Detailed balance

Summary

Detailed balance is a property of a distribution $\pi$ and a Markov chain. It says that, in equilibrium, the probability flow from each state $i$ to each state $j$ equals the flow from $j$ back to $i$. Detailed balance implies that $\pi$ is a stationary distribution, and a chain with such a $\pi$ is called reversible. It is the usual design principle of Markov chain Monte Carlo algorithms.