Biophysical models that attempt to infer real-world quantities from data usually have many free parameters. This over-parameterisation can result in degeneracies in model inversion and render parameter estimation ill-posed. However, in many applications, we are not interested in quantifying the parameters per se, but rather in identifying changes in parameters between experimental conditions (e.g. patients vs controls). Here we present a Bayesian framework to make inference on changes in the parameters of biophysical models even when model inversion is degenerate, which we refer to as Bayesian EstimatioN of CHange (BENCH). We infer the parameter changes in two steps; First, we train models that can estimate the pattern of change in the measurements given any hypothetical direction of change in the parameters using simulations. Next, for any pair of real data sets, we use these pre-trained models to estimate the probability that an observed difference in the data can be explained by each model of change. BENCH is applicable to any type of data and models and particularly useful for biophysical models with parameter degeneracies, where we can assume the change is sparse. In this paper, we apply the approach in the context of microstructural modelling of diffusion MRI data, where the models are usually over-parameterised and not invertible without injecting strong assumptions. Using simulations, we show that in the context of the standard model of white matter our approach is able to identify changes in microstructural parameters from conventional multi-shell diffusion MRI data. We also apply our approach to a subset of subjects from the UK-Biobank Imaging to identify the dominant standard model parameter change in areas of white matter hyperintensities under the assumption that the standard model holds in white matter hyperintensities.