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This paper concerns the approximation of probability measures on $mathbf{R}^d$ with respect to the Kullback-Leibler divergence. Given an admissible target measure, we show the existence of the best approximation, with respect to this divergence, from certain sets of Gaussian measures and Gaussian mixtures. The asymptotic behavior of such best approximations is then studied in the small parameter limit where the measure concentrates; this asymptotic behaviour is characterized using $Gamma$-convergence. The theory developed is then applied to understanding the frequentist consistency of Bayesian inverse problems. For a fixed realization of noise, we show the asymptotic normality of the posterior measure in the small noise limit. Taking into account the randomness of the noise, we prove a Bernstein-Von Mises type result for the posterior measure.
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We consider random walks on the group of orientation-preserving homeomorphisms of the real line ${mathbb R}$. In particular, the fundamental question of uniqueness of an invariant measure of the generated process is raised. This problem was already studied by Choquet and Deny (1960) in the context of random walks generated by translations of the line. Nowadays the answer is quite well understood in general settings of strongly contractive systems. Here we focus on broader class of systems satisfying the conditions: recurrence, contraction and unbounded action. We prove that under these conditions the random process possesses a unique invariant Radon measure on ${mathbb R}$. Our work can be viewed as a subsequent paper of Babillot et al. (1997) and Deroin et al. (2013).
This paper is concerned with transition paths within the framework of the overdamped Langevin dynamics model of chemical reactions. We aim to give an efficient description of typical transition paths in the small temperature regime. We adopt a variational point of view and seek the best Gaussian approximation, with respect to Kullback-Leibler divergence, of the non-Gaussian distribution of the diffusion process. We interpret the mean of this Gaussian approximation as the most likely path and the covariance operator as a means to capture the typical fluctuations around this most likely path. We give an explicit expression for the Kullback-Leibler divergence in terms of the mean and the covariance operator for a natural class of Gaussian approximations and show the existence of minimisers for the variational problem. Then the low temperature limit is studied via $Gamma$-convergence of the associated variational problem. The limiting functional consists of two parts: The first part only depends on the mean and coincides with the $Gamma$-limit of the Freidlin-Wentzell rate functional. The second part depends on both, the mean and the covariance operator and is minimized if the dynamics are given by a time-inhomogenous Ornstein-Uhlenbeck process found by linearization of the Langevin dynamics around the Freidlin-Wentzell minimizer.
In the paper, we investigate the following fundamental question. For a set $mathcal{K}$ in $mathbb{L}^0(mathbb{P})$, when does there exist an equivalent probability measure $mathbb{Q}$ such that $mathcal{K}$ is uniformly integrable in $mathbb{L}^1(mathbb{Q})$. Specifically, let $mathcal{K}$ be a convex bounded positive set in $mathbb{L}^1(mathbb{P})$. Kardaras [6] asked the following two questions: (1) If the relative $mathbb{L}^0(mathbb{P})$-topology is locally convex on $mathcal{K}$, does there exist $mathbb{Q}sim mathbb{P}$ such that the $mathbb{L}^0(mathbb{Q})$- and $mathbb{L}^1(mathbb{Q})$-topologies agree on ${mathcal{K}}$? (2) If $mathcal{K}$ is closed in the $mathbb{L}^0(mathbb{P})$-topology and there exists $mathbb{Q}sim mathbb{P}$ such that the $mathbb{L}^0(mathbb{Q})$- and $mathbb{L}^1(mathbb{Q})$-topologies agree on $mathcal{K}$, does there exist $mathbb{Q}sim mathbb{P}$ such that $mathcal{K}$ is $mathbb{Q}$-uniformly integrable? In the paper, we show that, no matter $mathcal{K}$ is positive or not, the first question has a negative answer in general and the second one has a positive answer. In addition to answering these questions, we establish probabilistic and topological characterizations of existence of $mathbb{Q}simmathbb{P}$ satisfying these desired properties. We also investigate the peculiar effects of $mathcal{K}$ being positive.
Correcting for skewness can result in more accurate tail probability approximations in the central limit theorem for sums of independent random variables. In this paper, we extend the theory to sums of local statistics of independent random variables and apply the result to $k$-runs, U-statistics, and subgraph counts in the Erdos-Renyi random graph. To prove our main result, we develop exponential concentration inequalities and higher-order Cramer-type moderate deviations via Steins method.
We study in depth the nesting graph and volume distribution of the nodal domains of a Gaussian field, which have been shown in previous works to exhibit asymptotic laws. A striking link is established between the asymptotic mean connectivity of a nodal domain (i.e. the vertex degree in its nesting graph) and the positivity of the percolation probability of the field, along with a direct dependence of the average nodal volume on the percolation probability. Our results support the prevailing ansatz that the mean connectivity and volume of a nodal domain is conserved for generic random fields in dimension $d=2$ but not in $d ge 3$, and are applied to a number of concrete motivating examples.
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