57 research outputs found
Convergence of Gaussian Process Regression with Estimated Hyper-parameters and Applications in Bayesian Inverse Problems
This work is concerned with the convergence of Gaussian process regression. A
particular focus is on hierarchical Gaussian process regression, where
hyper-parameters appearing in the mean and covariance structure of the Gaussian
process emulator are a-priori unknown, and are learnt from the data, along with
the posterior mean and covariance. We work in the framework of empirical Bayes,
where a point estimate of the hyper-parameters is computed, using the data, and
then used within the standard Gaussian process prior to posterior update. We
provide a convergence analysis that (i) holds for any continuous function
to be emulated; and (ii) shows that convergence of Gaussian process regression
is unaffected by the additional learning of hyper-parameters from data, and is
guaranteed in a wide range of scenarios. As the primary motivation for the work
is the use of Gaussian process regression to approximate the data likelihood in
Bayesian inverse problems, we provide a bound on the error introduced in the
Bayesian posterior distribution in this context
Quasi-Monte Carlo and Multilevel Monte Carlo Methods for Computing Posterior Expectations in Elliptic Inverse Problems
We are interested in computing the expectation of a functional of a PDE solution under a Bayesian posterior distribution. Using Bayes's rule, we reduce the problem to estimating the ratio of two related prior expectations. For a model elliptic problem, we provide a full convergence and complexity analysis of the ratio estimator in the case where Monte Carlo, quasi-Monte Carlo, or multilevel Monte Carlo methods are used as estimators for the two prior expectations. We show that the computational complexity of the ratio estimator to achieve a given accuracy is the same as the corresponding complexity of the individual estimators for the numerator and the denominator. We also include numerical simulations, in the context of the model elliptic problem, which demonstrate the effectiveness of the approach
Posterior Consistency for Gaussian Process Approximations of Bayesian Posterior Distributions
We study the use of Gaussian process emulators to approximate the parameter-to-observation map or the negative log-likelihood in Bayesian inverse problems. We prove error bounds on the Hellinger distance between the true posterior distribution and various approximations based on the Gaussian process emulator. Our analysis includes approximations based on the mean of the predictive process, as well as approximations based on the full Gaussian process emulator. Our results show that the Hellinger distance between the true posterior and its approximations can be bounded by moments of the error in the emulator. Numerical results confirm our theoretical findings
Are there approximate relations among transverse momentum dependent distribution functions?
Certain exact relations among transverse momentum dependent parton
distribution functions due to QCD equations of motion turn into approximate
ones upon the neglect of pure twist-3 terms. On the basis of available data
from HERMES we test the practical usefulness of one such
``Wandzura-Wilczek-type approximation'', namely of that connecting
h_{1L}^{\perp(1)a}(x) to h_L^a(x), and discuss how it can be further tested by
future CLAS and COMPASS data.Comment: 9 pages, 3 figure
How Deep Are Deep Gaussian Processes?
Recent research has shown the potential utility of Deep Gaussian Processes.
These deep structures are probability distributions, designed through
hierarchical construction, which are conditionally Gaussian. In this paper, the
current published body of work is placed in a common framework and, through
recursion, several classes of deep Gaussian processes are defined. The
resulting samples generated from a deep Gaussian process have a Markovian
structure with respect to the depth parameter, and the effective depth of the
resulting process is interpreted in terms of the ergodicity, or non-ergodicity,
of the resulting Markov chain. For the classes of deep Gaussian processes
introduced, we provide results concerning their ergodicity and hence their
effective depth. We also demonstrate how these processes may be used for
inference; in particular we show how a Metropolis-within-Gibbs construction
across the levels of the hierarchy can be used to derive sampling tools which
are robust to the level of resolution used to represent the functions on a
computer. For illustration, we consider the effect of ergodicity in some simple
numerical examples
A Hierarchical Multilevel Markov Chain Monte Carlo Algorithm with Applications to Uncertainty Quantification in Subsurface Flow
In this paper we address the problem of the prohibitively large computational
cost of existing Markov chain Monte Carlo methods for large--scale applications
with high dimensional parameter spaces, e.g. in uncertainty quantification in
porous media flow. We propose a new multilevel Metropolis-Hastings algorithm,
and give an abstract, problem dependent theorem on the cost of the new
multilevel estimator based on a set of simple, verifiable assumptions. For a
typical model problem in subsurface flow, we then provide a detailed analysis
of these assumptions and show significant gains over the standard
Metropolis-Hastings estimator. Numerical experiments confirm the analysis and
demonstrate the effectiveness of the method with consistent reductions of more
than an order of magnitude in the cost of the multilevel estimator over the
standard Metropolis-Hastings algorithm for tolerances
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An overview of ADSL Homed Nepenthes Honeypots In Western Australia
This paper outlines initial analysis from research in progress into ADSL homed Nepenthes honeypots. One of the Nepenthes honeypots prime objective in this research was the collection of malware for analysis and dissection. A further objective is the analysis of risks that are circulating within ISP networks in Western Australian. What differentiates Nepenthes from many traditional honeypot designs it that is has been engineered from a distributed network philosophy. The program allows distribution of results across a network of sensors and subsequent aggregation of malware statistics readily within a large network environment
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