LLMs excel at predictive tasks and complex reasoning tasks, but many high-value deployments rely on decisions under uncertainty, for example, which tool to call, which expert to consult, or how many resources to invest. While the usefulness and feasibility of Bayesian approaches remain unclear for LLM inference, this position paper argues that the control layer of an agentic AI system (that orchestrates LLMs and tools) is a clear case where Bayesian principles should shine. Bayesian decision theory provides a framework for agentic systems that can help to maintain beliefs over task-relevant latent quantities, to update these beliefs from observed agentic and human-AI interactions, and to choose actions. Making LLMs themselves explicitly Bayesian belief-updating engines remains computationally intensive and conceptually nontrivial as a general modeling target. In contrast, this paper argues that coherent decision-making requires Bayesian principles at the orchestration level of the agentic system, not necessarily the LLM agent parameters. This paper articulates practical properties for Bayesian control that fit modern agentic AI systems and human-AI collaboration, and provides concrete examples and design patterns to illustrate how calibrated beliefs and utility-aware policies can improve agentic AI orchestration.

External priors of unknown reliability create a brittle trade-off in causal discovery: blind trust amplifies errors, blind rejection wastes signal. Real priors are also heterogeneously reliable -- physical laws are trustworthy, LLM-suggested edges are speculative -- yet existing methods either ignore priors or impose them through globally uniform trust. We propose PRCD-MAP, a soft prior-consumption layer that assigns per-edge trust to an imperfect prior and uses it to modulate a prior-aware $\ell_1$ and prior-weighted $\ell_2$ regularizer in a MAP objective. Trust is calibrated by empirical Bayes on a Laplace-approximated marginal likelihood and propagated along the prior graph by an MLP, so data-confirmed neighborhoods boost trust and contradictions suppress it. PRCD-MAP enjoys a population-level safety guarantee: it is $\varepsilon$-safe in expectation over the prior-generation distribution, with $\varepsilon\leq C\cdot\mathrm{acc}(1{-}\mathrm{acc})\cdot d^2/T$ at the parametric $T^{-1}$ rate and vanishing at the prior-quality endpoints. When the prior is uninformative, learned trust provably collapses to its floor and the method recovers a no-prior baseline. Empirically, on real CausalTime data PRCD-MAP exploits informative LLM priors (LLM-prior gain $+0.067/+0.089$ AUROC on AQI/Medical over a no-prior PRCD-MAP backbone; combined backbone+prior lead $+0.123/+0.043$ over PCMCI+), auto-attenuates on the anonymous-variable Traffic stress test, and retains a lead at $d{=}300$; against BayesDAG, the closest soft-Bayesian baseline, PRCD-MAP wins on every CausalTime dataset under a matched $W_0$-only protocol. A four-way ablation isolates each component: EB calibration and MLP trust propagation jointly carry the plurality of the gain, with positive sign on every dataset. Extensions to nonlinear (NAM) and cross-sectional settings show the calibrated-trust principle is setting-agnostic.

Commercial Microwave Links (CMLs) offer dense spatial coverage for rainfall sensing but produce path-integrated measurements that make accurate ground-level reconstruction challenging. Existing methods typically oversimplify CMLs as point sensors and neglect line integration relating rainfall to signal attenuation, resulting in degraded performance under heterogeneous precipitation. In this work, we view rain field reconstruction as a Bayesian inverse problem with Diffusion Models (DMs) as high-fidelity spatial priors. We show that diffusion models better preserve key rainfall statistics compared to censored Gaussian processes. Framing rainfall estimation as a Bayesian inverse problem with a DM prior enables training-free posterior sampling using a broad family of methods, including Plug-and-Play, Sequential Monte Carlo, and Replica Exchange methods. Experiments on synthetic and real-world datasets demonstrate consistent improvements over established CML-based reconstruction baselines.

