Thread: Metropolis Monte Carlo Autocorrelation

Hello,

I do not quite understand yet how to assess the convergence of a metropolis run using the autocorrelation. I think I get the principle, that if we have less autocorrelation in the Markov chain, the obtained samples are more independent.
For each parameter in my model, I calculate the autocorrelation at time-lag as follows:



where is the mean of .

Like this I calculate the autocorrelations for each parameter for each time lag . So when I do 100 steps in my Metropolis-Algorithm, I calculate all autocorrelations for . Of course, the autocorrelation at lag 0 is 1, and drops to around zero. But in all cases, it drops to around zero, no matter how many steps I do. When I plot the autocorrelation of a MC-Simulation with 1000 steps, then I see that each parameter converges after about 1/4 of the steps. But if I do only 10 steps, I see the same, after about 1/4 of the steps each parameter converges, which cannot be. Where is my mistake in thinking? How can I really use autocorrelation to assess that my ensemble converges?

Many thanks in advance,
Greetings Hans

How are the thetas distributed? They obviously are not independent otherwise you won't be calculating the autocorrelations. I suppose they are identically guassian distributed? If so, then what is the joint pdf that you are using to simulate the thetas?

Originally Posted by straussvogel

Hello,

I do not quite understand yet how to assess the convergence of a metropolis run using the autocorrelation. I think I get the principle, that if we have less autocorrelation in the Markov chain, the obtained samples are more independent.
For each parameter in my model, I calculate the autocorrelation at time-lag as follows:



where is the mean of .

Like this I calculate the autocorrelations for each parameter for each time lag . So when I do 100 steps in my Metropolis-Algorithm, I calculate all autocorrelations for . Of course, the autocorrelation at lag 0 is 1, and drops to around zero. But in all cases, it drops to around zero, no matter how many steps I do. When I plot the autocorrelation of a MC-Simulation with 1000 steps, then I see that each parameter converges after about 1/4 of the steps. But if I do only 10 steps, I see the same, after about 1/4 of the steps each parameter converges, which cannot be. Where is my mistake in thinking? How can I really use autocorrelation to assess that my ensemble converges?

Many thanks in advance,
Greetings Hans