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I have not be able to find clear answers to the following two questions regarding MCMC:

  1. If I were to evaluate the expected value (more generally, expected value of a function) of a target distribution using MCMC, my first thought of the procedure would be as follows:

    • given a Markov chain with the target distribution being its equilibrium distribution, I simulate many times, say N, the chain up to a predefined number of steps, say K. In each simulation, the output is a sample of X_K. So my total sample will be X^(1)_K, X^(2)_K,..,X^(N)_K.

-Then, I averaged the sample above to obtain the expected value estimate

Is this what actually being done in practice?

  1. Can someone provide an example situation where MCMC is a better simulation method to use than the conventional MC (simulate many independent samples from the distribution).

Thanks

TW

user8677843
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  • 1. No, you don't have to create N chains and always take the last sample to get the average. You just simulate one chain, (discard the first few-hundred samples until the chain has reached stationarity) and take the average over all the samples in this chain to get your expected value estimate. 2. For instance, if you want to sample from a conditional distribution. An example would be you want to sample from a standard Gaussian, but with the condition 'X > 2.7'. Standard MC would be very inefficient in this case. – misterkugelblitz Oct 19 '17 at 08:09
  • Thanks for the comments. Further to your answers, I have the following questions: 1) if I only use samples in the same chain, they are dependent. Can the convergence be justified mathematically? 2) MCMC seems only works for finite state process? If I have a continuous time process, say some diffusion process, can MCMC still be applied? I would imagine this can be very slow as typically we want to simulate values over a multiple time points for financial applications. – user8677843 Oct 25 '17 at 14:16
  • 1) Yes, they are dependent, but you can still estimate the mean. This can be justified mathematically (keywords: stationarity, reversibility). If you want to increase the dependence you can e.g. just save every k-th value. 2) I am no expert in random processes, but I think you need to discretize every numerical solution unless you have an analtical expression of your process. – misterkugelblitz Oct 31 '17 at 19:02

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