Bayesian Inference
Bayesian inference updates a prior probability distribution over model parameters with the likelihood of observed data to obtain a posterior distribution, from which uncertainties and credible intervals follow. Because posteriors are rarely available in closed form, approximate methods such as variational inference and Markov chain Monte Carlo are widely used. See also Uncertainty quantification and Gaussian process regression.
Bayesian regression for equation of state parameters (2022)
Zhan and Kitchin cast nonlinear regression of seven analytical equations of state as a probabilistic graphical model in pyro and approximated the posterior with stochastic variational inference and with Hamiltonian Monte Carlo (No-U-Turn sampler) (Zhan 2022). HMC gave tighter posteriors closer to the least squares optimum and was more robust to initialization, but took about 20 minutes per model versus about 5 for SVI. The credible intervals were quantitatively close to frequentist delta method intervals, so the authors recommend ordinary nonlinear regression when parameters are roughly normal and reserve Bayesian regression for non-normal parameter distributions. Summary: zhan2022a.