Bayesian Multiple Correspondence Analysis (BMCA)

Classical MCA performs a deterministic weighted SVD of the Burt or indicator matrix and reports a single set of category coordinates with no measure of uncertainty. Bayesian MCA models the same association structure probabilistically and outputs a posterior distribution over coordinates, enabling credible intervals and principled model comparison for the number of dimensions.

Bayesian latent class analysis assigns each respondent to one of K discrete hidden classes and estimates class-conditional category probabilities. Bayesian MCA does not assign respondents to classes; instead it embeds categories and respondents in a continuous low-dimensional space, retaining the geometric interpretability of classical correspondence analysis while adding Bayesian inference.

There is no single standard package. Implementations have been built in Stan, PyMC, and JAGS using custom Bayesian matrix-factorization models informed by MCA geometry. R packages such as FactoMineR handle classical MCA, but Bayesian extensions require additional probabilistic programming code.

Use marginal likelihood (model evidence) comparisons or Bayes factors across models with different numbers of retained dimensions. Spike-and-slab priors on the singular values can also automatically shrink irrelevant dimensions toward zero, yielding an implicit dimension selection.

The classical MCA framework treats all categorical variables nominally, ignoring the ordering of ordinal levels. If ordinal structure is important, consider ordinal correspondence analysis or Bayesian probabilistic models that explicitly encode the ordering through monotonicity constraints on category scores.