Detailed Balance

A Markov chain with transition matrix $P$ satisfies detailed balance with respect to a distribution $\pi$ if ${\pi }_i{p}_{ij}={\pi }_j{p}_{ji}$ for every pair of states $i,j$. A chain satisfying this condition is called reversible with respect to $\pi$, meaning a trajectory of the chain, run backward in time, has the same statistical behavior as one run forward. Detailed balance is a sufficient condition for $\pi$ to be a stationary distribution: summing both sides over $i$ recovers exactly the stationarity equation $\pi P=\pi$. It is not necessary, however, some chains have a stationary distribution without satisfying detailed balance for any pairing (these are non-reversible chains). Detailed balance is the design principle behind the Metropolis-Hastings algorithm, which constructs an acceptance probability specifically to enforce this pairwise balance for an arbitrary target distribution $\pi$.

• Detailed balance is easier to verify and construct than stationarity directly: checking a single pairwise equation for every state pair is often more tractable than solving the full system $\pi P=\pi$, which is exactly why MCMC algorithm designers lean on it.
• Not every Markov chain with a valid stationary distribution is reversible, non-reversible chains (which fail detailed balance for their stationary $\pi$) still converge correctly but sometimes mix faster in practice, an active area of MCMC algorithm research.
• The Metropolis-Hastings acceptance probability, $\min \left(1,\frac{\pi (j)q(i\mid j)}{\pi (i)q(j\mid i)}\right)$, is derived by algebraically forcing detailed balance to hold for an arbitrary target $\pi$ and an arbitrary proposal distribution $q$, one of the most consequential applications of this identity in applied statistics.