Order-invariant prior specification in Bayesian factor analysis

In (exploratory) factor analysis, the loading matrix is identified only up to orthogonal rotation. For identifiability, one thus often takes the loading matrix to be lower triangular with positive diagonal entries. In Bayesian inference, a standard practice is then to specify a prior under which the loadings are independent, the off-diagonal loadings are normally distributed, and the diagonal loadings follow a truncated normal distribution. This prior specification, however, depends in an important way on how the variables and associated rows of the loading matrix are ordered. We show how a minor modification of the approach allows one to compute with the identifiable lower triangular loading matrix but maintain invariance properties under reordering of the variables.