Global balance and detailed balance
Section of Classes, recurrence, and reversibility, part of Stochastic processes
Global balance and detailed balance
| Type | Section |
| Taught in | Classes, recurrence, and reversibility |
| Part | Stochastic processes |
| Questions using it | 1 |
The stationarity equation at state j,
\pi_j=\sum_i \pi_i p_{ij},
equates the probability mass at j with the mass that one step of P brings into j. That is global balance.
Definition 1 (Detailed balance) A probability \pi and a transition matrix P satisfy detailed balance when
\pi_i p_{ij}=\pi_j p_{ji}\qquad\text{for all }i,j.
Sum the pairwise equations over i. The right-hand side becomes \pi_j\sum_i p_{ji}=\pi_j, and the left-hand side becomes \sum_i\pi_i p_{ij}. Detailed balance implies global balance, and therefore stationarity. The same implication for a general transition kernel is on Markov chains, Gibbs sampling, and Metropolis–Hastings. What this page adds is the process reading: time reversal, the cycle test, and the chains that are stationary without being reversible.
A chain that satisfies detailed balance is reversible when it is started in \pi. A stationary path has the same distribution when time is read backwards. Stationarity alone does not give reversibility. If \pi_i>0, the reversed transition probabilities are
p_{ij}^*=\frac{\pi_j p_{ji}}{\pi_i}.
Detailed balance says P^*=P.
For a finite irreducible chain, Kolmogorov’s cycle criterion is an equivalent test that does not require \pi in advance. Around every directed cycle, the product of the transition probabilities equals the product around the reversed cycle. A missing reverse edge counts as probability 0, and then a positive forward cycle product fails the test.
Used in
- Taught in Classes, recurrence, and reversibility
- Also used in Continuous-time Markov chains · Markov chains, Gibbs sampling, and Metropolis–Hastings
- Questions that use it (1) — Two steps and the stationary law (answer)
- All 3 questions on Classes, recurrence, and reversibility
See also
Detailed balance implies stationarity · Stationarity and detailed balance · Recurrence and period · Finite irreducible chains · Detailed balance · Communicating classes · Return and period · Two families with an explicit stationary law