When are Markov processes non-reversible?

[carllacan]

I had to solve a problem in which I had a Markov process described by a transition matrix with elements $a _{ij} = P(X_t =x_j | X_{t-1} = x_i)$, where $X_t$ is the state at time t and $x_n$ are the possible states of the system.

I was asked to find, given the state of the system at a time t, the probability of the system having been in a certain state at time t-1. Using the Bayes Rule I found how to write a matrix $B$ that represents this reversed process:
$b_{ij} = P(X_{t-1} = x_i| X_t =x_j) =\frac{P(X_t =x_j | X_{t-1} = x_i) P(X_{t-1} = x_i)}{(P(X_t =x_j )} = \frac{a_{ji} \pi _j}{\pi _i}$, where $\vec{\pi}$ is the stationary distribution vector.

Trying to check my result I searched on google for "reversed markov processes" and I found that $a_{ij}\pi_{i} = a_{ji} \pi _j$ it the condition for a Markov process being reversible, and agrees with my solution when A = B. Should I interpret this as that the mean that a Markov process is reversible only if the matrix describing it is the same that describes the reversed process?

 

[mmwave]

it is important to be critical and analyze the information you find, especially when it comes to interpreting mathematical concepts like Markov processes. In this case, it is important to understand the concept of reversibility in relation to Markov processes.

First, let's define a reversible Markov process. A Markov process is considered reversible if the probability of transitioning from state i to state j is equal to the probability of transitioning from state j to state i. In other words, the process is reversible if the transition matrix is symmetrical.

Now, let's look at your solution and the condition you found on Google. Your solution shows that for a Markov process to be reversible, the matrix A (describing the forward process) must be equal to the matrix B (describing the reversed process). This is because the condition for reversibility is that a_{ij}\pi_{i} = a_{ji} \pi _j, which is exactly what your solution shows.

However, this does not mean that a Markov process is only reversible if the transition matrix is the same for both the forward and reversed processes. It is possible for a Markov process to be reversible even if the transition matrix is different for the forward and reversed processes. In fact, there are many examples of reversible Markov processes where the transition matrix is not symmetrical.

So, to answer your question, you should not interpret this as the only condition for a Markov process to be reversible. Reversibility is a concept that applies to the transition matrix of a Markov process, and it can have various conditions and implications. It is important to thoroughly understand the concept and its applications before making any interpretations.