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?
Best Answer
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.