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Chapter 5: Problem 51

(a) Show that the limiting probabilities of the reversed Markov chain are the same as for the forward chain by showing that they satisfy the equations $$ \pi_{j}=\sum_{i} \pi_{i} Q_{u} $$ (b) Give an intuitive explanation for the result of part (a).

Short Answer

Expert verified
To show that the limiting probabilities of the reversed Markov chain are the same as for the forward chain, we first recall the equation for the forward chain \(\pi_{j} = \sum_{i} \pi_{i} P_{ij}\) and the equation for the reversed chain \(\pi_{j}=\sum_{i} \pi_{i} Q_{ij}\). We then establish the relationship between the transition probability matrices \(P\) and \(Q\): \(\pi_{i} P_{ij} = \pi_{j} Q_{ij}\), and substitute this into the forward chain equation. After the substitution, we find that the equation for the forward chain matches the equation for the reversed chain, verifying that the limiting probabilities are the same for both chains. Intuitively, this occurs because the equilibrium state doesn't depend on the direction of the transitions, resulting in the same stationary distribution for both forward and reverse chains.

Step by step solution

01

Write the equation for the forward Markov chain

Recall the relationship for the forward Markov chain: \(\pi P = \pi\), where \(\pi\) is the limiting probability vector and \(P\) is the transition probability matrix. We can also describe it element-wise, as follows: $$ \pi_{j} = \sum_{i} \pi_{i} P_{ij}, $$ where \(\pi_{j}\) is the probability of being in state \(j\) and \(P_{ij}\) is the probability of transition from state \(i\) to state \(j\).
02

Write the equation for the reversed Markov chain

Now, let's consider the reversed Markov chain, with transition probability matrix \(Q\) (obtained by reversing the direction of the transitions in the forward chain). The limiting probabilities for this reversed chain should satisfy: $$ \pi Q = \pi, $$ which in element-wise form, is: $$ \pi_{j}=\sum_{i} \pi_{i} Q_{ij}. $$
03

Derive the relationship between P and Q

The transition probability matrices \(P\) and \(Q\) for the forward and reversed chains, respectively, are related through the following equation: $$ \pi_{i} P_{ij} = \pi_{j} Q_{ij}. $$ We'll use this relationship to show that the limiting probabilities are the same for both chains.
04

Substitute the relationship into the forward chain equation

By substituting the relationship from Step 3 into the equation for the forward chain from Step 1, we have: $$ \pi_{j} = \sum_{i} \frac{\pi_{i} P_{ij}}{Q_{ij}} Q_{ij}. $$ This equation allows us to connect the forward and reverse chains.
05

Show that the limiting probabilities are the same for both chains

Now, let's compare the equations for the forward and reversed chains from Step 1 and Step 2, respectively: $$ \pi_{j} = \sum_{i} \pi_{i} P_{ij}, \quad\text{and}\quad \pi_{j} = \sum_{i} \pi_{i} Q_{ij}. $$ By substituting the relationship derived in Steps 3 and 4, we get: $$ \pi_{j} = \sum_{i} \frac{\pi_{i} P_{ij}}{Q_{ij}} Q_{ij}, $$ which is equal to the equation for the reversed chain. This shows that the limiting probabilities for the reversed chain are the same as those for the forward chain.
06

Intuitive explanation

Intuitively, the reason for this result is that a stationary distribution, which is what the limiting probabilities represent, is an equilibrium state of the Markov chain. An equilibrium doesn't depend on the direction of the transitions. In other words, once in equilibrium, both forward and reverse chains reach the same stationary distribution. This confirms that the limiting probabilities of the reversed chain are indeed the same as those for the forward chain.

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Most popular questions from this chapter

Prove that if the number of states in a Markov chain is \(M\), and if state \(J\) can be reached from state \(i\), then it can be reached in \(M\) steps or less.

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