Maximum Likelihood vs. Bayesian estimation of uncertainty

When a physical or mathematical model is inferred from experimental data, it is essential to assess uncertainties in model parameters, if only because highly uncertain parameters effectively have not been learned from the data. This discussion compares two frameworks for estimating uncertainty: maximum likelihood (ML) and Bayesian inference (BI). We see that the ML framework is an approximation to the BI approach, in that ML uses a subset of the likelihood information whereas BI uses all of it. Interestingly, both approaches start from the same likelihood-based probabilistic framework. Both approaches require prior assumptions, which may only remain implicit in the case of ML. Both approaches require numerical care in complex systems with rough parameter-space landscapes.