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Diffusion-sensitized magnetic resonance imaging probes the cellular structure of the human brain, but the primary microstructural information gets lost in averaging over higher-level, mesoscopic tissue organization such as different orientations of neuronal fibers. While such averaging is inevitable due to the limited imaging resolution, we propose a method for disentangling the microscopic cell properties from the effects of mesoscopic structure. We further avoid the classical fitting paradigm and use supervised machine learning in terms of a Bayesian estimator to estimate the microstructural properties. The method finds detectable parameters of a given microstructural model and calculates them within seconds, which makes it suitable for a broad range of neuroscientific applications.