detailed balance and reversible chains

Imagine filming a chain at equilibrium and then running the movie backwards. If the reversed film is statistically indistinguishable from the forward one — the same transition statistics — the chain is called reversible, and it satisfies a stronger, prettier balance than mere stationarity. Detailed balance is the condition that, at equilibrium, the flow of probability from i to j exactly matches the flow back from j to i, for every single pair of states.

The detailed balance equation is pi(i) P(i, j) = pi(j) P(j, i) for all states i and j. The left side is the equilibrium rate at which probability crosses from i to j; the right side is the rate back from j to i. Demanding these be equal for every pair is far stronger than ordinary balance (which only requires the TOTAL inflow to each state to equal its outflow). The key payoff: if a distribution pi satisfies detailed balance with P, then pi is automatically stationary — summing the detailed-balance equations over i recovers pi P = pi for free.

This gives a wonderfully easy way to verify or design equilibria. Rather than solving the full linear system pi P = pi, you just check the pairwise equations, which is often much simpler. Reversibility is the engine behind the Metropolis-Hastings algorithm: it constructs a chain by deliberately enforcing detailed balance with respect to a desired target pi, guaranteeing pi is the stationary distribution. Be aware, though: many perfectly good stationary chains are NOT reversible — detailed balance is sufficient for stationarity but not necessary.