In many acoustical applications, porous materials may be stratified or physically anisotropic along their depth direction. In order to better understand the sound absorbing mechanisms of these porous media, the depth-dependent anisotropy can be approximated as a multilayer combination of finite-thickness porous materials with each layer being considered as isotropic. The uniqueness of this work is that it applies Bayesian probabilistic inference to determine the number of constituent layers in a multilayer porous specimen and macroscopic properties of their pores. This is achieved through measurement of the acoustic surface impedance and subsequent transfer-matrix analysis based on a valid theoretical model for the acoustical properties of porous media. The number of layers considered in the transfer-matrix analysis is varied, and Bayesian model selection is applied to identify individual layers present in the porous specimen and infer the parameters of their microstructure. Nested sampling is employed in this process to solve the computationally intensive inversion problem.