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1. Exponential ergodicity and steady-state approximations for a class of markov processes under fast regime switching

   https://doi.org/10.1017/apr.2020.47  

   Arapostathis, Ari; Pang, Guodong; Zheng, Yi (March 2021, Advances in Applied Probability)

   null (Ed.)

   Abstract We study ergodic properties of a class of Markov-modulated general birth–death processes under fast regime switching. The first set of results concerns the ergodic properties of the properly scaled joint Markov process with a parameter that is taken to be large. Under very weak hypotheses, we show that if the averaged process is exponentially ergodic for large values of the parameter, then the same applies to the original joint Markov process. The second set of results concerns steady-state diffusion approximations, under the assumption that the ‘averaged’ fluid limit exists. Here, we establish convergence rates for the moments of the approximating diffusion process to those of the Markov-modulated birth–death process. This is accomplished by comparing the generator of the approximating diffusion and that of the joint Markov process. We also provide several examples which demonstrate how the theory can be applied. 

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2. An adapted particle swarm optimization algorithm as a model for exploring premyofibril formation

   https://doi.org/10.1063/1.5145010  

   Sherman, William; Grosberg, Anna (April 2020, AIP Advances)

   While the fundamental steps outlining myofibril formation share a similar scheme for different cell and species types, various granular details involved in the development of a functional contractile muscle are not well understood. Many studies of myofibrillogenesis focus on the protein interactions that are involved in myofibril maturation with the assumption that there is a fully formed premyofibril at the start of the process. However, there is little known regarding how the premyofibril is initially constructed. Fortunately, the protein α-actinin, which has been consistently identified throughout the maturation process, is found in premyofibrils as punctate aggregates known as z-bodies. We propose a theoretical model based on the particle swarm optimization algorithm that can explore how these α-actinin clusters form into the patterns observed experimentally. Our algorithm can produce different pattern configurations by manipulating specific parameters that can be related to α-actinin mobility and binding affinity. These patterns, which vary experimentally according to species and muscle cell type, speak to the versatility of α-actinin and demonstrate how its behavior may be altered through interactions with various regulatory, signaling, and metabolic proteins. The results of our simulations invite speculation that premyofibrils can be influenced toward developing different patterns by altering the behavior of individual α-actinin molecules, which may be linked to key differences present in different cell types. 

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3. A Gaussian Process Framework for Overlap and Causal Effect Estimation with High-Dimensional Covariates

   https://doi.org/10.1515/jci-2018-0024  

   Ghosh, Debashis; Cruz-Cortés, Efrén (January 2019, Journal of causal inference)

   A powerful tool for the analysis of nonrandomized observational studies has been the potential outcomes model. Utilization of this framework allows analysts to estimate average treatment effects. This article considers the situation in which high-dimensional covariates are present and revisits the standard assumptions made in causal inference. We show that by employing a flexible Gaussian process framework, the assumption of strict overlap leads to very restrictive assumptions about the distribution of covariates, results for which can be characterized using classical results from Gaussian random measures as well as reproducing kernel Hilbert space theory. In addition, we propose a strategy for data-adaptive causal effect estimation that does not rely on the strict overlap assumption. These findings reveal under a focused framework the stringency that accompanies the use of the treatment positivity assumption in high-dimensional settings. 

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4. Diffusion Models With Learned Adaptive Noise

   Sahoo, Subham; Gokaslan, Aaron; De_Sa, Christopher; Kuleshov, Volodymyr (December 2024, NeurIPS 2024)

   Diffusion models have gained traction as powerful algorithms for synthesizing high-quality images. Central to these algorithms is the diffusion process, a set of equations which maps data to noise in a way that can significantly affect performance. In this paper, we explore whether the diffusionprocess can be learned from data.Our work is grounded in Bayesian inference and seeks to improve log-likelihood estimation by casting the learned diffusion process as an approximate variational posterior that yields a tighter lower bound (ELBO) on the likelihood.A widely held assumption is that the ELBO is invariant to the noise process: our work dispels this assumption and proposes multivariate learned adaptive noise (MuLAN), a learned diffusion process that applies noise at different rates across an image. Our method consists of three components: a multivariate noise schedule, adaptive input-conditional diffusion, and auxiliary variables; these components ensure that the ELBO is no longer invariant to the choice of the noise schedule as in previous works. Empirically, MuLAN sets a new state-of-the-art in density estimation on CIFAR-10 and ImageNet while matching the performance of previous state-of-the-art models with 50% fewer steps. We provide the code, along with a blog post and video tutorial on the project page: https://s-sahoo.com/MuLAN 

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5. Second‐order preserving point process permutations

   https://doi.org/10.1002/sta4.558  

   Mohler, George; Mateu, Jorge (March 2023, Stat)

   While random permutations of point processes are useful for generating counterfactuals in bivariate interaction tests, such permutations require that the underlying intensity be separable. In many real‐world datasets where clustering or inhibition is present, such an assumption does not hold. Here, we introduce a simple combinatorial optimization algorithm that generates second‐order preserving (SOP) point process permutations, for example, permutations of the times of events such that the function of the permuted process matches the function of the data. We apply the algorithm to synthetic data generated by a self‐exciting Hawkes process and a self‐avoiding point process, along with data from Los Angeles on earthquakes and arsons and data from Indianapolis on law enforcement drug seizures and overdoses. In all cases, we are able to generate a diverse sample of permuted point processes where the distribution of the functions closely matches that of the data. We then show how SOP point process permutations can be used in two applications: (1) bivariate Knox tests and (2) data augmentation to improve deep learning‐based space‐time forecasts. 

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