Kernel quadrature (KQ) is a kernel-based approach to numerical integration, closely related to Bayesian quadrature (BQ) and probabilistic integration [38, 39, 10]. For sufficiently regular integrands, KQ can exploit spectral structure in a reproducing kernel Hilbert space (RKHS) that is invisible to plain Monte Carlo and thereby converge faster than the usual O(N 1/2) rate in the number of points [3, 28]. Unconstrained kernel-based rules, however, may produce numerically unstable weights, motivating longstanding interest in positively weighted rules [13, 21, 29, 46]. In this paper, positive weights mean nonnegative weights that sum to one, i.e., simplex or convex-combination weights. Whether positive-weight KQ can systematically improve over Monte Carlo is a subtle question. Kernel herding and related constructions suggested fast rates under favorable assumptions [13], but the conditional-gradient viewpoint of Bach et al. [4] clarified that the strongest such assumptions are not generally available in infinite-dimensional RKHSs. Subsequent herding-type analyses in broad RKHS settings have therefore mostly remained at the Monte-Carlo scale, except under additional structure or modified algorithms such as sparse herding variants [31, 44, 43]. Beyond herding, subsampling-based positive KQ methods such as thinning [16, 15] and recombination [21, 24] have obtained rates beyond Monte Carlo, but a general mechanism for such improvement in the simple i.i.d.

In this paper, we investigate the supremum-norm generalization error and the uniform inference for a specific class of kernel regression methods, namely the kernel gradient flows. Under the widely adopted capacity-source condition framework in the kernel regression literature, we first establish convergence rates for the supremum norm generalization error of both continuous and discrete kernel gradient flows under the source condition $s>α_0$, where $α_0\in(0,1)$ denotes the embedding index of the kernel function. Moreover, we show that these rates match the minimax optimal rates. Building on this result, we then construct simultaneous confidence bands for both continuous and discrete kernel gradient flows. Notably, the widths of the proposed confidence bands are also optimal, in the sense that their shrinkage rates are greater than, while can be arbitrarily close to, the minimax optimal rates.

Trimming suspicious calibration points is a common response to contamination in conformal prediction. Its effect on clean-target coverage, however, is governed by the retained law induced by trimming, not by the contamination level alone. We analyse fixed-threshold trimming as conditioning rather than purification. It replaces the contaminated calibration law with a retained law, reducing clean-target coverage to a one-dimensional score-CDF transfer problem with an exact finite-sample identity. A componentwise bound on the transfer gap gives a population-level diagnostic. This separates a clean-side covariance cost from a retained-contamination cost, governed by the dirty-to-clean retention ratio. Trimming helps when the anomaly score separates retention probabilities while remaining score-neutral on the clean population. Otherwise, it cannot substantially reduce contamination through the retained mixture coefficient. We also give finite-sample certificate templates that provide numerical guarantees under independent audit.

Despite the prevalence of the attention sink phenomenon in Large Language Models (LLMs), where initial tokens disproportionately monopolize attention scores, its structural origins remain elusive. This work provides a \textit{mechanistic explanation} for this phenomenon. First, we trace its root to the value aggregation process inherent in self-attention, which induces a systematic variance discrepancy. We further demonstrate that this discrepancy is drastically amplified by the activation of super neurons within Feed-Forward Network (FFN) layers. Specifically, the channel-sparse down-projections trigger a dimension disparity of the first-token representation, necessitating the formation of attention sinks as a structural anchor. Then, we validate this causal chain through two controlled interventions: (i) isolating the aggregation effect via attention mask modifications and (ii) amplifying the variance of targeted token representations. Both interventions can replicate attention sinks at arbitrary positions. Our mechanistic understanding offers a foundation for the systematic control of sink formation. Finally, as a proof of concept, we propose \textit{head-wise RMSNorm}, an architectural modification that stabilizes value aggregation outputs during pre-training. Our experiments demonstrate that restoring statistical parity across positions significantly accelerates convergence.

