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2. Metropolis Algorithm Example

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Metropolis Algorithm belongs to Markov Chain Monte Carlo (MCMC) methods. Markov Chain Monte Carlo (MCMC) methods are an effective means of sampling from the posterior distribution of interest even when the posterior has no known closed algebraic form. The basic idea behind MCMC is to generate samples from the posterior distribution and to use these samples to approximate expectations of quantities of interest. Here is the algorithm flow chart:


As an example, you can see how the Metropolis algorithm works step by step:


Markov Chain Monte Carlo methods (MCMC) enable researchers to directly sample sequences of values from the posterior distribution of interest, foregoing the need for closed form analytic solutions. With MCMC, you use these samples to estimate the posterior distribution’s quantities of interest. MCMC methods sample successively from a target distribution. Each sample depends on the previous one, hence the notion of the Markov chain. You can think of a Markov chain applied to sampling as a mechanism that traverses randomly through a target distribution without having any memory of where it has been given the immediate past value. Where it moves next is entirely dependent on where it is now.

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