Despite the flexibility of these methods, their inference is based on the computationally intensive thinning algorithm.

A Friday report from The Washington Post warned that small New York City landlords are "at their breaking point" under Mayor Zohran Mamdani's housing policies.

The endangered pygmy slow loris is the only known venomous primate on Earth. Breakthroughs, discoveries, and DIY tips sent six days a week. As 2025 drew to a close, the Bronx Zoo in New York welcomed one of the most adorable animals you could imagine into the world: a pygmy slow loris (). In the picture shared by the zoo, the tiny endangered primate baby stares out with its giant dark eyes so intensely you'd think it was born with its eyes open. Indeed, that's exactly how slow lorises come out--as well as completely covered in fur.

Empirical Bayes (EB) improves the accuracy of simultaneous inference "by learning from the experience of others" (Efron, 2012). Classical EB theory focuses on latent variables that are iid draws from a fitted prior (Efron, 2019). Modern applications, however, feature complex structure, like arrays, spatial processes, or covariates. How can we apply EB ideas to these settings? We propose a generalized approach to empirical Bayes based on the notion of probabilistic symmetry. Our method pairs a simultaneous inference problem-with an unknown prior-to a symmetry assumption on the joint distribution of the latent variables. Each symmetry implies an ergodic decomposition, which we use to derive a corresponding empirical Bayes method. We call this methodBayesian empirical Bayes (BEB). We show how BEB recovers the classical methods of empirical Bayes, which implicitly assume exchangeability. We then use it to extend EB to other probabilistic symmetries: (i) EB matrix recovery for arrays and graphs; (ii) covariate-assisted EB for conditional data; (iii) EB spatial regression under shift invariance. We develop scalable algorithms based on variational inference and neural networks. In simulations, BEB outperforms existing approaches to denoising arrays and spatial data. On real data, we demonstrate BEB by denoising a cancer gene-expression matrix and analyzing spatial air-quality data from New York City.

Proven methods for teaching the readers who struggle most have been known for decades. Why do we often fail to use them? "There's a window of opportunity to intervene," Mark Seidenberg, a cognitive neuroscientist, said. "You don't want to let that go." In 2024, my niece Caroline received a Ph.D. in gravitational-wave physics. Her research interests include "the impact of model inaccuracies on biases in parameters recovered from gravitational wave data" and "Petrov type, principal null directions, and Killing tensors of slowly rotating black holes in quadratic gravity." I watched a little of her dissertation defense, on Zoom, and was lost as soon as she'd finished introducing herself. She and her husband now live in Italy, where she has a postdoctoral appointment. Caroline's academic achievements seem especially impressive if you know that until third grade she could barely read: to her, words on a page looked like a pulsing mass. She attended a private school in Connecticut, and there was a set time every day when students selected books to read on their own. "I can't remember how long that lasted, but it felt endless," she told me. She hid her disability by turning pages when her classmates did, and by volunteering to draw illustrations during group story-writing projects. One day, she told her grandmother that she could sound out individual letters but when she got to "the end of a row" she couldn't remember what had come before. A psychologist eventually identified her condition as dyslexia. Fluent readers sometimes think of dyslexia as a tendency to put letters in the wrong order or facing the wrong direction, but it's more complicated than that.