6.5 MCMC

• Curse of dimensionality -> Grid approximation is limited to cases with only a few parameters.

• Stochastic Process: sequence of random variables

• Markov Chain: dependence only on previous element

\[\theta^{(i+1)} \sim f(\theta^{(i+1)} | \theta^{(i)}, y)\]

• Monte Carlo: Random samples from chain

• Markov Chain Monte Carlo produces a Markov chain of samples to approximate posterior.

  • Samples are not directly from the posterior and are not independent!

  • More on how it works in next chapter. But we can just jump in with rstan