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Gaussian particles provide a flexible framework for modelling and simulating three-dimensional star-shaped random sets. In our framework, the radial function of the particle arises from a kernel smoothing, and is associated with an…
A full-rank lattice in the Euclidean space is a discrete set formed by all integer linear combinations of a basis. Given a probability distribution on $\mathbb{R}^n$, two operations can be induced by considering the quotient of the space by…
Galaxy cluster counts at high redshift, binned into spatial pixels and binned into ranges in an observable proxy for mass, contain a wealth of information on both the dark energy equation of state and the mass selection function required to…
In order to quantify the error budget in the measured probability distribution functions of cell densities, the two-point statistics of cosmic densities in concentric spheres is investigated. Bias functions are introduced as the ratio of…
We present a Bayesian inference approach to estimating the cumulative mass profile and mean squared velocity profile of a globular cluster given the spatial and kinematic information of its stars. Mock globular clusters with a range of…
Random tessellations of the space represent a class of prototype models of heterogeneous media, which are central in several applications in physics, engineering and life sciences. In this work, we investigate the statistical properties of…
In this paper, two parametric probability distributions capable to describe the statistics of X-ray photon detection by a CCD are presented. They are formulated from simple models that account for the pile-up phenomenon, in which two or…
Cosmic voids have been shown to be an effective probe of cosmology, complementary to galaxy clusters. In this work, we present a simple theoretical framework for predicting of the size of the largest voids expected within a given redshift…
This paper introduces a comprehensive framework for complex-valued probability measures and explores their novel applications in information theory and statistical analysis. We define a complex probability measure as a phase-modulated…
Latent space geometry has shown itself to provide a rich and rigorous framework for interacting with the latent variables of deep generative models. The existing theory, however, relies on the decoder being a Gaussian distribution as its…
We investigate properties of the correlation function of clusters of galaxies using geometrical models. On small scales the correlation function depends on the shape and the size of superclusters. On large scales it describes the geometry…
Using the generalized entropies which depend on two parameters we propose a set of quantitative characteristics derived from the Information Geometry based on these entropies. Our aim, at this stage, is modest, as we are first constructing…
The spatial cosmic matter distribution on scales of a few up to more than a hundred Megaparsec displays a salient and pervasive foamlike pattern. Voronoi tessellations are a versatile and flexible mathematical model for such weblike spatial…
Deep generative models provide a systematic way to learn nonlinear data distributions, through a set of latent variables and a nonlinear "generator" function that maps latent points into the input space. The nonlinearity of the generator…
Information geometry is the application of differential geometry in statistics, where the Fisher-Rao metric serves as the Riemannian metric on the statistical manifold, providing an intrinsic property for parameter sensitivity. In this…
We discuss probabilistic models of random covariance structures defined by distributions over sparse eigenmatrices. The decomposition of orthogonal matrices in terms of Givens rotations defines a natural, interpretable framework for…
We study random graphs with latent geometric structure, where the probability of each edge depends on the underlying random positions corresponding to the two endpoints. We focus on the setting where this conditional probability is a…
In this chapter we review some examples, methods, and recent results involving comparison of clustering properties of point processes. Our approach is founded on some basic observations allowing us to consider void probabilities and moment…
First-order statistics of scattered light is described using the representation of probability density cloud which visualizes a two-dimensional distribution for complex amplitude. The geometric parameters of the cloud are studied in detail…
We analyze the geometry of a joint distribution over a set of discrete random variables. We briefly review Shannon's entropy, conditional entropy, mutual information and conditional mutual information. We review the entropic information…