Here's why the coaches association's 24-team College Football Playoff could ruin the sport Boston Celtics star Jaylen Brown tells ESPN's Stephen A Smith to'be quiet and retire' President Trump on $1,000 World Cup ticket prices: 'I wouldn't pay it either, to be honest' Pirates vs. Diamondbacks betting preview targets the under as both offenses go cold in series Former LSU coach Brian Kelly uses AI to prepare for job interviews, proving he's just like the rest of us Newsom office source responds to planned protest against trans athlete at state playoff girls' track meet'This can touch anyone': Gorman family speaks following loss of Sheridan'Project Freedom' could soon resume: Report Iranian people are not citizens, but'subjects' of the regime: Middle East expert Vice Admiral Robert Harward weighs in on restarting'Project Freedom' in Strait of Hormuz Largest teachers' union accused of antisemitism in federal civil rights complaint McEnany's URGENT plea: 'Be Spencer Pratt!' WHO doesn't expect large Hantavirus outbreak US blockade keeps stranglehold on Iran's economy What And Who Is Next In The Russini Saga? | OutKick Hot Mic TMZ Sports obtained documents this week showing that New England Patriots coach Mike Vrabel and former NFL reporter Dianna Russini signed a waiver for a private boating trip during her pregnancy in 2021. The report adds that Vrabel and Russini were the only two people on board for the two-to-three-hour excursion. On Thursday, the outlet obtained video and photos of the two on a dock in Putnam County, TN. See TMZ Sports' video below: ARE WE SURE MIKE VRABEL WILL SURVIVE RUSSINI SCANDAL AND COACH PATRIOTS THIS SEASON? According to the report, Russini gave birth to her first child with her husband, Kevin, later that summer. She was about six to seven months pregnant during the boating trip with Vrabel.

Boston Celtics star Jaylen Brown tells ESPN's Stephen A Smith to'be quiet and retire' President Trump on $1,000 World Cup ticket prices: 'I wouldn't pay it either, to be honest' Pirates vs. Diamondbacks betting preview targets the under as both offenses go cold in series Former LSU coach Brian Kelly uses AI to prepare for job interviews, proving he's just like the rest of us Newsom office source responds to planned protest against trans athlete at state playoff girls' track meet'This can touch anyone': Gorman family speaks following loss of Sheridan'Project Freedom' could soon resume: Report Iranian people are not citizens, but'subjects' of the regime: Middle East expert Vice Admiral Robert Harward weighs in on restarting'Project Freedom' in Strait of Hormuz Largest teachers' union accused of antisemitism in federal civil rights complaint McEnany's URGENT plea: 'Be Spencer Pratt!' WHO doesn't expect large Hantavirus outbreak US blockade keeps stranglehold on Iran's economy OutKick Here's why the coaches association's 24-team College Football Playoff could ruin the sport College Football Playoff Eying 24 Team Expansion: Is Bigger Better? The college football playoff could expand to 24 teams -- and a lot of people aren't happy about it. After just moving to 12, this feels like overkill. Is the sport trying to fix something that wasn't broken? When college football first moved to a playoff format, there were just four teams included.

Here's why the coaches association's 24-team College Football Playoff could ruin the sport President Trump on $1,000 World Cup ticket prices: 'I wouldn't pay it either, to be honest' Pirates vs. Diamondbacks betting preview targets the under as both offenses go cold in series Former LSU coach Brian Kelly uses AI to prepare for job interviews, proving he's just like the rest of us Newsom office source responds to planned protest against trans athlete at state playoff girls' track meet Iranians are fearful of facing the regime's frustration and anger after the war, activist says'This can touch anyone': Gorman family speaks following loss of Sheridan'Project Freedom' could soon resume: Report Iranian people are not citizens, but'subjects' of the regime: Middle East expert Vice Admiral Robert Harward weighs in on restarting'Project Freedom' in Strait of Hormuz Largest teachers' union accused of antisemitism in federal civil rights complaint McEnany's URGENT plea: 'Be Spencer Pratt!' WHO doesn't expect large Hantavirus outbreak OutKick Boston Celtics star Jaylen Brown tells ESPN's Stephen A Smith to'be quiet and retire' The viral exchange on X adds Brown to a list of NBA stars, including LeBron James and Kevin Durant, who've feuded with Smith ESPN commentator Stephen A. Smith is no stranger to having beef with NBA stars. It's time to add Celtics guard Jaylen Brown to the list. The latest dust-up started, naturally, on First Take, where Smith took aim at Brown for his comments following Boston's playoff collapse . Brown recently said this was his favorite year, despite the Celtics blowing a 3-1 series lead to the Philadelphia 76ers and getting eliminated in the first round of the NBA playoffs. That didn't sit well with Smith, who made it very clear Thursday that he thought Brown should have kept that to himself.