The author's preface gives an outline: "This book is about weakconvergence methods in metric spaces, with applications sufficient to show their power and utility. The Introduction motivates the definitions and indicates how the theory will yield solutions to problems arising outside it. Chapter 1 sets out the basic general theorems, which are then specialized in Chapter 2 to the space C[0, l ] of continuous functions on the unit interval and in Chapter 3 to the space D [0, 1 ] of functions with discontinuities of the first kind. The results of the first three chapters are used in Chapter 4 to derive a variety of limit theorems for dependent sequences of random variables. " The book develops and expands on Donsker's 1951 and 1952 papers on the invariance principle and empirical distributions. The basic random variables remain real-valued although, of course, measures on C[0, l ] and D[0, l ] are vitally used. Within this framework, there are various possibilities for a different and apparently better treatment of the material. More of the general theory of weak convergence of probabilities on separable metric spaces would be useful. Metrizability of the convergence is not brought up until late in the Appendix. The close relation of the Prokhorov metric and a metric for convergence in probability is (hence) not mentioned (see V. Strassen, Ann. Math. Statist. 36 (1965), 423-439; the reviewer, ibid. 39 (1968), 1563-1572). This relation would illuminate and organize such results as Theorems 4.1, 4.2 and 4.4 which give isolated, ad hoc connections between weak convergence of measures and nearness in probability. In the middle of p. 16, it should be noted that C*(S) consists of signed measures which need only be finitely additive if 5 is not compact. On p. 239, where the author twice speaks of separable subsets having nonmeasurable cardinal, he means "discrete" rather than "separable." Theorem 1.4 is Ulam's theorem that a Borel probability on a complete separable metric space is tight. Theorem 1 of Appendix 3 weakens completeness to topological completeness. After mentioning that probabilities on the rationals are tight, the author says it is an

Convergence of probability measures

Let's read! We will often find out this sentence everywhere. When still being a kid, mom used to order us to always read, so did the teacher. Some books are fully read in a week and we need the obligation to support reading. What about now? Do you still love reading? Is reading only for you who have obligation? Absolutely not! We here offer you a new book enPDFd preventing chronic diseases a vital investment to read.

Preventing Chronic Diseases: A Vital Investment

චත්තාරික සමය හා බැඳි සාම්ප්රධායික පසන් ගායන ශෛලිය පිළිබඳ අධ්යයනයක් (Unpublished doctoral dissertation)

Diffusion models have emerged as a powerful new family of deep generative models with record-breaking performance in many applications, including image synthesis, video generation, and molecule design. In this survey, we provide an overview of the rapidly expanding body of work on diffusion models, categorizing the research into three key areas: efficient sampling, improved likelihood estimation, and handling data with special structures. We also discuss the potential for combining diffusion models with other generative models for enhanced results. We further review the wide-ranging applications of diffusion models in fields spanning from computer vision, natural language generation, temporal data modeling, to interdisciplinary applications in other scientific disciplines. This survey aims to provide a contextualized, in-depth look at the state of diffusion models, identifying the key areas of focus and pointing to potential areas for further exploration. Github: https://github.com/YangLing0818/Diffusion-Models-Papers-Survey-Taxonomy.

Diffusion Models: A Comprehensive Survey of Methods and Applications

https://www.boeckler.de/pdf/v_2015_10_23_anaya.pdf

Spillovers of U.S. unconventional monetary policy to emerging markets: The role of capital flows

Generative models have been shown to provide a powerful mechanism for anomaly detection by learning to model healthy or normal reference data which can subsequently be used as a baseline for scoring anomalies. In this work we consider denoising diffusion probabilistic models (DDPMs) for unsupervised anomaly detection. DDPMs have superior mode coverage over generative adversarial networks (GANs) and higher sample quality than variational autoencoders (VAEs). However, this comes at the expense of poor scalability and increased sampling times due to the long Markov chain sequences required. We observe that within reconstruction-based anomaly detection a full-length Markov chain diffusion is not required. This leads us to develop a novel partial diffusion anomaly detection strategy that scales to high-resolution imagery, named AnoDDPM. A secondary problem is that Gaussian diffusion fails to capture larger anomalies; therefore we develop a multi-scale simplex noise diffusion process that gives control over the target anomaly size. AnoDDPM with simplex noise is shown to significantly outperform both f-AnoGAN and Gaussian diffusion for the tumorous dataset of 22 T1-weighted MRI scans (CCBS Edinburgh) qualitatively and quantitatively (improvement of +25.5% Sørensen–Dice coefficient, +17.6% IoU and +7.4% AUC).

