Mcmc

Terms

-MCMC: class of algorithms that draw samples form a probability distribution.

Markov Chain: stochastic process that describes a series of events where each event is dependent entirely on the event preceding it.
Moment: (of a function) quantitative measures related to a function's graph (like x,y graph). So in probability it could have the first moment as the expected value, then the second moment as the variance, and the third moment as the skewness.
Credible interval: Interval that characterizes probability distribution. Example: $\gamma = 0.95$ where $\gamma$ is the probability that a value falls within a range. So if $\mu$ is a value with $\gamma = 0.95$ probability to be between (35,45): the credible interval would be $35\leq \mu \leq 45$ .
Posterior probability: By applying Baye's rule to update the prior probability. So the posterior probability would everything that can be known about an uncertain proposition (like a scientific hypothesis).

MCMC create samples from a random variable, that have a probability density proportional to the known function.

S_n(h) = \frac{1}{n}\sum_{i=1}^n h(X_i)

MCMC is used in bayesian stats to calculate moments and credible intervals of posterior probability distributions.