Métodos alternativos de inferência para modelos psicométricos de variáveis latentes de alta dimensão

Abstract

Cognitive diagnostic models (CDMs), such as DINA and DINO, classify respondents according to multiple discrete latent attributes, providing interpretable diagnostic feedback. This interpretability comes at a computational cost: classical estimation by marginal maximum likelihood with the EM algorithm requires a sum over the 2^K latent profiles, which grows exponentially with the number of attributes K and makes inference impractical precisely in the high-dimensional settings where diagnosis would be most valuable. This thesis develops and evaluates scalable variational inference methods, based on automatic-differentiation optimisation, for the DINA and DINO models in high dimension, while preserving the psychometric structure of these models: the Q matrix, the ideal response function, the interpretable guessing gj and slipping sj item parameters, and the item-level monotonicity constraint gj + sj < 1. Two methodological families are proposed. The first is a relaxed, amortised variational autoencoder (VAE), in which a shared inference network maps the response vector to posterior mastery probabilities, and a structured decoder implements the DINA or DINO response function in differentiable form. The second is a local, non-amortised black-box variational inference (BBVI), with per-respondent variational parameters, considered in two variants: a score-function (REINFORCE) variant, which preserves binary samples, and a continuous-relaxation (Binary-Concrete) variant. The differentiable decoders are built from the product t-norm, for the conjunctive DINA rule, and the probabilistic t-conorm, for the disjunctive DINO rule. Replacing the categorical posterior over 2^K profiles with K Bernoulli factors per respondent, together with stochastic gradient estimation, avoids explicit enumeration and enables estimation in dimensions where enumeration is impractical. The methods are evaluated through simulation studies that vary the numbers of respondents, items and attributes, including high-dimensional and stress scenarios, assessing the recovery of item parameters and latent attributes, the variability across replications, and the computational time; a practical comparison with classical estimators implemented in R packages, at feasible dimensions, is also included. The results show that item-parameter recovery is strong in the evaluated scenarios with more items per attribute and that marginal attribute accuracy remains high, although full-profile recovery decays with K. Among the BBVI variants, the relaxed one shows shorter runtime and lower dispersion across replications in the evaluated configurations, whereas the score-function one avoids the continuous-relaxation bias at the cost of higher variance; the VAE, in turn, amortises inference for new respondents. The two families prove to be complementary: the work preserves the interpretability of CDMs while enabling scalable inference through variational approximations, automatic differentiation and accelerated computation.