We conclude that equality in Lemma 3.1 holds if and only if w

Maximum Likelihood Estimators (MLE) has many good properties. For example, the asymptotic variance of MLE solution attains equality of the asymptotic Cram{\'e}r-Rao lower bound (efficiency bound), which is the minimum possible variance for an unbiased estimator. However, obtaining such MLE solution requires calculating the likelihood function which may not be tractable due to the normalization term of the density model. In this paper, we derive a Discriminative Likelihood Estimator (DLE) from the Kullback-Leibler divergence minimization criterion implemented via density ratio estimation and a Stein operator. We study the problem of model inference using DLE. We prove its consistency and show that the asymptotic variance of its solution can attain the equality of the efficiency bound under mild regularity conditions. We also propose a dual formulation of DLE which can be easily optimized. Numerical studies validate our asymptotic theorems and we give an example where DLE successfully estimates an intractable model constructed using a pre-trained deep neural network.

UK launches taskforce to'break down barriers' for women in technology The government has launched a new taskforce it says will help women enter, stay and lead in the UK tech sector. Led by technology secretary Liz Kendall, it will see female leaders from tech companies and organisations advise the government on how to boost diversity and economic growth in the industry. BCS, the Chartered Institute for IT, recently suggested women accounted for only 22% of those working in IT specialist roles in the UK. Ms Kendall said the Women in Tech group would break down the barriers that still hold too many people back. When women are inspired to take on a role in tech and have a seat at the table, the sector can make more representative decisions, build products that serve everyone, she said.

Translating human-written mathematical theorems and proofs from natural language (NL) into formal languages (FLs) like Lean 4 has long been a significant challenge for AI. Most state-of-the-art methods either focus on theorem-only NL-to-FL auto-formalization or on FL proof synthesis from FL theorems. In practice, auto-formalization of both theorem and proof still requires human intervention, as seen in AlphaProof's silver-medal performance at the 2024 IMO, where problem statements were manually translated before automated proof synthesis. Our training ensures that NL-FL theorems (and their proofs) are mapped close together in this space if and only if the NL-FL pairs are semantically equivalent. Experiments show substantial improvements in proof auto-formalization over strong baselines (including GPT -5, Gemini-2.5, In mathematics, ensuring the correctness of proofs is a crucial yet inherently difficult task. Traditionally, mathematicians rely on the peer-review process for proof verification, yet as proofs grow increasingly complex, even careful human scrutiny can overlook subtle errors. For instance, in 1989, Kapranov and V oevodsky published a proof connecting -groupoids and homotopy types, which was later disproven by Carlos Simpson in 1998; more recently, while formalizing his 2023 paper (Tao, 2023) on the Maclaurin-type inequality, Terence Tao discovered a non-trivial bug. To mitigate challenges of verifying complex proofs, proof assistants and formal mathematical languages like Coq (Barras et al., 1999), Isabelle (Nipkow et al., 2002), HOL Light (Harrison, 2009), Meta-math (Megill & Wheeler, 2019), Lean 4 (Moura & Ullrich, 2021), and Peano (Poesia & Goodman, 2023) have been developed, offering a way to create computer-verifiable formal proofs. Such formal language (FL) proofs, defined by strict syntax and symbolic logic, enable reliable automated verification guarantees that resolve the inherent ambiguity of natural language (NL) proofs.

Machine learning technologies have permeated everyday life and it is common nowadays that an automated system makes decisions for/about us, such as who is going to get bank credit.

In this paper we construct a non-parametric learning algorithm to estimate the clinical state of a patient.

We thank the reviewers for their time and effort reviewing our paper. We are pleased that you found our work to "solve At the latest, source code will be made publicly available upon publication. When we write "greedy," we are referring to the sequential setting When we say "naive batch" we We will add a thorough discussion on this topic to a future version of the paper. We will update the section accordingly. The main issue is how to do this efficiently for complex, non-linear models.