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The framework of Inferential Models (IMs) has recently been developed in search of what is referred to as the holy grail of statistical theory, that is, prior-free probabilistic inference. Its method of Conditional IMs (CIMs) is a critical…

While hidden class models of various types arise in many statistical applications, it is often difficult to establish the identifiability of their parameters. Focusing on models in which there is some structure of independence of some of…

The ability to derive underlying principles from a handful of observations and then generalize to novel situations -- known as inductive reasoning -- is central to human intelligence. Prior work suggests that language models (LMs) often…

Model theoretic results such as Characterization and Definability give important information about different logics. It is well known that the proofs of those results for several modal logics have, somehow, the same 'taste'. A general proof…

Latent class models have wide applications in social and biological sciences. In many applications, pre-specified restrictions are imposed on the parameter space of latent class models, through a design matrix, to reflect practitioners'…

Language models (LMs) now excel at many tasks such as few-shot learning, question answering, reasoning, and dialog. However, they sometimes generate unsupported or misleading content. A user cannot easily determine whether their outputs are…

The detection of sophisticated hallucinations in Large Language Models (LLMs) is hampered by a ``Detection Dilemma'': methods probing internal states (Internal State Probing) excel at identifying factual inconsistencies but fail on logical…

As large language models (LLMs) often generate plausible but incorrect content, error detection has become increasingly critical to ensure truthfulness. However, existing detection methods often overlook a critical problem we term as…

Shojaee et al. (2025) report that Large Reasoning Models (LRMs) exhibit "accuracy collapse" on planning puzzles beyond certain complexity thresholds. We demonstrate that their findings primarily reflect experimental design limitations…

Although a unique solution is guaranteed in the Linear complementarity problem (LCP) when the matrix $\mathbf{M}$ is positive definite, practical applications often involve cases where $\mathbf{M}$ is only positive semi-definite, leading to…

A fundamental problem with nonlinear models is that maximum likelihood estimates are not guaranteed to exist. Though nonexistence is a well known problem in the binary response model literature, it presents significant challenges for other…

We challenge the prevailing assumption that complex reasoning in large language models (LLMs) necessitates massive training data. We demonstrate that sophisticated mathematical reasoning can emerge with only a few examples. Specifically,…

The CAP theorem states that a distributed system cannot simultaneously guarantee consistency, availability, and partition tolerance under network partition. Inspired by this result, this paper formulates a CAP-like conjecture for Large…

The fundamental theorem of statistical learning states that for binary classification problems, any Empirical Risk Minimization (ERM) learning rule has close to optimal sample complexity. In this paper we seek for a generic optimal learner…

Large language models (LLMs) solve reasoning problems by first generating a rationale and then answering. We formalize reasoning as a latent variable model and derive a reward-based filtered expectation-maximization (FEM) objective for…

Self-correction is an approach to improving responses from large language models (LLMs) by refining the responses using LLMs during inference. Prior work has proposed various self-correction frameworks using different sources of feedback,…

Self-improvement is a mechanism in Large Language Model (LLM) pre-training, post-training and test-time inference. We explore a framework where the model verifies its own outputs, filters or reweights data based on this verification, and…

We provide new sufficient conditions under which Ryser's conjecture holds.

In a landmark contribution to the structural vector autoregression (SVARs) literature, Rubio-Ramirez, Waggoner, and Zha (2010, `Structural Vector Autoregressions: Theory of Identification and Algorithms for Inference,' Review of Economic…

We prove a PCP theorem for the existential theory of the reals, showing that MAX-ETR-INV is $\exists\mathbb{R}$-hard to approximate to within some constant factor. The existential theory of the reals (ETR) is a decision problem asking if…