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Markov Chain Monte Carlo Without All the Bullshit (2015)
I have a little secret: I don’t like the terminology, notation, and style of writing in statistics. I find it unnecessarily complicated. This shows up when trying to read about Markov Chain Monte Carlo methods. Take, for example, the abstract to the Markov Chain Monte Carlo article in the Encyclopedia of Biostatistics. Markov chain Monte Carlo (MCMC) is a technique for estimating by simulation the expectation of a statistic in a complex model.
If we additionally require the stupid edge-case-catcher that no edge can have zero probability, then strong connectivity (of one component of a graph) is equivalent to the following property: Finally, in order to describe the stationary distribution in a more familiar manner (using linear algebra), we will write the transition probabilities as a matrix $ A$ where entry $ a_{j,i} = p_{(i,j)}$ if there is an edge $ (i,j) \in E$ and zero otherwise. There is an extra technical condition one can add to strong connectivity, called aperiodicity, which allows one to beef up the theorem so that $ x_t$ itself converges to the stationary distribution.
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