Bayesian model inversion using stochastic spectral embedding

•We present the SSLE approach for Bayesian model inversion based on SSE and SLE.•Posterior quantities are obtained analytically from the SSLE coefficients.•The original SSE algorithm is enhanced with an active learning enrichment scheme.•This method generalizes and drastically improves the efficiency of the SLE approach.•We showcase the method on three problems of different complexity and dimensionality.