Detailed Balance

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Detailed Balance

Definition (Detailed balance)

A transition matrix $P$ and a pmf $\pi$ are said to be in detailed balance if $\pi_{i}p_{ij}=\pi_{j}p_{ji}, \ \mbox{for all }i,j\in S$

Theorem (Sufficient Condition for Identifying Invariant Distribution)

If $P$ and $\pi$ are in detailed balance, then $\pi$ is invariant in $P$ .

Remark

To find an invariant distribution, we could try to solve $\lambda_{i}p_{ij}=\lambda_{j}p_{ji}\mbox{ for all }i,j\in S$

Remark

Detailed balance is a sufficient condition for identifying invariant distribution, not a necessary one. Hence, there may be an invariant distribution but detailed balance does not guarantee finding it.

Definition (Detailed balance — continuous)

A Q-Matrix $Q$ and a distribution $\pi$ are said to be in detailed balance if $\pi_{i}q_{ij}=\pi_{j}q_{ji}$ for all $i,j\in I$ .

Theorem

If $Q$ and $\pi$ are in detailed balance then $\pi Q = 0$ .

Linked from

Metropolis Hastings Algorithm

Detailed Balance

Invariant Distribution