6,088 research outputs found
Discretizing Distributions with Exact Moments: Error Estimate and Convergence Analysis
The maximum entropy principle is a powerful tool for solving underdetermined
inverse problems. This paper considers the problem of discretizing a continuous
distribution, which arises in various applied fields. We obtain the
approximating distribution by minimizing the Kullback-Leibler information
(relative entropy) of the unknown discrete distribution relative to an initial
discretization based on a quadrature formula subject to some moment
constraints. We study the theoretical error bound and the convergence of this
approximation method as the number of discrete points increases. We prove that
(i) the theoretical error bound of the approximate expectation of any bounded
continuous function has at most the same order as the quadrature formula we
start with, and (ii) the approximate discrete distribution weakly converges to
the given continuous distribution. Moreover, we present some numerical examples
that show the advantage of the method and apply to numerically solving an
optimal portfolio problem.Comment: 20 pages, 14 figure
Multilevel Double Loop Monte Carlo and Stochastic Collocation Methods with Importance Sampling for Bayesian Optimal Experimental Design
An optimal experimental set-up maximizes the value of data for statistical
inferences and predictions. The efficiency of strategies for finding optimal
experimental set-ups is particularly important for experiments that are
time-consuming or expensive to perform. For instance, in the situation when the
experiments are modeled by Partial Differential Equations (PDEs), multilevel
methods have been proven to dramatically reduce the computational complexity of
their single-level counterparts when estimating expected values. For a setting
where PDEs can model experiments, we propose two multilevel methods for
estimating a popular design criterion known as the expected information gain in
simulation-based Bayesian optimal experimental design. The expected information
gain criterion is of a nested expectation form, and only a handful of
multilevel methods have been proposed for problems of such form. We propose a
Multilevel Double Loop Monte Carlo (MLDLMC), which is a multilevel strategy
with Double Loop Monte Carlo (DLMC), and a Multilevel Double Loop Stochastic
Collocation (MLDLSC), which performs a high-dimensional integration by
deterministic quadrature on sparse grids. For both methods, the Laplace
approximation is used for importance sampling that significantly reduces the
computational work of estimating inner expectations. The optimal values of the
method parameters are determined by minimizing the average computational work,
subject to satisfying the desired error tolerance. The computational
efficiencies of the methods are demonstrated by estimating the expected
information gain for Bayesian inference of the fiber orientation in composite
laminate materials from an electrical impedance tomography experiment. MLDLSC
performs better than MLDLMC when the regularity of the quantity of interest,
with respect to the additive noise and the unknown parameters, can be
exploited
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