### Markov chain Monte Carlo

**(MCMC)**
(model technique to approximate multi-dimensional integrals)

**Markov chain Monte Carlo** is a technique in calculating an approximation
of a multi-dimensional definite integral. It is useful in
determining approximations of probabilities that are impractical
to calculate directly. The development of the technique (beginning in
the 1950s) and its use with computers is one of the keys to the
increased popularity of Bayesian statistics.

A **Markov chain** is a process that goes from state to state according
to given probabilities that do not depend upon history, i.e., the
probabilities regarding which state will follow the current state
are fixed, and, for example, if it returned to the same state, the
probabilities would be the same as at present. (Note that some
processes that seemingly don't fit this pattern can be modeled as
a Markov chain through inclusion of more states.)

In computational mathematics/science, a Monte Carlo method is the
determination of the probability of various outcomes by trying many
random examples, and counting and/or recording the various outcomes.
It depends upon the randomization of input data, depending on
probability to produce good results, and was given the name
*Monte Carlo* because of the place's association with gambling,
thus *chance*.

Combined, they can be used to approximate the solution to a
multi-dimensional integral, effectively solving sets of differential
equations too complex for more straight-forward **numerical integration**
techniques. For statistics, the general idea is that a Markov chain
can be devised that generates simulated samples of a probability
distribution, generating these sample instances with this distribution's
frequency. An approximation of the distribution is gathered by
counting the samples generated, e.g., binning them to make a
histogram. The advantage is that such a Markov chain can at times
be devised in cases where determining the distribution through
solving equations analytically or through (other) numerical
methods is impossible or impractical. In determining such a distribution,
it is effectively evaluating a complicated integral, thus can be
adapted to other applications requiring similar integral evaluations.

(*technique,models,mathematics,computation*)
http://en.wikipedia.org/wiki/Markov_chain_Monte_Carlo

https://towardsdatascience.com/a-zero-math-introduction-to-markov-chain-monte-carlo-methods-dcba889e0c50

**Referenced by:**

Bayesian statistics

forward model

Monte Carlo method

numerical analysis

spectral energy distribution (SED)

index