Editing Detailed balance
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:<math>\pi_{i} P_{ij} = \pi_{j} P_{ji}\,,</math> | :<math>\pi_{i} P_{ij} = \pi_{j} P_{ji}\,,</math> | ||
where <math>P</math> is the Markov transition matrix (transition probability), ''i.e.'', ''P''<sub>''ij''</sub> = ''P''(''X''<sub>''t''</sub> = ''j'' | ''X''<sub>''t'' − 1</sub> = ''i''); and π<sub>''i''</sub> and π<sub>''j''</sub> are the equilibrium probabilities of being in states ''i'' and ''j'', respectively.<ref name=OHagan>Anthony O'Hagan and Jonathan J. Forster "Kendall's Advanced Theory of Statistics, Volume 2B: Bayesian Inference", | where <math>P</math> is the Markov transition matrix (transition probability), ''i.e.'', ''P''<sub>''ij''</sub> = ''P''(''X''<sub>''t''</sub> = ''j'' | ''X''<sub>''t'' − 1</sub> = ''i''); and π<sub>''i''</sub> and π<sub>''j''</sub> are the equilibrium probabilities of being in states ''i'' and ''j'', respectively.<ref name=OHagan>Anthony O'Hagan and Jonathan J. Forster "Kendall's Advanced Theory of Statistics, Volume 2B: Bayesian Inference", Oxford University Press (2004) ISBN 0340807520</ref> When Pr(''X''<sub>''t''−1</sub> = ''i'') = π<sub>''i''</sub> for all ''i'', this is equivalent to the joint probability matrix, Pr(''X''<sub>''t''−1</sub> = ''i'', ''X''<sub>''t''</sub> = ''j'') being symmetric in ''i'' and ''j''; or symmetric in ''t'' − 1 and ''t''. | ||
The definition carries over straightforwardly to continuous variables, where π becomes a probability density, and ''P''(''s''′, ''s'') a transition kernel probability density from state ''s''′ to state ''s'': | The definition carries over straightforwardly to continuous variables, where π becomes a probability density, and ''P''(''s''′, ''s'') a transition kernel probability density from state ''s''′ to state ''s'': |