Surface waves extracted from ambient noise cross-correlations can be used to study depth variations of azimuthal anisotropy in the crust and upper mantle, complementing XKS splitting observations. In this work, we propose a novel approach based on beamforming to estimate azimuthal anisotropy of Rayleigh wave phase velocities extracted from ambient noise cross-correlations. This allows us to identify and remove measurements biased by wavefront deformation due to 3D heterogeneities, and to properly estimate uncertainties associated with observed phase velocities. In a second step, phase velocities measured at different periods can be inverted at depth with a transdimensional Bayesian algorithm where the presence or absence of anisotropy at different depths is a free variable. This yields a comprehensive probabilistic solution that can be exploited in different ways, in particular by projecting it onto a lower dimensional space, appropriate for interpretation. For example, we show the probability distribution of the integrated anisotropy over a given depth range (e.g. upper crust, lower crust). We apply this approach to recent data acquired across the AlpArray network and surrounding permanent stations. We show that only the upper crust has a large-scale coherent azimuthal anisotropy at the scale of the Alps with fast axis directions parallel to the Alpine arc, while such large-scale patterns are absent in the lower crust and uppermost mantle. This suggests that the recent Alpine history has only overridden the anisotropic signature in the upper crust, and that the deeper layers carry the imprint of older processes. In the uppermost mantle, fast directions of anisotropy are oriented broadly north-south, which is different from results from XKS-splitting measurements or long-period surface waves. Our results therefore suggest that XKS observations are related to deeper layers, the asthenosphere and/or subduction slabs. The area north-west of the Alps shows strong anisotropy in the lower crust and uppermost mantle with a fast axis in the north-east direction that could be related to Variscan deformation.