Detailed Balance Condition

Definition

$\forall i,j\in E:{\mu }_i{p}_{ij}={\mu }_j{p}_{ji}$ where $\{X_t\}$ is IRR HMC, $\mathbf{P}$ is the Transition Matrix of the $\{X_t\}$, and $\mathbf{\mu }$ is a Distribution

The probability of the transition from the state $i$ to $j$ is equal to the probability from the $j$ to $i$

Facts

$\forall i,j\in E:{\mu }_i{p}_{ij}={\mu }_j{p}_{ji}\Rightarrow {\mathbf{\mu }}^{⊺}\mathbf{P}={\mathbf{\mu }}^{⊺}$ The detailed balanced condition implies the stationarity of the distribution $\mathbf{\mu }$ In other words, the distribution $\mathbf{\mu }$ satisfying the detailed balanced condition is a Stationary Distribution.