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Chapter 4: Problem 68

(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 $$ \pi_{j}=\sum_{i} \pi_{i} Q_{i j} $$ (b) Give an intuitive explanation for the result of part (a).

Short Answer

Expert verified
In summary, the limiting probabilities of the reversed Markov chain are the same as those for the forward chain because they both satisfy the equation \(\pi_j = \sum_{i} \pi_i Q_{ij}\). Intuitively, this is because the long-term behavior of the Markov chain is not affected by the direction in which it is traversed, and the limiting probabilities represent the proportion of time spent in each state, which is independent of the direction of the chain.

Step by step solution

01

Understanding the Forward Markov Chain

For a (discrete-time) forward Markov chain with state space \(S\), we have a transition probability matrix \(P\), where the element \(P_{ij}\) is the probability of transitioning from state \(i\) to state \(j\). The limiting probabilities \(\pi = (\pi_i)_{i\in S}\) are the long-term probabilities that the chain is in each state \(i\), known as the stationary distribution, which satisfy \(\pi_i \geq 0\) for all \(i\), \(\sum_{i} \pi_i = 1\), and \[\pi = \pi P .\]
02

Understanding the Reversed Markov Chain

For a reversed Markov chain, we still have the state space \(S\) and stationary distribution \(\pi\). We define the transition probability matrix \(Q\) for the reversed chain by: \[Q_{ij} = \frac{\pi_j}{\pi_i} P_{ji},\] which represents the probability of transitioning from state \(j\) to state \(i\), reversing the direction of the forward chain.
03

Show that the Limiting Probabilities Satisfy the Equations

Now we need to show that the limiting probabilities of the reversed Markov chain, represented by the vector \(\pi\), satisfy the given equation: \[\pi_j = \sum_{i} \pi_i Q_{ij} .\] Let's compute the right-hand side of this equation and show that it indeed equals \(\pi_j\): \[\sum_{i} \pi_i Q_{ij} = \sum_{i} \pi_i \left(\frac{\pi_j}{\pi_i}P_{ji}\right) = \pi_j \sum_{i} P_{ji} .\] Recall that each row in the transition probability matrix \(P\) must sum to 1, so we get \[\pi_j \sum_{i} P_{ji} = \pi_j (1) = \pi_j .\] Thus, the limiting probabilities of the reversed Markov chain are the same as those for the forward chain.
04

Intuitive Explanation

The reversed Markov chain is essentially "playing the forward Markov chain backward". Since the stationary distribution reflects long-term behavior, it is reasonable that the limiting probabilities should be the same for both the forward and reversed chains. At a high level, this is because the long-term behavior of the chain is not affected by the direction in which it is traversed. The limiting probabilities represent the proportion of time spent in each state over the long run, and these proportions are independent of the direction of the chain.

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Most popular questions from this chapter

Three white and three black balls are distributed in two urns in such a way that each contains three balls. We say that the system is in state \(i, i=0,1,2,3\), if the first urn contains \(i\) white balls. At each step, we draw one ball from each urn and place the ball drawn from the first urn into the second, and conversely with the ball from the second urn. Let \(X_{n}\) denote the state of the system after the \(n\) th step. Explain why \(\left\\{X_{n}, n=0,1,2, \ldots\right\\}\) is a Markov chain and calculate its transition probability matrix.

A particle moves among \(n+1\) vertices that are situated on a circle in the following manner. At each step it moves one step either in the clockwise direction with probability \(p\) or the counterclockwise direction with probability \(q=1-p\). Starting at a specified state, call it state 0 , let \(T\) be the time of the first return to state \(0 .\) Find the probability that all states have been visited by time \(T\).

A professor continually gives exams to her students. She can give three possible types of exams, and her class is graded as either having done well or badly. Let \(p_{i}\) denote the probability that the class does well on a type \(i\) exam, and suppose that \(p_{1}=0.3, p_{2}=0.6\), and \(p_{3}=0.9\). If the class does well on an exam, then the next exam is equally likely to be any of the three types. If the class does badly, then the next exam is always type 1 . What proportion of exams are type \(i, i=1,2,3 ?\)

\(M\) balls are initially distributed among \(m\) urns. At each stage one of the balls is selected at random, taken from whichever urn it is in, and then placed, at random, in one of the other \(M-1\) urns. Consider the Markov chain whose state at any time is the vector \(\left(n_{1}, \ldots, n_{m}\right)\) where \(n_{i}\) denotes the number of balls in urn \(i\). Guess at the limiting probabilities for this Markov chain and then verify your guess and show at the same time that the Markov chain is time reversible.

It follows from the argument made in Exercise 38 that state \(i\) is null recurrent if it is recurrent and \(\pi_{i}=0 .\) Consider the one-dimensional symmetric random walk of Example \(4.15\). (a) Argue that \(\pi_{i}=\pi_{0}\) for all \(i\). (b) Argue that all states are null recurrent.
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