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Doubly-intractable posterior distributions arise in many applications ofstatistics concerned with discrete and dependent data, including physics,spatial statistics, machine learning, the social sciences, and other fields. Aspecific example is psychometrics, which has adapted high-dimensional Isingmodels from machine learning, with a view to studying the interactions amongbinary item responses in educational assessments. To estimate high-dimensionalIsing models from educational assessment data, $\ell_1$-penalized nodewiselogistic regressions have been used. Theoretical results in high-dimensionalstatistics show that $\ell_1$-penalized nodewise logistic regressions canrecover the true interaction structure with high probability, provided thatcertain assumptions are satisfied. Those assumptions are hard to verify inpractice and may be violated, and quantifying the uncertainty about theestimated interaction structure and parameter estimators is challenging. Wepropose a Bayesian approach that helps quantify the uncertainty about theinteraction structure and parameters without requiring strong assumptions, andcan be applied to Ising models with thousands of parameters. We demonstrate theadvantages of the proposed Bayesian approach compared with $\ell_1$-penalizednodewise logistic regressions by simulation studies and applications to smalland large educational data sets with up to 2,485 parameters. Among otherthings, the simulation studies suggest that the Bayesian approach is morerobust against model misspecification due to omitted covariates than$\ell_1$-penalized nodewise logistic regressions.

Bayesian Model Selection for High-Dimensional Ising Models, With Applications to Educational Data