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Markov Chain Monte Carlo (MCMC) methods attempt to simulate realizations from some complex distribution of interest. MCMC approaches are so-named because one uses the previous sample value to randomly generate the next sample value, creating a Markov chain on the way (as the transition probability from x to x' depends on x only). The Markov chain has to converge to the *equilibrium distribution*. Only then the simulated values can be used. The speed at which the Markov chain converges to the equilibrium distribution is called the *mixing speed*. A classic example of Markov Chain Monte Carlo is the Metropolis-Hastings algorithm.

Major topics in the Markov Chain Monte Carlo research are the following: 1) designing a Markov chain with high mixing speed, keeping low number of supplementary calculations; 2) designing a Markov chain which is relatively insensitive to the choice of initial values; 3) designing convergence diagnostics. Markov Chain Monte Carlo methods are used in two major contexts: 1) if the distribution is known, approximate some complex characteristic of the distribution by simulating it; 2) if the parameters of the distribution of X are unknown, estimate them using Bayesian analysis: assume that the parameters are randomly distributed according to some *prior distribution*; simulate them jointly with X in such a way that the simulated values at a given iteration depend on the simulated values at the previous iteration; after the Markov chain has converged estimate the parameters by averaging their simulated values.

Oftentimes, Markov Chain Monte Carlo exploits Gibbs sampler - a way of simulating a random vector if we know the conditional distribution of each coordinate given the other coordinates. We initialize the coordinates at a reasonable set of values. We fix all them but one. We simulate the remaining coordinate conditional on the values of other coordinates. In a similar fashion, we go through all the coordinates multiple times, until the joint distribution of the simulated coordinates has converged to the true distribution.

**MARKOV CHAIN MONTE CARLO REFERENCES
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Robert, C. P., & Casella, G. (2004). Monte Carlo Statistical Methods (2nd ed.). New York: Springer.

Liu, J. S. (2008). Monte Carlo Strategies in Scientific Computing. Springer.

Brooks, S., Gelman, A., Jones, G., & Meng, X. (2011). Handbook of Markov Chain Monte Carlo. Chapman & Hall / CRC.

Gilks, W. R., Richardson, S., & Spiegelhalter, D. (ed) (1995). Markov Chain Monte Carlo in Practice. Chapman & Hall / CRC.

Berg, B. A. (2004). Markov Chain Monte Carlo Simulations and Their Statistical Analysis (With Web-Based Fortran Code). Hackensack, NJ: World Scientific.

Lemieux, C. (2009). Monte Carlo and Quasi-Monte Carlo Sampling. Springer.

McElreath, R. (2015). Statistical Rethinking: A Bayesian Course with Examples in R and Stan. Chapman & Hall / CRC.

- Monte Carlo Methods in Dynamic and Stochastic Volatility Models, implementation in R, Hedibert.org

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