Psychometrics — the science of measuring psychological attributes such as ability, knowledge, and personality — was among the earliest fields to adopt Bayesian methods. The core problem is inferring a latent trait (a student's mathematical ability, a patient's depression severity) from a set of observed responses to test items. Bayesian inference treats the latent trait as a random variable with a posterior distribution, providing not just a point estimate of ability but a full characterization of measurement uncertainty.
Bayesian Item Response Theory
Item response theory (IRT) models the probability of a correct response as a function of the person's latent ability and the item's characteristics. The two-parameter logistic (2PL) model uses discrimination (how sharply the item differentiates between ability levels) and difficulty (the ability level at which the probability of a correct response is 50%).
Bayesian Priors θᵢ ~ N(0, 1) [person ability]
aⱼ ~ LogNormal(0, σ²_a) [item discrimination]
bⱼ ~ N(0, σ²_b) [item difficulty]
Bayesian IRT estimation via MCMC simultaneously calibrates all item parameters and person abilities, properly accounting for the joint uncertainty. This is particularly important for new items with few responses, where maximum-likelihood estimates may not exist or may be wildly unstable. Bayesian priors regularize these estimates, borrowing strength from the distribution of item parameters across the test.
Computerized Adaptive Testing
Computerized adaptive testing (CAT) selects items in real time based on the test-taker's evolving ability estimate. After each response, the posterior distribution of ability is updated, and the next item is chosen to maximally reduce posterior uncertainty — typically the item whose information function peaks near the current ability estimate. This sequential Bayesian procedure can achieve the same measurement precision as a fixed-length test with 50-70% fewer items.
The Graduate Record Examination (GRE), the Graduate Management Admission Test (GMAT), and many professional licensure exams use computerized adaptive testing. Each test-taker receives a unique sequence of items tailored to their ability level. The final score is the posterior mean (or EAP — expected a posteriori) of the ability distribution, and the posterior standard deviation provides an individualized standard error of measurement. Bayesian stopping rules terminate the test when the posterior credible interval is narrow enough for a reliable pass/fail decision.
Multidimensional and Hierarchical IRT
Many assessments measure multiple correlated abilities — a math test may tap algebraic reasoning, geometric intuition, and quantitative literacy simultaneously. Multidimensional IRT (MIRT) models estimate a vector of latent abilities for each person, with Bayesian methods handling the high-dimensional posterior through MCMC or variational inference. Hierarchical IRT models allow item parameters to vary across test forms, administrations, or populations, with hierarchical priors ensuring comparability.
"Every test item is a Bayesian experiment: it provides evidence about the test-taker's ability, and the posterior after all items have been answered is the principled summary of everything the test reveals." — Wim J. van der Linden, Handbook of Item Response Theory
Diagnostic Classification Models
Diagnostic classification models (DCMs) — also called cognitive diagnostic models — classify students into mastery profiles for a set of discrete skills. The DINA model, for example, requires mastery of all relevant skills for a high probability of a correct response. Bayesian estimation of DCMs produces posterior probabilities of mastery for each skill and each student, enabling targeted remediation. The posterior probability that a student has mastered "fraction addition" is more actionable than a total test score.
Learning Analytics and Growth Modeling
Bayesian models track student learning over time through dynamic IRT models, where ability evolves as students engage with instructional material. These models underpin intelligent tutoring systems that adjust difficulty, provide hints, and select practice problems based on the student's current posterior ability estimate. The Bayesian framework naturally handles the irregular timing and missing data that characterize real learning environments.