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This short paper introduces a novel approach to global sensitivity analysis, grounded in the variance-covariance structure of random variables derived from random measures. The proposed methodology facilitates the application of information-theoretic rules for uncertainty quantification, offering several advantages. Specifically, the approach provides valuable insights into the decomposition of variance within discrete subspaces, similar to the standard ANOVA analysis. To illustrate this point, the method is applied to datasets obtained from the analysis of randomized controlled trials on evaluating the efficacy of the COVID-19 vaccine and assessing clinical endpoints in a lung cancer study., Comment: 10 pages

Induction-transduction of activating-deactivating points are fundamental mechanisms of action that underlie innumerable systems and phenomena, mathematical, natural, and anthropogenic, and can exhibit complex behaviors such as self-excitation, phase transitions, hysteresis, polarization, periodicity, chaos, wave behavior, geometry, and energy transfer. We describe a class of primitives for induction-transduction based on dynamics on images of marked random counting measures under graphical random transformations. We derive a field equation for the law of the activation-deactivation (Bernoulli) process on an arbitrary measurable space and describe some mechanisms of action on the unit interval., Comment: 15 pages main (4 figures, 1 table), 80 page appendix (76 figures, 1 table), comments welcome

We form the Jacobi theta distribution through discrete integration of exponential random variables over an infinite inverse square law surface. It is continuous, supported on the positive reals, has a single positive parameter, is unimodal, positively skewed, and leptokurtic. Its cumulative distribution and density functions are expressed in terms of the Jacobi theta function. We describe asymptotic and log-normal approximations, inference, and a few applications of such distributions to modeling., Comment: 16 pages, 8 figures

We present a general framework for uncertainty quantification that is a mosaic of interconnected models. We define global first and second order structural and correlative sensitivity analyses for random counting measures acting on risk functionals of input-output maps. These are the ANOVA decomposition of the intensity measure and the decomposition of the random measure variance, each into subspaces. Orthogonal random measures furnish sensitivity distributions. We show that the random counting measure may be used to construct positive random fields, which admit decompositions of covariance and sensitivity indices and may be used to represent interacting particle systems. The first and second order global sensitivity analyses conveyed through random counting measures elucidate and integrate different notions of uncertainty quantification, and the global sensitivity analysis of random fields conveys the proportionate functional contributions to covariance. This framework complements others when used in conjunction with for instance algorithmic uncertainty and model selection uncertainty frameworks., Comment: 65 pages, 20 figures, 14 tables

We show that the sequence of thinned uniform random counting measures converges weakly to the Poisson random measure. We call such measures 'orthogonal dice' due to their natural connection with certain counting problems and link the sizes of orthogonal dice with the primes by constructing the largest known prime orthogonal die. We give many examples of possible use of the construct of orthogonal dice in various areas of applications including gambling, cosmology, random matrices, approximation theory and circuits. The gambling example reveals for instance that fair six-sided die numbered $\{1,2,3,4,5,6\}$ generate negative covariance in point representations of hands across players, in contrast to seven-sided dice numbered $\{1,2,3,4,5,6,7\}$ that generate zero covariance and are orthogonal. The cosmology example suggests that a 'supermassive' orthogonal die underlies the galaxy point patterns of the Universe. The random matrix application reveals that the variance of the spectral gap of the Gaussian orthogonal ensemble exhibits non-monotone scaling with thinning for the Dirac (empirical) random measure, in contrast to the orthogonal dice where the spectral gap variance is monotone. Finally, the approximation application shows that the discrete Legendre polynomials converge to the Charlier polynomials in $L^p$ for integers $p\ge1$ whereas the electronic circuit/shot noise application identifies electric current as the Ornstein-Uhlenbeck process driven by an orthogonal die random measure., Comment: 31 pages, 6 figures, 1 table

Prediction and explanation are key objects in supervised machine learning, where predictive models are known as black boxes and explanatory models are known as glass boxes. Explanation provides the necessary and sufficient information to interpret the model output in terms of the model input. It includes assessments of model output dependence on important input variables and measures of input variable importance to model output. High dimensional model representation (HDMR), also known as the generalized functional ANOVA expansion, provides useful insight into the input-output behavior of supervised machine learning models. This article gives applications of HDMR in supervised machine learning. The first application is characterizing information leakage in ``big-data'' settings. The second application is reduced-order representation of elementary symmetric polynomials. The third application is analysis of variance with correlated variables. The last application is estimation of HDMR from kernel machine and decision tree black box representations. These results suggest HDMR to have broad utility within machine learning as a glass box representation., Comment: 54 pages, 23 figures, 5 tables

We show that in a broad class of random counting measures one may identify only three that are rescaled versions of themselves when restricted to a subspace. These are Poisson, binomial and negative binomial random measures. We provide some simple examples of possible applications of such measures., Comment: 11 pages, 1 table