An Efficient Rotation-Sign-Permutation Algorithm to Solve Rotational Indeterminacy in Bayesian Exploratory Factor Analysis

Posterior samples of factor loadings in Bayesian exploratory factor analysis are not directly comparable across MCMC iterations due to rotational indeterminacy. As a result, their posterior means are typically close to zero because they cancel across rotationally equivalent orientations, yielding uninterpretable factor loading estimates. Alignment-based post-processing is therefore required, but existing approaches face a trade-off: exact alignment methods do not scale well beyond low-dimensional settings, whereas scalable alternatives rely on approximations that can reduce accuracy. We introduce an efficient version of the Rotation-Sign-Permutation (RSP) algorithm of Papastamoulis and Ntzoufras (2022) that overcomes these limitations by making exact alignment scalable to a large number of factors. Simulations and empirical examples demonstrate that this algorithm achieves higher alignment accuracy in high-dimensional settings with negligible computational overhead. An optimized C++ implementation is available in the open-source R package BayesianEFA.