This paper compares methods for the numerical computation of multivariate t-probabilities for hyperrectangular integration regions. Methods based on acceptance-rejection, spherical-radial transformations and separation-of-variables transformations are considered. Tests using randomly chosen problems show that the most efficient numerical methods use a transformation developed by Genz (1992) for multivariate normal probabilities. These methods allow moderately accurate multivariate t-probabilities to be quickly computed for problems with as many as twenty variables. Methods for the non-central multivariate t-distribution are also described. Key Words: multivariate t-distribution, non-central distribution, numerical integration, statistical computation. 1 Introduction A common problem in many statistics applications is the numerical computation of the multivariate t (MVT) distribution function (see Tong, 1990) defined by T(a; b; \Sigma; ) = \Gamma( +m 2 ) \Gamma( 2 ) p j\Sigma...

We examine a cascade encoding model for neural response in which a linear filtering stage is followed by a noisy, leaky, integrate-and-fire spike generation mechanism. This model provides a biophysically more realistic alternative to models based on Poisson (memoryless) spike generation, and can effectively reproduce a variety of spiking behaviors seen in vivo. We describe the maximum likelihood estimator for the model parameters, given only extracellular spike train responses (not intracellular voltage data). Specifically, we prove that the log likelihood function is concave and thus has an essentially unique global maximum that can be found using gradient ascent techniques. We develop an efficient algorithm for computing the maximum likelihood solution, demonstrate the effectiveness of the resulting estimator with numerical simulations, and discuss a method of testing the model’s validity using time-rescaling and density evolution techniques. Paninski et al., November 30, 2004 2 1

This paper is a survey of the major techniques and approaches available for the numerical approximation of integrals in statistics. We classify these into five broad categories; namely, asymptotic methods, importance sampling, adaptive importance sampling, multiple quadrature and Markov chain methods. Each method is discussed giving an outline of the basic supporting theory and particular features of the technique. Conclusions are drawn concerning the relative merits of the methods based on the discussion and their application to three examples. The following broad recommendations are made. Asymptotic methods should only be considered in contexts where the integrand has a dominant peak with approximate ellipsoidal symmetry. Importance sampling, and preferably adaptive importance sampling, based on a multivariate Student should be used instead of asymptotics methods in such a context. Multiple quadrature, and in particular subregion adaptive integration, are the algorithms of choice for...

Recursive binary partitioning is a popular tool for regression analysis. Two fundamental problems of exhaustive search procedures usually applied to fit such models have been known for a long time: overfitting and a selection bias towards covariates with many possible splits or missing values. While pruning procedures are able to solve the overfitting problem, the variable selection bias still seriously affects the interpretability of tree-structured regression models. For some special cases unbiased procedures have been suggested, however lacking a common theoretical foundation. We propose a unified framework for recursive partitioning which embeds tree-structured regression models into a well defined theory of conditional inference procedures. Stopping criteria based on multiple test procedures are implemented and it is shown that the predictive performance of the resulting trees is as good as the performance of established exhaustive search procedures. It turns out that the partitions and therefore the models induced by both approaches are structurally different, confirming the need for an unbiased variable selection. Moreover, it is shown that the prediction accuracy of trees with early stopping is equivalent to the prediction accuracy of pruned trees with unbiased variable selection. The methodology presented here is applicable to all kinds of regression problems, including nominal, ordinal, numeric, censored as well as multivariate response variables and arbitrary measurement scales of the covariates. Data from studies on glaucoma classification, node positive breast cancer survival and mammography experience are re-analyzed.

Quasi-Monte Carlo quadrature methods have been used for several decades. Their accuracy ranges from excellent to poor, depending on the problem. This article discusses how quasi-Monte Carlo quadrature error can be assessed, and what are the factors that influence it.

The problem of colouring a k-colourable graph is well-known to be NP-complete, for k 3. The MAX-k-CUT approach to approximate k-colouring is to assign k colours to all of the vertices in polynomial time such that the fraction of `defect edges' (with endpoints of the same colour) is provably small. The best known approximation was obtained by Frieze and Jerrum [9], using a semidefinite programming (SDP) relaxation which is related to the Lovasz #-function. In a related work, Karger et al. [18] devised approximation algorithms for colouring k-colourable graphs exactly in polynomial time with as few colours as possible. They also used an SDP relaxation related to the #-function.

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Abstract—Gaussian process classifiers (GPCs) are Bayesian probabilistic kernel classifiers. In GPCs, the probability of belonging to a certain class at an input location is monotonically related to the value of some latent function at that location. Starting from a Gaussian process prior over this latent function, data are used to infer both the posterior over the latent function and the values of hyperparameters to determine various aspects of the function. Recently, the expectation propagation (EP) approach has been proposed to infer the posterior over the latent function. Based on this work, we present an approximate EM algorithm, the EM-EP algorithm, to learn both the latent function and the hyperparameters. This algorithm is found to converge in practice and provides an efficient Bayesian framework for learning hyperparameters of the kernel. A multiclass extension of the EM-EP algorithm for GPCs is also derived. In the experimental results, the EM-EP algorithms are as good or better than other methods for GPCs or Support Vector Machines (SVMs) with cross-validation. Index Terms—Gaussian process classification, Bayesian methods, kernel methods, expectation propagation, EM-EP algorithm. 1

Abstract. An L2-type discrepancy arises in the average- and worst-case error analyses for multidimensional quadrature rules. This discrepancy is uniquely defined by K(x, y), which serves as the covariance kernel for the space of random functions in the average-case analysis and a reproducing kernel for the space of functions in the worst-case analysis. This article investigates the asymptotic order of the root mean square discrepancy for randomized (0,m,s)-nets in base b. For moderately smooth K(x, y) the discrepancy is O(N −1 [log(N)] (s−1)/2), and for K(x, y) with greater smoothness the discrepancy is O(N −3/2 [log(N)] (s−1)/2), where N = b m is the number of points in the net. Numerical experiments indicate that the (t, m, s)-nets of Faure, Niederreiter and Sobol ′ do not necessarily attain the higher order of decay for sufficiently smooth kernels. However, Niederreiter nets may attain the higher order for kernels corresponding to spaces of periodic functions. 1.

We introduce quadratically gated mixture of experts (QGME), a statistical model for multi-class nonlinear classification. The QGME is formulated in the setting of incomplete data, where the data values are partially observed. We show that the missing values entail joint estimation of the data manifold and the classifier, which allows adaptive imputation during classifier learning. The expectation maximization (EM) algorithm is derived for joint likelihood maximization, with adaptive imputation performed analytically in the E-step. The performance of QGME is evaluated on three benchmark data sets and the results show that the QGME yields significant improvements over competing methods. 1.