The proposed permutation-invariant aggregation schemes can be extended to handle private latent variables by incorporating permutation-equivariance into the encoding distribution. This extension ensures that the encoding distribution is invariant with respect to the ordering of the encoded features of each modality, including both shared and private latent variables. By enforcing permutation-equivariance, the encoding distribution approximates the posterior distribution in a way that is unaffected by the inputs from the private latent variables when encoding the shared latent variables. This approach allows for a more flexible and robust modeling of multi-modal data, where the private latent variables can be effectively integrated into the overall variational framework.