Markov chain methods consist of repeatedly applying stochastic updates until eventually the state begins to yield samples from the equilibrium. However, samples drawn from the equilibrium distribution are identically distributed, any two successive samples will be highly correlated with each other. We should return only every n successive sample to avoid the bias of the correlation. Thus Markov chains require time to burn in to the equilibrium distribution and the time to transition from one sample to another reasonably decorrelated sample after reaching equilibrium.

State University of New York at Stony Brook

There are some drawbacks of using Markov chain Monte Carlo methods. It is expensive to use because it required more time, and it is hard to evaluate the length of running time because we do not know in advance how many steps the Markov chain must run before reaching its equilibrium distribution. 


Drawbacks of MCMC

Goodfellow, I., Bengio, Y., & Courville, A. (2016). $\mathit{Deep \ Learning.}$ MIT Press. Retrieved from [www.deeplearningbook.org](https://www.deeplearningbook.org) 

Deep Learning

Expensive to Use

We do not know in advance how many steps the Markov chain must run before reaching its equilibrium distribution. This length of time is called the mixing time. Testing whether a Markov chain has reached equilibrium is also diﬃcult. We do not have a precise enough theory for guiding us in answering this question. So, we simply run the Markov chain for an amount of time that we roughly estimate to be suﬃcient, and use heuristic methods to determine whether the chain has mixed. These heuristic methods include manually inspecting samples or measuring correlations between successive samples.

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