2,404 research outputs found
Asymptotic optimality of the cross-entropy method for Markov chain problems
The correspondence between the cross-entropy method and the zero-variance
approximation to simulate a rare event problem in Markov chains is shown. This
leads to a sufficient condition that the cross-entropy estimator is
asymptotically optimal.Comment: 13 pager; 3 figure
Variance Reduction Techniques in Monte Carlo Methods
Monte Carlo methods are simulation algorithms to estimate a numerical quantity in a statistical model of a real system. These algorithms are executed by computer programs. Variance reduction techniques (VRT) are needed, even though computer speed has been increasing dramatically, ever since the introduction of computers. This increased computer power has stimulated simulation analysts to develop ever more realistic models, so that the net result has not been faster execution of simulation experiments; e.g., some modern simulation models need hours or days for a single ’run’ (one replication of one scenario or combination of simulation input values). Moreover there are some simulation models that represent rare events which have extremely small probabilities of occurrence), so even modern computer would take ’for ever’ (centuries) to execute a single run - were it not that special VRT can reduce theses excessively long runtimes to practical magnitudes.common random numbers;antithetic random numbers;importance sampling;control variates;conditioning;stratied sampling;splitting;quasi Monte Carlo
Distributed Binary Detection with Lossy Data Compression
Consider the problem where a statistician in a two-node system receives
rate-limited information from a transmitter about marginal observations of a
memoryless process generated from two possible distributions. Using its own
observations, this receiver is required to first identify the legitimacy of its
sender by declaring the joint distribution of the process, and then depending
on such authentication it generates the adequate reconstruction of the
observations satisfying an average per-letter distortion. The performance of
this setup is investigated through the corresponding rate-error-distortion
region describing the trade-off between: the communication rate, the error
exponent induced by the detection and the distortion incurred by the source
reconstruction. In the special case of testing against independence, where the
alternative hypothesis implies that the sources are independent, the optimal
rate-error-distortion region is characterized. An application example to binary
symmetric sources is given subsequently and the explicit expression for the
rate-error-distortion region is provided as well. The case of "general
hypotheses" is also investigated. A new achievable rate-error-distortion region
is derived based on the use of non-asymptotic binning, improving the quality of
communicated descriptions. Further improvement of performance in the general
case is shown to be possible when the requirement of source reconstruction is
relaxed, which stands in contrast to the case of general hypotheses.Comment: to appear on IEEE Trans. Information Theor
Semiparametric Cross Entropy for rare-event simulation
The Cross Entropy method is a well-known adaptive importance sampling method
for rare-event probability estimation, which requires estimating an optimal
importance sampling density within a parametric class. In this article we
estimate an optimal importance sampling density within a wider semiparametric
class of distributions. We show that this semiparametric version of the Cross
Entropy method frequently yields efficient estimators. We illustrate the
excellent practical performance of the method with numerical experiments and
show that for the problems we consider it typically outperforms alternative
schemes by orders of magnitude
Linear PDEs and eigenvalue problems corresponding to ergodic stochastic optimization problems on compact manifolds
We consider long term average or `ergodic' optimal control poblems with a
special structure: Control is exerted in all directions and the control costs
are proportional to the square of the norm of the control field with respect to
the metric induced by the noise. The long term stochastic dynamics on the
manifold will be completely characterized by the long term density and
the long term current density . As such, control problems may be
reformulated as variational problems over and . We discuss several
optimization problems: the problem in which both and are varied
freely, the problem in which is fixed and the one in which is fixed.
These problems lead to different kinds of operator problems: linear PDEs in the
first two cases and a nonlinear PDE in the latter case. These results are
obtained through through variational principle using infinite dimensional
Lagrange multipliers. In the case where the initial dynamics are reversible we
obtain the result that the optimally controlled diffusion is also
symmetrizable. The particular case of constraining the dynamics to be
reversible of the optimally controlled process leads to a linear eigenvalue
problem for the square root of the density process
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