When Respondents Are Scarce: LLM-Derived Semantic Priors for Bayesian Confirmatory Factor Analysis

The items of a reflective psychometric scale can be read as a proposal about a construct's structure. Language models make that proposal measurable by converting each item into an embedding, a list of numbers, similar for similar meanings. We introduce semanticfa, an R package for semantic factor analysis, which factors item-embedding similarities. It also interprets factors, flags redundant items, builds short forms, and screens new items. A worked example analyzes the 50-item International Personality Item Pool Big-Five Factor Markers, embedded with Qwen3-Embedding-8B, and compares the results with a factor analysis of 874,434 respondents. Retention criteria tied between five and six factors, and the tie rule kept five. Three corresponded mainly to single domains, and two blended most Extraversion items with five Agreeableness items. Each factor's congruence with its best-matching human factor (1 = identical) ranged from .92 for Openness to .52 for Agreeableness. Item pairs more similar in meaning tended to be answered more alike (r = .46 with raw responses), but same-domain reverse-keyed opposites, answered oppositely, were about as similar as cross-domain pairs (.58 vs. .57). Sign-flipping reverse-keyed items' embeddings did not improve agreement with raw responses and worsened it with reverse-scored ones. With five factors, a 25-item short form matched the domains better than all 200 random five-per-domain sets, but parallel analysis kept only three. We propose semantic-behavioral coherence, the agreement between semantic and response structures, as a measurable property of a scale, given an embedding model. Semantic factor analysis complements, and does not replace, response-based factor analysis.

Large language models (LLMs) have been proposed as alternatives to human experts for estimating unknown quantities with associated uncertainty, a process known as Bayesian elicitation. We test this by asking eleven LLMs to estimate population statistics, such as health prevalence rates, personality trait distributions, and labor market figures, and to express their uncertainty as 95% credible intervals. We vary each model's reasoning effort (low, medium, high) to test whether more "thinking" improves results. Our findings reveal three key results. First, larger, more capable models produce more accurate estimates, but increasing reasoning effort provides no consistent benefit. Second, all models are severely overconfident: their 95% intervals contain the true value only 9--44% of the time, far below the expected 95%. Third, a statistical recalibration technique called conformal prediction can correct this overconfidence, expanding the intervals to achieve the intended coverage. In a preliminary experiment, giving models web search access degraded predictions for already-accurate models, while modestly improving predictions for weaker ones. Models performed well on commonly discussed topics but struggled with specialized health data. These results indicate that LLM uncertainty estimates require statistical correction before they can be used in decision-making.

Large Language Models can generate synthetic survey responses at low cost, but their accuracy varies unpredictably across questions. We study the design problem of allocating a fixed budget of human respondents across estimation tasks when cheap LLM predictions are available for every task. Our framework combines three components. First, building on Prediction-Powered Inference, we characterize a question-specific rectification difficulty that governs how quickly the estimator's variance decreases with human sample size. Second, we derive a closed-form optimal allocation rule that directs more human labels to tasks where the LLM is least reliable. Third, since rectification difficulty depends on unobserved human responses for new surveys, we propose a meta-learning approach, trained on historical data, that predicts it for entirely new tasks without pilot data. The framework extends to general M-estimation, covering regression coefficients and multinomial logit partworths for conjoint analysis. We validate the framework on two datasets spanning different domains, question types, and LLMs, showing that our approach captures 61-79% of the theoretically attainable efficiency gains, achieving 11.4% and 10.5% MSE reductions without requiring any pilot human data for the target survey.