Marginals of multivariate Gibbs distributions with applications in Bayesian species sampling

Gibbs partition models are the largest class of infinite exchangeable partitions of the positive integers generalizing the product form of the probability function of the two-parameter Poisson-Dirichlet family. Recently those models have been investigated in a Bayesian nonparametric approach to species sampling problems as alternatives to the Dirichlet and the Pitman-Yor process priors. Here we derive marginals of conditional and unconditional multivariate distributions arising from exchangeable Gibbs partitions to obtain explicit formulas for joint falling factorial moments of corresponding conditional and unconditional Gibbs sampling formulas. Our proofs rely on a known result on factorial moments of sum of non independent indicators. We provide an application to a Bayesian nonparametric estimation of the predictive probability to observe a species already observed a certain number of times.Comment: 24 pages; corrected typos, corrected equation numbering and labels referencing, added some comment

The Falling Factorial Basis and Its Statistical Applications

We study a novel spline-like basis, which we name the "falling factorial basis", bearing many similarities to the classic truncated power basis. The advantage of the falling factorial basis is that it enables rapid, linear-time computations in basis matrix multiplication and basis matrix inversion. The falling factorial functions are not actually splines, but are close enough to splines that they provably retain some of the favorable properties of the latter functions. We examine their application in two problems: trend filtering over arbitrary input points, and a higher-order variant of the two-sample Kolmogorov-Smirnov test.Comment: Full version for the ICML paper with the same titl

Shifted symmetric functions and multirectangular coordinates of Young diagrams

In this paper, we study shifted Schur functions $S_\mu^\star$, as well as a new family of shifted symmetric functions $\mathfrak{K}_\mu$ linked to Kostka numbers. We prove that both are polynomials in multi-rectangular coordinates, with nonnegative coefficients when written in terms of falling factorials. We then propose a conjectural generalization to the Jack setting. This conjecture is a lifting of Knop and Sahi's positivity result for usual Jack polynomials and resembles recent conjectures of Lassalle. We prove our conjecture for one-part partitions.Comment: 2nd version: minor modifications after referee comment

Calculation of some determinants using the s-shifted factorial

Several determinants with gamma functions as elements are evaluated. This kind of determinants are encountered in the computation of the probability density of the determinant of random matrices. The s-shifted factorial is defined as a generalization for non-negative integers of the power function, the rising factorial (or Pochammer's symbol) and the falling factorial. It is a special case of polynomial sequence of the binomial type studied in combinatorics theory. In terms of the gamma function, an extension is defined for negative integers and even complex values. Properties, mainly composition laws and binomial formulae, are given. They are used to evaluate families of generalized Vandermonde determinants with s-shifted factorials as elements, instead of power functions.Comment: 25 pages; added section 5 for some examples of application

Bayesian Estimation of the Size of a Population

We consider the following problem: estimate the size of a population marked with serial numbers after only a sample of the serial numbers has been observed. Its simplicity in formulation and the inviting possibilities of application make this estimation well suited for an undergraduate level probability course. Our contribution consists in a Bayesian treatment of the problem. For an improper uniform prior distribution, we show that the posterior mean and variance have nice closed form expressions and we demonstrate how to compute highest posterior density intervals. Maple and R code is provided on the authors’ web-page to allow students to verify the theoretical results and experiment with data

Generalized Heine Identity for Complex Fourier Series of Binomials

In this paper we generalize an identity first given by Heinrich Eduard Heine in his treatise, {\it Handbuch der Kugelfunctionen, Theorie und Anwendungen (1881), which gives a Fourier series for $1/[z-\cos\psi]^{1/2}$, for $z,\psi\in\R$, and $z>1$, in terms of associated Legendre functions of the second kind with odd-half-integer degree and vanishing order. In this paper we give a generalization of this identity as a Fourier series of $1/[z-\cos\psi]^\mu$, where z,\mu\in\C, $|z|>1$, and the coefficients of the expansion are given in terms of the same functions with order given by $\frac12-\mu$. We are also able to compute certain closed-form expressions for associated Legendre functions of the second kind.Comment: 12 page