Porous media represent a vast class of materials, encountered in, i.e., geology, medicine and acoustics. Knowledge of the porous microstructure is necessary for the use of the different models describing the acoustical behavior of porous media. The focus of this work is put on inverse identification methods, that allow the retrieval of several microstructural parameters of a porous material. The inverse method is based on the measurement of an acoustic field transmitted and/or reflected by the porous material, and is recast as a Bayesian inference problem. This allows for an uncertainty quantification of the identified parameters, and model selection, in case of competing models for the acoustic representation of a porous material. Different materials are tested in impedance tubes, and their properties are inferred using different acoustical models. Samples of multiple thicknesses are backed with airgaps of different length, which allows for a densification of the test matrix for a given material: this ultimately improves knowledge on the porous microstructure properties, described by posterior probability density functions in a Bayesian context.