Let Q be a symmetric transition probability matrix on states 1 , . . . , n; that is $q_{i,j} = q_{j,i}$ for all i , j.

Consider a markov chain defined as follows. Whenever the chain is in state i, the value of a random variable $X_i$ with probability mass function $P(X_i = j)= q_{i,j}$ is generated; if $X_i = j,$ then the chain either moves to state j with probability $b_j/(b_i + b_j),$ or it remains in state i otherwise, where $b_1,...,b_n $ are specified positive numbers.

Now how to show that the Markov chain is time reversible with stationary probabilities $$\pi_j = \frac{b_j}{\sum_{i=1}^n b_i}, j = 1,...,n$$

Any computer science help will be accepted.

Note: This question is taken from PROBABILITY MODELS for Computer Science written by Sheldon M. Ross.


1 Answer 1


For this chain, the transition probabilities for $j \ne i$ are are $$p_{ij}=q_{ij}\,\frac{b_j}{b_i+b_j}\,.$$ Thus $\pi_i=b_i/Z$ satisfy $$\pi_i p_{ij}=\frac{b_ib_j}{Z(b_i+b_j)}=\pi_j p_{ji}\,,$$ which is the equation defining reversibility.


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