8 research outputs found
Split Sampling: Expectations, Normalisation and Rare Events
In this paper we develop a methodology that we call split sampling methods to
estimate high dimensional expectations and rare event probabilities. Split
sampling uses an auxiliary variable MCMC simulation and expresses the
expectation of interest as an integrated set of rare event probabilities. We
derive our estimator from a Rao-Blackwellised estimate of a marginal auxiliary
variable distribution. We illustrate our method with two applications. First,
we compute a shortest network path rare event probability and compare our
method to estimation to a cross entropy approach. Then, we compute a
normalisation constant of a high dimensional mixture of Gaussians and compare
our estimate to one based on nested sampling. We discuss the relationship
between our method and other alternatives such as the product of conditional
probability estimator and importance sampling. The methods developed here are
available in the R package: SplitSampling
Tail approximations of integrals of Gaussian random fields
This paper develops asymptotic approximations of as
for a homogeneous smooth Gaussian random field, ,
living on a compact -dimensional Jordan measurable set . The integral of
an exponent of a Gaussian random field is an important random variable for many
generic models in spatial point processes, portfolio risk analysis, asset
pricing and so forth. The analysis technique consists of two steps: 1. evaluate
the tail probability over a small domain
depending on , where as and is the Lebesgue measure; 2. with
appropriately chosen, we show that
.Comment: Published in at http://dx.doi.org/10.1214/10-AOP639 the Annals of
Probability (http://www.imstat.org/aop/) by the Institute of Mathematical
Statistics (http://www.imstat.org
Extreme Analysis of a Non-convex and Nonlinear Functional of Gaussian Processes -- On the Tail Asymptotics of Random Ordinary Differential Equations
In this paper, we consider a stochastic system described by a differential
equation admitting a spatially varying random coefficient.
The differential equation has been employed to model various static physics
systems such as elastic deformation, water flow, electric-magnetic fields,
temperature distribution, etc.
A random coefficient is introduced to account for the system's uncertainty
and/or imperfect measurements.
This random coefficient is described by a Gaussian process (the input
process) and thus the solution to the differential equation (under certain
boundary conditions) is a complexed functional of the input Gaussian process.
In this paper, we focus the analysis on the one-dimensional case and derive
asymptotic approximations of the tail probabilities of the solution to the
equation that has various physics interpretations under different contexts.
This analysis rests on the literature of the extreme analysis of Gaussian
processes (such as the tail approximations of the supremum) and extends the
analysis to more complexed functionals.Comment: supplementary material is include
Efficient simulation for tail probabilities of Gaussian random fields
We are interested in computing tail probabilities for the max-ima of Gaussian random fields. In this paper, we discuss two special cases: random fields defined over a finite num-ber of distinct point and fields with finite Karhunen-Loève expansions. For the first case we propose an importance sampling estimator which yields asymptotically zero rela-tive error. Moreover, it yields a procedure for sampling the field conditional on it having an excursion above a high level with a complexity that is uniformly bounded as the level increases. In the second case we propose an estimator which is asymptotically optimal. These results serve as a first step analysis of rare-event simulation for Gaussian random fields.