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- ItemStein variational gradient descent: Many-particle and long-time asymptotics(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2021) Nüsken, Nikolas; Renger, D. R. MichielStein variational gradient descent (SVGD) refers to a class of methods for Bayesian inference based on interacting particle systems. In this paper, we consider the originally proposed deterministic dynamics as well as a stochastic variant, each of which represent one of the two main paradigms in Bayesian computational statistics: emphvariational inference and emphMarkov chain Monte Carlo. As it turns out, these are tightly linked through a correspondence between gradient flow structures and large-deviation principles rooted in statistical physics. To expose this relationship, we develop the cotangent space construction for the Stein geometry, prove its basic properties, and determine the large-deviation functional governing the many-particle limit for the empirical measure. Moreover, we identify the emphStein-Fisher information (or emphkernelised Stein discrepancy) as its leading order contribution in the long-time and many-particle regime in the sense of $Gamma$-convergence, shedding some light on the finite-particle properties of SVGD. Finally, we establish a comparison principle between the Stein-Fisher information and RKHS-norms that might be of independent interest.
- ItemA large-deviations principle for all the components in a sparse inhomogeneous random graph(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2021) Andreis, Luisa; König, Wolfgang; Langhammer, Heide; Patterson, Robert I. A.We study an inhomogeneous sparse random graph, GN, on [N] = { 1,...,N } as introduced in a seminal paper [BJR07] by Bollobás, Janson and Riordan (2007): vertices have a type (here in a compact metric space S), and edges between different vertices occur randomly and independently over all vertex pairs, with a probability depending on the two vertex types. In the limit N → ∞ , we consider the sparse regime, where the average degree is O(1). We prove a large-deviations principle with explicit rate function for the statistics of the collection of all the connected components, registered according to their vertex type sets, and distinguished according to being microscopic (of finite size) or macroscopic (of size ≈ N). In doing so, we derive explicit logarithmic asymptotics for the probability that GN is connected. We present a full analysis of the rate function including its minimizers. From this analysis we deduce a number of limit laws, conditional and unconditional, which provide comprehensive information about all the microscopic and macroscopic components of GN. In particular, we recover the criterion for the existence of the phase transition given in [BJR07].
- ItemAnisothermal chemical reactions: Onsager--Machlup and macroscopic fluctuation theory(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2021) Renger, D. R. MichielWe study a micro and macroscopic model for chemical reactions with feedback between reactions and temperature of the solute. The first result concerns the quasipotential as the large-deviation rate of the microscopic invariant measure. The second result is an application of modern Onsager-Machlup theory to the pathwise large deviations, in case the system is in detailed balance. The third result is an application of macroscopic fluctuation theory to the reaction flux large deviations, in case the system is in complex balance.
- ItemA large-deviations principle for all the cluster sizes of a sparse Erdős-Rényi graph(New York, NY [u.a.] : Wiley, 2021) Andreis, Luisa; König, Wolfgang; Patterson, Robert I. A.[For Abstract, see PDF]