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Variable-exponent fractional models attract increasing attentions in various applications, while the rigorous analysis is far from well developed. This work provides general tools to address these models. Specifically, we first develop a…
Mixture models are widely used in Bayesian statistics and machine learning, in particular in computational biology, natural language processing and many other fields. Variational inference, a technique for approximating intractable…
As the frontiers of applied statistics progress through increasingly complex experiments we must exploit increasingly sophisticated inferential models to analyze the observations we make. In order to avoid misleading or outright erroneous…
String theory predicts that the couplings of Nature descend from dynamical fields. All known string-motivated particle physics models also come with a wide range of possible extra sectors. It is common to posit that such moduli are frozen…
A generalization of exterior calculus is considered by allowing the partial derivatives in the exterior derivative to assume fractional orders. That is, a fractional exterior derivative is defined. This is found to generate new vector…
We study a generalization of conditional probability for arbitrary ordered vector spaces. A related problem is that of assigning a numerical value to one vector relative to another. We characterize the groups for which these generalized…
We present new results on the classical algorithm of variable elimination, which underlies many algorithms including for probabilistic inference. The results relate to exploiting functional dependencies, allowing one to perform inference…
A generalization of stable and casual stable probability distribution is proposed. The notion of $\go G$-casual stability can be used to introduce discrete analogues of stable distributions on the sent $\mathbb Z$ of integers. In contrary…
In this paper, we develop a general theory of truncated inverse binomial sampling. In this theory, the fixed-size sampling and inverse binomial sampling are accommodated as special cases. In particular, the classical Chernoff-Hoeffding…
We derive a covariance formula for the number of excursion or level set components of a smooth stationary Gaussian field on $\mathbb{R}^d$ contained in compact domains. We also present two applications of this formula: (1) for fields whose…
This paper focuses on inverse problems to identify parameters by incorporating information from measurements. These generally ill-posed problems are formulated here in a probabilistic setting based on Bayes's theorem because it leads to a…
We study the consistency and consistency strength of various configurations concerning the cardinal characteristics $\mathfrak{s}_\theta,\mathfrak{p}_\theta,\mathfrak{g}_\theta,\mathfrak{r}_\theta,\mathfrak{t}_\theta$ at uncountable regular…
We consider the problem of extending an acyclic binary relation that is invariant under a given family of transformations into an invariant preference. We show that when a family of transformations is commutative, every acyclic invariant…
The emergent field of probabilistic numerics has thus far lacked clear statistical principals. This paper establishes Bayesian probabilistic numerical methods as those which can be cast as solutions to certain inverse problems within the…
We consider the problem of predicting the covariance of a zero mean Gaussian vector, based on another feature vector. We describe a covariance predictor that has the form of a generalized linear model, i.e., an affine function of the…
A gauge invariant partition function is defined for gauge theories which leads to the standard quantization. It is shown that the descent equations and consequently the consistent anomalies and Schwinger terms can be extracted from this…
Every conditionally convergent series of real numbers has a subseries that diverges. The subseries numbers, previously studied in arXiv:1801.06206 , answer the question how many subsets of the natural numbers are necessary, such that every…
This chapter provides a overview of Bayesian inference, mostly emphasising that it is a universal method for summarising uncertainty and making estimates and predictions using probability statements conditional on observed data and an…
It is known that some cosmological perturbations are conformal invariant. This facilitates the studies of perturbations within some gravitational theories alternative to general relativity, for example the scalar-tensor theory, because it…
Consequences of the basic and most evident consistency requirement-that measured events cannot happen and not happen at the same time-are shortly reviewed. Particular emphasis is given to event forecast and event control. As a consequence,…