The Detailed Balance Condition and Time Reversibility

Solving $\pi p = \pi$ directly, as the previous lesson did, requires setting up and solving a linear system — feasible for three states, tedious for thirty. Detailed balance is a stronger, purely local condition — balance the flow between every pair of states, not just the flow into and out of each state overall — that, when it holds, both guarantees stationarity and often makes $\pi$ trivial to write down by hand. It also has a striking interpretive payoff: a chain satisfying detailed balance looks statistically identical whether you watch it run forwards or backwards in time.

Learning objectives. By the end of this note you should be able to: