stated transition probabilities for the reversed chain in particularfor reversible markov chain we have (xq( xf (yq( yfor all xy ( these are the detailed (or localbalance equations note that the detailed balance equations imply the global balance equations henceif markov chain is irreducible and there exists pdf such that ( holdsthen (xmust be the limiting pdf in the discrete state space case an additional condition is that the chain must be aperiodicmeaning that the return times to the same state cannot always be multiple of some integer local balance equations aperiodic example (random walk on graphconsider markov chain that performs "random walkon the graph in figure at each step jumping from the current vertex (nodeto one of the adjacent verticeswith equal probability clearly this markov chain is reversible it is also irreducible and aperiodic let (xdenote the limiting probability that the chain is in vertex by symmetryf ( ( ( ( ) ( ( and ( ( moreoverby the detailed balance equationsf ( )/ ( )/ and ( )/ ( )/ it follows that ( ( ( / ( / ( ( so that ( / ( / and ( / figure the random walk on this graph is reversible statistics statistics deals with the gatheringsummarizationanalysisand interpretation of data the two main branches of statistics are classical or frequentist statisticshere the observed data is viewed as the outcome of random data described by probabilistic model -usually the model is specified up to (multidimensionalparameterthat ist (thfor some th the statistical inference is then purely concerned with the model and in particular with the parameter th for exampleon the basis of the data one may wish to (aestimate the parameter(bperform statistical tests on the parameteror frequentist statistics