On the Distribution of Quadratic Expressions in Various Types of Random Vectors

Several approximations to the distribution of indefinite quadratic expressions in possibly singular Gaussian random vectors and ratios thereof are obtained in this dissertation. It is established that such quadratic expressions can be represented in their most general form as the difference of two positive definite quadratic forms plus a linear combination of Gaussian random variables. New advances on the distribution of quadratic expressions in elliptically contoured vectors, which are expressed as scalar mixtures of Gaussian vectors, are proposed as well. Certain distributional aspects of Hermitian quadratic expressions in complex Gaussian vectors are also investigated. Additionally, approximations to the distributions of quadratic forms in uniform, beta, exponential and gamma random variables as well as order statistics thereof are determined from their exact moments, for which explicit representations are derived. Closed form representations of the approximations to the density functions of the various types of quadratic expressions being considered herein are obtained by adjusting the base density functions associated with the quadratic forms appearing in the decompositions of the expressions by means of polynomials whose coefficients are determined from the moments of the target distributions. Quadratic forms being ubiquitous in Statistics, the proposed distributional results should prove eminently useful

On the Efficient Calculation of a Linear Combination of Chi-Square Random Variables with an Application in Counting String Vacua

Linear combinations of chi square random variables occur in a wide range of fields. Unfortunately, a closed, analytic expression for the pdf is not yet known. As a first result of this work, an explicit analytic expression for the density of the sum of two gamma random variables is derived. Then a computationally efficient algorithm to numerically calculate the linear combination of chi square random variables is developed. An explicit expression for the error bound is obtained. The proposed technique is shown to be computationally efficient, i.e. only polynomial in growth in the number of terms compared to the exponential growth of most other methods. It provides a vast improvement in accuracy and shows only logarithmic growth in the required precision. In addition, it is applicable to a much greater number of terms and currently the only way of computing the distribution for hundreds of terms. As an application, the exponential dependence of the eigenvalue fluctuation probability of a random matrix model for 4d supergravity with N scalar fields is found to be of the asymptotic form exp(-0.35N).Comment: 21 pages, 19 figures. 3rd versio

Deep Exponential Families

We describe \textit{deep exponential families} (DEFs), a class of latent variable models that are inspired by the hidden structures used in deep neural networks. DEFs capture a hierarchy of dependencies between latent variables, and are easily generalized to many settings through exponential families. We perform inference using recent "black box" variational inference techniques. We then evaluate various DEFs on text and combine multiple DEFs into a model for pairwise recommendation data. In an extensive study, we show that going beyond one layer improves predictions for DEFs. We demonstrate that DEFs find interesting exploratory structure in large data sets, and give better predictive performance than state-of-the-art models