As systems approaches to the event of biological models become more mature, attention is increasingly that specialize in the matter of inferring parameter values within those models from experimental data. However, particularly for nonlinear models, it's not obvious, either from inspection of the model or from the experimental data, that the inverse problem of parameter fitting will have a singular solution, or maybe a non-unique solution that constrains the parameters to lie within a plausible physiological range. Where parameters can't be constrained they're termed 'unidentifiable'. We specialise in gaining insight into the causes of unidentifiability using inference-based methods, and compare a recently developed measure-theoretic approach to inverse sensitivity analysis to the favored Markov chain Monte Carlo and approximate Bayesian computation techniques for Bayesian inference. All three approaches map the uncertainty in quantities of interest within the output space to the probability of sets of parameters within the input space. The geometry of those sets demonstrates how unidentifiability are often caused by parameter compensation and provides an intuitive approach to inference-based experimental design.