59,244 research outputs found
Calculation of aggregate loss distributions
Estimation of the operational risk capital under the Loss Distribution
Approach requires evaluation of aggregate (compound) loss distributions which
is one of the classic problems in risk theory. Closed-form solutions are not
available for the distributions typically used in operational risk. However
with modern computer processing power, these distributions can be calculated
virtually exactly using numerical methods. This paper reviews numerical
algorithms that can be successfully used to calculate the aggregate loss
distributions. In particular Monte Carlo, Panjer recursion and Fourier
transformation methods are presented and compared. Also, several closed-form
approximations based on moment matching and asymptotic result for heavy-tailed
distributions are reviewed
Approximating Probability Densities by Iterated Laplace Approximations
The Laplace approximation is an old, but frequently used method to
approximate integrals for Bayesian calculations. In this paper we develop an
extension of the Laplace approximation, by applying it iteratively to the
residual, i.e., the difference between the current approximation and the true
function. The final approximation is thus a linear combination of multivariate
normal densities, where the coefficients are chosen to achieve a good fit to
the target distribution. We illustrate on real and artificial examples that the
proposed procedure is a computationally efficient alternative to current
approaches for approximation of multivariate probability densities. The
R-package iterLap implementing the methods described in this article is
available from the CRAN servers.Comment: to appear in Journal of Computational and Graphical Statistics,
http://pubs.amstat.org/loi/jcg
Empirical Risk Minimization for Probabilistic Grammars: Sample Complexity and Hardness of Learning
Probabilistic grammars are generative statistical models that are useful for compositional and sequential structures. They are used ubiquitously in computational linguistics. We present a framework, reminiscent of structural risk minimization, for empirical risk minimization of probabilistic grammars using the log-loss. We derive sample complexity bounds in this framework that apply both to the supervised setting and the unsupervised setting. By making assumptions about the underlying distribution that are appropriate for natural language scenarios, we are able to derive distribution-dependent sample complexity bounds for probabilistic grammars. We also give simple algorithms for carrying out empirical risk minimization using this framework in both the supervised and unsupervised settings. In the unsupervised case, we show that the problem of minimizing empirical risk is NP-hard. We therefore suggest an approximate algorithm, similar to expectation-maximization, to minimize the empirical risk. Learning from data is central to contemporary computational linguistics. It is in common in such learning to estimate a model in a parametric family using the maximum likelihood principle. This principle applies in the supervised case (i.e., using annotate
Distribution of Mutual Information from Complete and Incomplete Data
Mutual information is widely used, in a descriptive way, to measure the
stochastic dependence of categorical random variables. In order to address
questions such as the reliability of the descriptive value, one must consider
sample-to-population inferential approaches. This paper deals with the
posterior distribution of mutual information, as obtained in a Bayesian
framework by a second-order Dirichlet prior distribution. The exact analytical
expression for the mean, and analytical approximations for the variance,
skewness and kurtosis are derived. These approximations have a guaranteed
accuracy level of the order O(1/n^3), where n is the sample size. Leading order
approximations for the mean and the variance are derived in the case of
incomplete samples. The derived analytical expressions allow the distribution
of mutual information to be approximated reliably and quickly. In fact, the
derived expressions can be computed with the same order of complexity needed
for descriptive mutual information. This makes the distribution of mutual
information become a concrete alternative to descriptive mutual information in
many applications which would benefit from moving to the inductive side. Some
of these prospective applications are discussed, and one of them, namely
feature selection, is shown to perform significantly better when inductive
mutual information is used.Comment: 26 pages, LaTeX, 5 figures, 4 table
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