Shoe print analysis is pivotal in forensic investigations, especially when shoe prints are recovered at crime scenes. These prints often display accidentals, such as cuts, scrapes, and wear patterns, which develop due to usage and vary based on factors like brand, model, and size. While some accidentals are common across shoe prints, others are highly distinctive and rare. Distinguishing these uncommon patterns from the regular ones is essential for strengthening forensic evidence. For example, a pattern with a probability of occurrence of 1/100 is relatively common, while one with a probability of 1/10\(^6\) is exceptionally rare and thus more significant as evidence. In this study, we develop a hierarchical Bayesian model with spatially varying coefficients to analyze shoe print data. Our model effectively removes common patterns to identify whether a suspect's shoe print exhibits significantly distinctive features compared to regular accidental patterns, by incorporating spatial information from a large collection of shoe prints. This probability serves as critical statistical evidence to aid the jury in making informed decisions. Our proposed method outperforms state-of-the-art techniques, enhancing accuracy and reliability in forensic shoe print analysis. We also developed a theory for posterior consistency for our proposed model.

Large Language Models (LLMs) have achieved remarkable success in modeling natural language, internalizing many grammatical and semantic regularities that linguists identify as characteristic of human language. Nevertheless, they remain expensive to train and sometimes struggle to match specific aspects of human linguistic ability. In this project, we ask whether insights from language theory can reveal systematic patterns in where LLMs succeed or fail, and whether these insights can guide more effective training methods.

We study Meta’s OPT model trained on a BabyLM dataset (100M words), which is “developmentally more plausible” than state-of-the-art LLMs. We evaluate the model under controlled grammatical interventions using the BLiMP benchmark, which spans 67 syntactic categories, each defined by sentence pairs differing in a targeted grammatical rule violation. For example, the sentence “Who did Alan realize he liked?” is natural, whereas “Who did Alan realize who liked?” violates a grammatical constraint known as an Island Constraint, which restricts which parts of hierarchical sentence structures are available for question formation.

Using the BLiMP database, we track the model’s preference for grammatical over ungrammatical sentences across training iterations and grammatical categories. Our results show that in nearly one-third of BLiMP categories—including Island Constraints—OPT fails to consistently assign higher likelihoods to grammatical sentences, even after extensive training. Interestingly, when the model fails, it often establishes an early but clear (erroneous) separation of likelihoods at a stage when other structural behaviors are still developing. We are investigating this transition point as a potential locus where alternative training strategies could improve learning efficiency and model performance.

This paper introduces Bayesian variable selection methods for linear regression models with autocorrelated errors, using spike-and-slab priors in a two-step MCMC procedure. The approach enables the simultaneous selection of predictors, including lagged variables, and lagged error terms, ensuring consistency and scalability. Applications to finance (S&P 500 prediction) and environmental science demonstrate its superior predictive performance and accuracy in autoregressive time series models.