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We introduce stochastic variational inference for Gaussian process models. This enables the application of Gaussian process (GP) models to data sets containing millions of data points. We show how GPs can be vari- ationally decomposed to…
This paper introduces graphemes for constructing and analyzing stochastic processes that describe the evolution of large dynamic graphs. Unlike graphons, which capture the static properties of dense graphs via exchangeability or subgraph…
Large ensembles of stochastically evolving interacting particles describe phenomena in diverse fields including statistical physics, neuroscience, biology, and engineering. In such systems, the infinitesimal evolution of each particle…
New proofs are given of the existence of the compensator (or dual predictable projection) of a locally integrable c\'adl\'ag adapted process of finite variation and of the existence of the quadratic variation process for a c\'adl\'ag local…
With a sequence of regressions, one may generate joint probability distributions. One starts with a joint, marginal distribution of context variables having possibly a concentration graph structure and continues with an ordered sequence of…
Random factor graphs provide a powerful framework for the study of inference problems such as decoding problems or the stochastic block model. Information-theoretically the key quantity of interest is the mutual information between the…
Many natural phenomena are effectively described by interacting particle systems, which can be modeled using either deterministic or stochastic differential equations (SDEs). In this study, we specifically investigate particle systems…
We observe that a lemma used in the study of even sets of nodes on surfaces applies almost verbatim to prove a celebrated formula of Gauss on the 2-torsion of the class group of a quadratic field.
A new class of generalized backward doubly stochastic differential equations (GBDSDEs in short) driven by Teugels martingales associated with L\'evy process are investigated. We establish a comparison theorem which allows us to derive an…
We develop a mean-field theory for large, non-exchangeable particle (agent) systems where the states and interaction weights co-evolve in a coupled system of SDEs. A first main result is the establishment of the propagation of…
The Nazarov-Sodin constant describes the average number of nodal set components of Gaussian fields on large scales. We generalise this to a functional describing the corresponding number of level set components for arbitrary levels. Using…
Continuous-time Bayesian networks is a natural structured representation language for multicomponent stochastic processes that evolve continuously over time. Despite the compact representation, inference in such models is intractable even…
We prove a central limit theorem for a random field generated by d commuting probability preserving transformations; the martingale is given by a commuting filtration (cf. D. Khosnevisan, Multiparameter Processes, Springer 2002). The result…
We consider Mckean-Vlasov type stochastic differential equations with multiplicative noise arising from the random vortex method. Such an equation can be viewed as the mean-field limit of interacting particle systems with singular…
In the previous paper we have shown analytically that, if the drift function of the d-dimensional Langevin equation is the Langevin function with a properly chosen scale factor, then the evolution of the drift function is a martingale…
Whittle-Mat\'ern fields are a recently introduced class of Gaussian processes on metric graphs, which are specified as solutions to a fractional-order stochastic differential equation. Unlike earlier covariance-based approaches for…
In this paper, we aim at characterizing generalized functionals of discrete-time normal martingales. Let $M=(M_n)_{n\in \mathbb{N}}$ be a discrete-time normal martingale that has the chaotic representation property. We first construct…
We offer a new proof of the classical law of large numbers for a general class of branching Markov processes based on the asymptotic behaviour of the moments developed in \cite{bmoments, gonzalez2022erratum}. Moreover, we show that the law…
In this article, we identify the necessary and sufficient conditions for the existence of a random field solution for some linear s.p.d.e.'s of parabolic and hyperbolic type. These equations rely on a spatial operator $\cL$ given by the…
We implement a version of radial conformal field theory in a family of statistical fields generated by central charge modification of the Gaussian free field and show that the correlation functions of such fields under the insertion of…