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We introduce IRIS, a geometric and heuristic-based scoring system for evaluating mathematical conjectures and theorems expressed as linear inequalities over numerical invariants. The IRIS score reflects multiple dimensions of significance—including sharpness, diversity, difficulty, and novelty—and enables the principled ranking of conjectures by their structural importance. As a tool for fully automated discovery, IRIS supports the generation and prioritization of high-value conjectures. We demonstrate its utility through case studies in convex geometry and graph theory, showing that IRIS can assist in both rediscovery of known results and proposal of novel, nontrivial conjectures. 

Assortment optimization has received active explorations in the past few decades due to its practical importance. Despite the extensive literature dealing with optimization algorithms and latent score estimation, uncertainty quantification for the optimal assortment still needs to be explored and is of great practical significance. Instead of estimating and recovering the complete optimal offer set, decision-makers may only be interested in testing whether a given property holds true for the optimal assortment, such as whether they should include several products of interest in the optimal set, or how many categories of products the optimal set should include. This paper proposes a novel inferential framework for testing such properties. We consider the widely adopted multinomial logit (MNL) model, where we assume that each customer will purchase an item within the offered products with a probability proportional to the underlying preference score associated with the product. We reduce inferring a general optimal assortment property to quantifying the uncertainty associated with the sign change point detection of the marginal revenue gaps. We show the asymptotic normality of the marginal revenue gap estimator, and construct a maximum statistic via the gap estimators to detect the sign change point. By approximating the distribution of the maximum statistic with multiplier bootstrap techniques, we propose a valid testing procedure. We also conduct numerical experiments to assess the performance of our method. 

We propose a novel combinatorial inference framework to conduct general uncertainty quantification in ranking problems. We consider the widely adopted Bradley-Terry-Luce (BTL) model, where each item is assigned a positive preference score that determines the Bernoulli distributions of pairwise comparisons’ outcomes. Our proposed method aims to infer general ranking properties of the BTL model. The general ranking properties include the “local” properties such as if an item is preferred over another and the “global” properties such as if an item is among the top K-ranked items. We further generalize our inferential framework to multiple testing problems where we control the false discovery rate (FDR) and apply the method to infer the top-K ranked items. We also derive the information-theoretic lower bound to justify the minimax optimality of the proposed method. We conduct extensive numerical studies using both synthetic and real data sets to back up our theory. 

Abstract Background Interstitial lung abnormalities (ILA) can be detected on computed tomography (CT) in lung cancer patients and have an association with mortality in advanced non-small cell lung cancer (NSCLC) patients. The aim of this study is to demonstrate the significance of ILA for mortality in patients with stage I NSCLC using Boston Lung Cancer Study cohort. Methods Two hundred and thirty-one patients with stage I NSCLC from 2000 to 2011 were investigated in this retrospective study (median age, 69 years; 93 males, 138 females). ILA was scored on baseline CT scans prior to treatment using a 3-point scale (0 = no evidence of ILA, 1 = equivocal for ILA, 2 = ILA) by a sequential reading method. ILA score 2 was considered the presence of ILA. The difference of overall survival (OS) for patients with different ILA scores were tested via log-rank test and multivariate Cox proportional hazards models were used to estimate hazard ratios (HRs) including ILA score, age, sex, smoking status, and treatment as the confounding variables. Results ILA was present in 22 out of 231 patients (9.5%) with stage I NSCLC. The presence of ILA was associated with shorter OS (patients with ILA score 2, median 3.85 years [95% confidence interval (CI): 3.36 – not reached (NR)]; patients with ILA score 0 or 1, median 10.16 years [95%CI: 8.65 - NR]; P  <  0.0001). In a Cox proportional hazards model, the presence of ILA remained significant for increased risk for death (HR = 2.88, P  = 0.005) after adjusting for age, sex, smoking and treatment. Conclusions ILA was detected on CT in 9.5% of patients with stage I NSCLC. The presence of ILA was significantly associated with a shorter OS and could be an imaging marker of shorter survival in stage I NSCLC.