# Convergence of MCMC for ill-behaved functions

We are doing MCMC sampling over an n-dimensional space with a population of MCMCs. The goal is to have them stop when they approximately converge. The problem is that the convergence, which is so easily visually seen is hard to code. Typical Gelman-Rubin/Geweke work well for well behaved functions, but for some nasty ones with weird features, they oscillate or have random peaks that do not indicate the steady slow convergence as wanted.

What other convergence statistics can you recommend? I was looking at using time series methods but cannot find any papers on time series convergence.

I was thinking would it be possible to take a discrete fourier and do something with that or.....?

Some already looked at: Cramer von Mises, Kolmogorov-Smirnov, Kuiper, Raftley-Lewis, Heinberg-Wurtley.

There are different routes suggested in the literature, but none of them is foolproof to check that

1. one has reached stationarity in the sense that $$X_t\sim\pi(x)$$ marginally;
2. one has explored the stationary distribution in the sense that$$\frac{1}{T}\sum_{t=1}^T h(X_t)\approx\mathbb{E}^\pi[h(X)]\tag{2}$$

(which are two different notions of convergence). Among these, for problem 1.

1. coupling the Markov chain with a perfect sampling version (expensive)
2. using renewal events (expensive)
3. comparing MCMC implementations with different parameterisations (no guarantee)
4. using control variates like the score function (no guarantee)
5. estimating the visited mass of the visited $$\pi$$ over the visited region (requires normalisation constant)
6. using tempering with particle systems to compare the range of the values of the target $$\pi$$ visited (no guarantee)

The convergence of the average (2) can be assessed by blocking or subsampling assuming the CLT holds.