We present a family of conformal test martingales based on shifted Legendre polynomials, which extends the Simple Jumper martingale. The Simple Legendre Jumper substitutes the linear betting function with a polynomial of arbitrary degree, thereby facilitating the detection of variance, skewness, and higher-order deviations from uniformity; the standard Simple Jumper is a specific instance of degree one. The Product Legendre Jumper integrates multiple polynomial degrees into a unified betting function, although its state space expands exponentially--a cost we refer to as the jumping tax. To address this issue, we introduce the Variational Legendre Jumper, which factorises the joint adaptation through a mean-field approximation, thereby reducing exponential scaling to linear time with minimal loss in power. Lastly, the Composite Legendre Jumper incorporates several jumping rates, ensuring a wealth floor under exchangeability and automatic adaptation to the shift's timescale. Empirical results from a real-world classification task demonstrate that the combined methods consistently surpass any single-degree martingale under distributional shift, and the composite variant is recommended as the default when the shift timescale is unknown.

Proponents and critics of artificial intelligence often compare the technology to industrial automation--really, it's more like an intern. One summer during high school, I took a temporary job writing computer programs for a consulting firm. Each morning, I drove through rush-hour traffic to an office park near Princeton, New Jersey, on the crowded Route 1 corridor. At a desk in some sort of equipment room, I coded quick-and-dirty database tools for internal use. One of my programs simplified the process of logging hours into timesheets.

Despite theoretical advantages, causal methods for Gene Regulatory Network (GRN) inference from single-cell RNA-seq data consistently fail to match or outperform correlation-based baselines in many realistic benchmarks, a persistent puzzle which casts doubt on the value of causality for this task. We argue that existing benchmarks are insufficiently controlled to answer this question because they evaluate on real or semi-real data where multiple pathologies co-occur, confounding failure modes, and obscuring the specific conditions under which different inference methods excel or fail. To address this gap, we introduce a controlled diagnostic framework that isolates seven biologically motivated pathologies (dropout, latent confounders, cell-type mixing, feedback loops, network density, sample size, and pseudotime drift) and measure how six representative methods spanning three inference paradigms degrade as each pathology intensifies. Across 6,120 controlled experiments, we find that causal methods genuinely dominate in clean and structurally favorable regimes, but specific pathologies (notably dropout and latent confounders) selectively neutralize their advantages. We further introduce an errortype decomposition that reveals methods with similar aggregate accuracy commit qualitatively different errors. To probe whether single-pathology effects persist when multiple stressors co-occur, we perform an interaction sweep over the three most impactful pathologies and find that their joint effects are sub-additive, while also exposing density-conditional cross-overs invisible to single-dial analysis. Our findings offer a nuanced understanding of when and why different methods succeed or fail for GRN inference, providing actionable insights for method development and practical guidance for practitioners.3

We derive a tighter inequality than Cauchy-Schwarz Inequality, leverage it to refine Pearson's

To be consistent with accuracy definition, we denote the correctness ofstj for instance t as sim(stj,rt) = ( 2 distance(stj,rt))/ 2 where sim(stj,rt) is in the range [0,1] and distance(stj,rt) is in range [0, 2], 2 is the largest Euclidean distance in the probability simplex. Given a test dataset I, the correctness of a learner SLj on I can be denoted as 2 corrSLj = 1n Pn t=1sim(stj,rt). In this section, we define multiple metrics for consistency, accuracy, and correct-consistency in detail. Figure 1 shows the metrics computation in our experiments. We have created a git repository for this work and will be posted upon the acceptance and publicationofthiswork.

Our analysis further exposes that IGR encourages neurons with the same activation states for natural samples and the corresponding perturbed samples.

We develop a method to generate prediction intervals that have a user-specified coverage level across all regions of feature-space, a property calledconditional coverage.

Australia's beloved weather website got a makeover - and infuriated users It was an unseasonably warm spring day in Sydney on 22 October, with a forecast of 39C (99F) - a real scorcher. The day before, the state of New South Wales had reported its hottest day in over a century, a high of 44.8C in the outback town of Bourke. But little did the team at the national Bureau of Meteorology foresee that they, in particular, would soon be feeling the heat. Affectionately known by Australians as the Bom, the agency's long-awaited website redesign went live that morning, more than a decade after the last update. Within hours, the Bom was flooded with a deluge of complaints.