16 Posterior Analysis
Stan uses Markov chain Monte Carlo (MCMC) techniques to generate samples from the posterior distribution for full Bayesian inference. Markov chain Monte Carlo (MCMC) methods were developed for situations in which it is not straightforward to make independent draws Metropolis et al. (1953).
Stan’s variational inference algorithm provides draws from the variational approximation to the posterior which may be analyzed just as any other MCMC output, despite the fact that it is not actually a Markov chain.
Stan’s Laplace algorithm produces a sample from a normal approximation centered at the mode of a distribution in the unconstrained space. If the mode is a maximum a posteriori (MAP) estimate, the samples provide an estimate of the mean and standard deviation of the posterior distribution. If the mode is a maximum likelihood estimate (MLE), the sample provides an estimate of the standard error of the likelihood.