Betr Promo Code FOXNEWS: Get a $200 Bonus Ahead of Today's MLB Action China accelerates its AI build-out, expert warns against a U.S. pause Can AI help create your food? 'Chernobyl moment': GOP lawmaker urges Congress to act on AI before it's too late Holdout Clancy juror says he'DIDN'T HAVE ANY DOUBTS' Nvidia is the'king of kings' in tech as AI chip demand surges Tim Curry remembered by'Mighty Max' co-star Rob Paulsen: 'He made us better' This page may contain affiliate links to legal sports betting partners. If you sign up or place a wager, FOX News may be compensated. This content was created by a team that works independently from the Fox newsroom. LSU Tigers head coach Lane Kiffin reacts to a play against Clemson during the first half at Tiger Stadium in Baton Rouge, La., on Sept. 5, 2026.

Reporter for The Athletic fumes over Arch Manning, silent on Josh Jacobs' alleged violence against woman Jordon Hudson dusts herself off and resumes her WAG duties, Belichick vs Sydney Sweeney & 'Gainesville Ripper' Mets are'trash' because they're not listening to former NJ Gov Chris Christie, Stephen A Smith says Orioles pitcher Shane Baz drilled with 105.1 mph line drive in scary scene NFL's battle with drinking and driving continues as Keenan Allen formally charged in Indianapolis Anthony Molina and Matthew Liberatore's struggles make over 8 runs the play for Giants-Cardinals Shota Imanaga's home run struggles and Wrigley wind make over 7.5 the play for Braves-Cubs Jaxson Dart's girlfriend Marissa Ayers shines in Week 1 win, a Giants-Cowboys fan fight & Ole Miss tension MLB's most electric slugger just had one of the slowest, most disrespectful walk-off home runs of all time James Franklin makes Virginia Tech press conference extremely awkward with bizarre'Hey Michael' moment Texans' Azeez Al-Shaair to be fined once again by NFL for wearing Pro-Palestinian personal message on eyeblack Stephen A. Smith calls out NBA star Kawhi Leonard for letting Clippers take the fall for salary cap scandal Sheriff decries Democrat's'disgusting display' at House hearing on sanctuary laws Sheriff decries Democrat's'disgusting display' at House hearing on sanctuary laws Sheriff calls Democrat's questioning at House hearing a'disgusting display' It's urgent for Congress to step in and protect college sports: Sen Ted Cruz Mike Pence warns this Trump move would send a'deafening' message to Putin'COMPLETE WASTE OF MONEY': Real estate celebrity calls out $200,000 housing mistake Rand Paul slams Congress over AI regulation: 'I wouldn't put Congress in charge of a McDonald's' Rand Paul slams Congress over AI regulation: 'I wouldn't put Congress in charge of a McDonald's' Ed Sheeran responds to Macklemore backlash amid'Free Palestine' controversy Landry said Ole Miss AD Keith Carter'could have saved some money and waited until Saturday' to find Umanmielen I think they're going to put them on their roster that has to be submitted by Friday afternoon. And then all eyes are going to be on, do they run out of the tunnel Saturday afternoon in Columbia, South Carolina, or in Baton Rouge, Louisiana? And if that's the case, Dan, the SEC is going to have a decision to make. On Tuesday afternoon, however, the rivalry received another jolt outside LSU's football facility. In a lawsuit filed this past July, the Ole Miss athletic department is currently suing former players, and current LSU Tigers, Devin Harper and Princewill Umanmielen, on charges of breach of contract tied to their exit from Oxford following the departure of head coach Lane Kiffin.

Bayesian filtering and smoothing for high-dimensional nonlinear dynamical systems are fundamental yet challenging problems in many areas of science and engineering. In this work, we propose AFSF, a unified amortized framework for filtering and smoothing with conditional normalizing flows. The core idea is to encode each observation history into a fixed-dimensional summary statistic and use this shared representation to learn both a forward flow for the filtering distribution and a backward flow for the backward transition kernel. Specifically, a recurrent encoder maps each observation history to a fixed-dimensional summary statistic whose dimension does not depend on the length of the time series. Conditioned on this shared summary statistic, the forward flow approximates the filtering distribution, while the backward flow approximates the backward transition kernel. The smoothing distribution over an entire trajectory is then recovered by combining the terminal filtering distribution with the learned backward flow through the standard backward recursion. By learning the underlying temporal evolution structure, AFSF also supports extrapolation beyond the training horizon. Moreover, by coupling the two flows through shared summary statistics, AFSF induces an implicit regularization across latent state trajectories and improves trajectory-level smoothing. In addition, we develop a flow-based particle filtering variant that provides an alternative filtering procedure and enables ESS-based diagnostics when explicit model factors are available. Numerical experiments demonstrate that AFSF provides accurate approximations of both filtering distributions and smoothing paths.

Recently, Forré (arXiv:2104.11547, 2021) introduced transitional conditional independence, a notion of conditional independence that provides a unified framework for both random and non-stochastic variables. The original paper establishes a strong global Markov property connecting transitional conditional independencies with suitable graphical separation criteria for directed mixed graphs with input nodes (iDMGs), together with a version of causal calculus for iDMGs in a general measure-theoretic setting. These notes aim to further illustrate the motivations behind this framework and its connections to the literature, highlight certain subtlies in the general measure-theoretic causal calculus, and extend the "one-line" formulation of the ID algorithm of Richardson et al. (Ann. Statist. 51(1):334--361, 2023) to the general measure-theoretic setting.

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The $β$-VAE is a foundational framework for unsupervised disentanglement, using $β$ to regulate the trade-off between latent factorization and reconstruction fidelity. Empirically, however, disentanglement performance exhibits a pervasive non-monotonic trend: benchmarks such as MIG and SAP typically peak at intermediate $β$ and collapse as regularization increases. We demonstrate that this collapse is a fundamental information-theoretic failure, where strong Kullback-Leibler pressure promotes marginal independence at the expense of the latent channel's semantic informativeness. By formalizing this mechanism in a linear-Gaussian setting, we prove that for $β> 1$, stationarity-induced dynamics trigger a spectral contraction of the encoder gain, driving latent-factor mutual information to zero. To resolve this, we introduce the $λβ$-VAE, which decouples regularization pressure from informational collapse via an auxiliary $L_2$ reconstruction penalty $λ$. Extensive experiments on dSprites, Shapes3D, and MPI3D-real confirm that $λ> 0$ stabilizes disentanglement and restores latent informativeness over a significantly broader range of $β$, providing a principled theoretical justification for dual-parameter regularization in variational inference backbones.

We introduced an adaptive feature bank update scheme to dynamically absorb new features and discard obsolete features.