Herein we show that a positive encoder gap exists for signals that are (approximately)k-sparse. Furthermore, considerZ0 as the set of m random variablesz that only differs fromZ in itsith variable, z0i = (y0i,x0i). We finally get expressions for the covering number as a function of . Thus,weneedλ/m 2λ/(1+ν)2, which is satisfied as long asm 2 (1+ν)2/2, which is satisfied in all relevant scenarios. For remarkc), denote x = x0 +v, and note that the minimizer of the above optimization problem satisfies (as follows from optimality of the minimizer [Mehta and Gray,2013, Lemma 13 of Supplementary]) 1 2 kx DϕD(x)k22+λkϕD(x)k1= 1 2 kxk22 1 2 kDϕD(x)k22.

Large Language Models (LLMs) are increasingly used in tasks such as psychological text analysis and decision-making in automated workflows. However, their reliability remains a concern due to potential biases inherited from their training process. In this study, we examine how different response format--binary versus continuous-- may systematically influence LLMs' judgments. In a value statement judgments task and a text sentiment analysis task, we prompted LLMs to simulate human responses and tested both formats across several models, including both open-source and commercial models. Our findings revealed a consistent negative bias: LLMs were more likely to deliver "negative" judgments in binary formats compared to continuous ones. Control experiments further revealed that this pattern holds across both tasks. Our results highlight the importance of considering response format when applying LLMs to decision tasks, as small changes in task design can introduce systematic biases.

Neural networks that can produce accurate, input-conditional uncertainty representations are critical for real-world applications. Recent progress on heteroscedastic continuous regression has shown great promise for calibrated uncertainty quantification on complex tasks, like image regression. However, when these methods are applied to discrete regression tasks, such as crowd counting, ratings prediction, or inventory estimation, they tend to produce predictive distributions with numerous pathologies. We propose to address these issues by training a neural network to output the parameters of a Double Poisson distribution, which we call the Deep Double Poisson Network (DDPN). In contrast to existing methods that are trained to minimize Gaussian negative log likelihood (NLL), DDPNs produce a proper probability mass function over discrete output. Additionally, DDPNs naturally model under-, over-, and equi-dispersion, unlike networks trained with the more rigid Poisson and Negative Binomial parameterizations. We show DDPNs 1) vastly outperform existing discrete models; 2) meet or exceed the accuracy and flexibility of networks trained with Gaussian NLL; 3) produce proper predictive distributions over discrete counts; and 4) exhibit superior out-of-distribution detection. DDPNs can easily be applied to a variety of count regression datasets including tabular, image, point cloud, and text data.

Understanding how changes in community parasite prevalence alter the rate and age distribution of severe malaria is essential for optimizing control efforts. Paton et al. assessed the incidence of pediatric severe malaria admissions from 13 hospitals in East Africa from 2006 to 2020 (see the Perspective by Taylor and Slutsker). Each 25% increase in community parasite prevalence shifted hospital admissions toward younger children. Low rates of lifetime infections appeared to confer some immunity to severe malaria in very young children. Children under the age of 5 years thus need to remain a focus of disease prevention for malaria control. Science , abj0089, this issue p. [926][1]; see also abk3443, p. [855][2] The relationship between community prevalence of Plasmodium falciparum and the burden of severe, life-threatening disease remains poorly defined. To examine the three most common severe malaria phenotypes from catchment populations across East Africa, we assembled a dataset of 6506 hospital admissions for malaria in children aged 3 months to 9 years from 2006 to 2020. Admissions were paired with data from community parasite infection surveys. A Bayesian procedure was used to calibrate uncertainties in exposure (parasite prevalence) and outcomes (severe malaria phenotypes). Each 25% increase in prevalence conferred a doubling of severe malaria admission rates. Severe malaria remains a burden predominantly among young children (3 to 59 months) across a wide range of community prevalence typical of East Africa. This study offers a quantitative framework for linking malaria parasite prevalence and severe disease outcomes in children. [1]: /lookup/doi/10.1126/science.abj0089 [2]: /lookup/doi/10.1126/science.abk3443

Frequentist Statistics tests whether an event (hypothesis) occurs or not. It calculates the probability of an event in the long run of the experiment. A very common flaw found in frequentist approach i.e. dependence of the result of an experiment on the number of times the experiment is repeated. Bayesian statistics is a mathematical procedure that applies probabilities to statistical problems. It provides people the tools to update their beliefs in the evidence of new data.