probabilities of Markov chains and computing value functions for
a class of stochastic optimal control problems.Comment: 38 pages, 4 figures, 1 table; to appear in Mathematics of Operations
Researc
Negative association, ordering and convergence of resampling methods
We study convergence and convergence rates for resampling schemes. Our first
main result is a general consistency theorem based on the notion of negative
association, which is applied to establish the almost-sure weak convergence of
measures output from Kitagawa's (1996) stratified resampling method. Carpenter
et al's (1999) systematic resampling method is similar in structure but can
fail to converge depending on the order of the input samples. We introduce a
new resampling algorithm based on a stochastic rounding technique of Srinivasan
(2001), which shares some attractive properties of systematic resampling, but
which exhibits negative association and therefore converges irrespective of the
order of the input samples. We confirm a conjecture made by Kitagawa (1996)
that ordering input samples by their states in yields a faster
rate of convergence; we establish that when particles are ordered using the
Hilbert curve in , the variance of the resampling error is
under mild conditions, where
is the number of particles. We use these results to establish asymptotic
properties of particle algorithms based on resampling schemes that differ from
multinomial resampling.Comment: 54 pages, including 30 pages of supplementary materials (a typo in
Algorithm 1 has been corrected
Bayesian learning of noisy Markov decision processes
We consider the inverse reinforcement learning problem, that is, the problem
of learning from, and then predicting or mimicking a controller based on
state/action data. We propose a statistical model for such data, derived from
the structure of a Markov decision process. Adopting a Bayesian approach to
inference, we show how latent variables of the model can be estimated, and how
predictions about actions can be made, in a unified framework. A new Markov
chain Monte Carlo (MCMC) sampler is devised for simulation from the posterior
distribution. This step includes a parameter expansion step, which is shown to
be essential for good convergence properties of the MCMC sampler. As an
illustration, the method is applied to learning a human controller
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