# 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 .

In addition to providing point estimates, Stan’s optimization algorithm provides a Laplace approximation from which it is easy to draw random values. Stan’s variational inference algorithm provides draws from the variational approximation to the posterior. Both of these outputs may be analyzed just as any other MCMC output, despite the fact that it is actually independent draws.

### References

Metropolis, N., A. Rosenbluth, M. Rosenbluth, M. Teller, and E. Teller. 1953. “Equations of State Calculations by Fast Computing Machines.” Journal of Chemical Physics 21: 1087–92.