Detailed balance

The term Detailed Balance ( detailed balance ) denotes a property of homogeneous Markov chains , a special stochastic process . A process in detailed equilibrium is clear if it is not clear whether it is moving forwards or backwards in time.

definition

A Markov chain with possible states and a transition matrix , where the probability for a transition from state to state denotes (i.e. the transition probability ), is called reversible with regard to the distribution , if ${\displaystyle (X_{z})_{z\in \mathbb {Z} }}$ ${\displaystyle (w_{ij})}$ ${\displaystyle w_{ij}}$ ${\displaystyle X_{i}}$ ${\displaystyle X_{j}}$ ${\displaystyle P}$

${\displaystyle P(X_{i})w_{ij}=P(X_{j})w_{ji}}$

applies to all . A Markov chain is called reversible if it has a distribution with respect to which it is reversible. ${\displaystyle i,j}$

The above equation is the condition of detailed equilibrium. If it is fulfilled, the system described by the Markov process is in detailed equilibrium or detailed balance.

properties

• The Metropolis algorithm is an example of a stochastic process that fulfills the property of detailed balance . It is used in Monte Carlo simulations to generate states of a system from previous states according to a transition probability.
• For stationary Markov chains with a transition matrix (i.e. especially for those chains that start in a stationary distribution ) this property is equivalent to temporal reversibility , that is, for the time-reversed process, applies to all ${\displaystyle (X_{z})_{z\in \mathbb {Z} }}$ ${\displaystyle (w_{ij})}$ ${\displaystyle {\tilde {X}}_{z}:=X_{-z}}$ ${\displaystyle t_{1},\dots ,t_{n}\in \mathbb {Z} }$

${\displaystyle (X_{t_{1}},\dots ,X_{t_{n}})\sim (X_{-t_{1}},\dots ,X_{-t_{n}})\,.}$

It is therefore irrelevant for each implementation in which direction it is run.

• Any distribution that meets the detailed balance condition is a stationary distribution. This follows by adding up both sides of the equation in the definition above : ${\displaystyle j}$

${\displaystyle P(X_{i})=P(X_{i})\underbrace {\sum _{j}w_{ij}} _{=1}=\sum _{j}P(X_{i})w_{ij}{\stackrel {\text{Def.}}{=}}\sum _{j}P(X_{j})w_{ji}=[(P(X_{j}))\cdot (w_{ji})]_{i}}$

The convergence of an arbitrary distribution against the stationary distribution is not given from this. The ergodic rate , for example, provides a sufficient criterion for this .

See also

• Gibbs sampling

literature

• G. Bhanot, The Metropolis algorithm , Rep. Prog. Phys. 51 (1988) 429
• Achim Klenke: Probability Theory. 2nd Edition. Springer-Verlag, Berlin Heidelberg 2008, ISBN 978-3-540-76317-8
• Hans-Otto Georgii: Stochastics: Introduction to Probability Theory and Statistics , 5th edition, de Gruyter 2015