# MCMC without detailed balance

The following paper provides a rather simple MCMC method that does not satisfy the detailed balance condition but rather only satisfies the balance condition: https://arxiv.org/pdf/1007.2262.pdf

They showed empirically that this method is able decrease the auto-correlation time of Potts model by a factor of 6 compared to conventional Metropolis-Hastings algorithm. My question is, since only the balanced condition is required for the MCMC to respect the stationary distribution in theory, why do people force the detailed balance condition in the practical implementation of most MCMC based algorithms. Is there a substantial advantage of an algorithm satisfying the detailed balance condition over one that only satisfies the balanced condition?

Why do people force the detailed balance condition in the practical implementation of most MCMC based algorithms.

The traditional MCMC which required the detail balance condition to ensure the Markov kernel is reversible. We also call this kind of algorithm reversible MCMC. The typical example is Metropolis hasting and Hamiltonian Monte Carlo. And the detail balance condition is a sufficient condition for the invariant distribution.

Is there a substantial advantage of an algorithm satisfying the detailed balance condition over one that only satisfies the balanced condition?

Yes, as I know, the reversible Markov chain's asymptotic property or convergence is easier to analysis compare with the non-reversible chain in math. However, as the paper you mentioned, the non-reversible chain becomes more and more popular now at least in statistic field, since it converge much faster than the reversible cases. Algorithms like PDMP(Piecewise-deterministic Markov process) and Zigzag are very hot topics now.

• do you have any introductory references to PMDMPs? The zig-zag process looks to be a good starting point.
– fool
Oct 11, 2020 at 20:47
• Dr. Joris Bierkens' home page provide a lot of materials about PDMP, it worth to read if you are interested in it. diamweb.ewi.tudelft.nl/~joris/pdmps.html Oct 12, 2020 at 2:48
• Awesome!!! Thank you, I will closer look :)!!!!
– fool
Oct 12, 2020 at 3:51