AnoDDPM: Anomaly Detection with Denoising Diffusion Probabilistic Models using Simplex Noise

SummaryThe pseudo-marginal algorithm is a variant of the Metropolis–Hastings algorithm which samples asymptotically from a probability distribution when it is only possible to estimate unbiasedly an unnormalized version of its density. Practically, one has to trade off the computational resources used to obtain this estimator against the asymptotic variances of the ergodic averages obtained by the pseudo-marginal algorithm. Recent works on optimizing this trade-off rely on some strong assumptions, which can cast doubts over their practical relevance. In particular, they all assume that the distribution of the difference between the log-density, and its estimate is independent of the parameter value at which it is evaluated. Under regularity conditions we show that as the number of data points tends to infinity, a space-rescaled version of the pseudo-marginal chain converges weakly to another pseudo-marginal chain for which this assumption indeed holds. A study of this limiting chain allows us to provide parameter dimension-dependent guidelines on how to optimally scale a normal random walk proposal, and the number of Monte Carlo samples for the pseudo-marginal method in the large-sample regime. These findings complement and validate currently available results.

Large-sample asymptotics of the pseudo-marginal method

Modern systems of official statistics require the timely estimation of area-specific densities of subpopulations. Ideally estimates should be based on precise geocoded information, which is not available because of confidentiality constraints. One approach for ensuring confidentiality is by rounding the geoco-ordinates. We propose multivariate non-parametric kernel density estimation that reverses the rounding process by using a measurement error model. The methodology is applied to the Berlin register of residents for deriving density estimates of ethnic minorities and aged people. Estimates are used for identifying areas with a need for new advisory centres for migrants and infrastructure for older people.

/pdf/estimating-the-density-of-ethnic-minorities-and-aged-people-4vvzj06mg5.pdf

Estimating the density of ethnic minorities and aged people in Berlin: multivariate kernel density estimation applied to sensitive georeferenced administrative data protected via measurement error

Aided by advances in neural density estimation, considerable progress has been made in recent years towards a suite of simulation-based inference (SBI) methods capable of performing flexible, black-box, approximate Bayesian inference for stochastic simulation models. While it has been demonstrated that neural SBI methods can provide accurate posterior approximations, the simulation studies establishing these results have considered only well-specified problems -- that is, where the model and the data generating process coincide exactly. However, the behaviour of such algorithms in the case of model misspecification has received little attention. In this work, we provide the first comprehensive study of the behaviour of neural SBI algorithms in the presence of various forms of model misspecification. We find that misspecification can have a profoundly deleterious effect on performance. Some mitigation strategies are explored, but no approach tested prevents failure in all cases. We conclude that new approaches are required to address model misspecification if neural SBI algorithms are to be relied upon to derive accurate scientific conclusions.

Investigating the Impact of Model Misspecification in Neural Simulation-based Inference

Complex simulators have become a ubiquitous tool in many scientific disciplines, providing high-fidelity, implicit probabilistic models of natural and social phenomena. Unfortunately, they typically lack the tractability required for conventional statistical analysis. Approximate Bayesian computation (ABC) has emerged as a key method in simulation-based inference, wherein the true model likelihood and posterior are approximated using samples from the simulator. In this paper, we draw connections between ABC and generalized Bayesian inference (GBI). First, we re-interpret the accept/reject step in ABC as an implicitly defined error model. We then argue that these implicit error models will invariably be misspecified. While ABC posteriors are often treated as a necessary evil for approximating the standard Bayesian posterior, this allows us to re-interpret ABC as a potential robustification strategy. This leads us to suggest the use of GBI within ABC, a use case we explore empirically.

Sebastian M. Schmon

Papers

AnoDDPM: Anomaly Detection with Denoising Diffusion Probabilistic Models using Simplex Noise

Large-sample asymptotics of the pseudo-marginal method

Estimating the density of ethnic minorities and aged people in Berlin: multivariate kernel density estimation applied to sensitive georeferenced administrative data protected via measurement error

Investigating the Impact of Model Misspecification in Neural Simulation-based Inference

Generalized Posteriors in Approximate Bayesian Computation