Lagmental Vicfred

Detailed Balance Makes a Chain Reversible by Vicfred

Last updated: Wed 22 June 2022

tags: mathematics, Markov chains, stationary distributions, random walks

Pairwise equality of stationary probability flow implies stationarity and time-reversal symmetry. A small computation will anchor the general statement before the abstraction takes over.

The mathematical object

A discrete-time Markov chain has transition matrix \(P=(p_{ij})\) and forgets the past after conditioning on the present. Matrix powers \(P^n\) give multi-step transition probabilities.

$$ \pi_ip_{ij}=\pi_jp_{ji} $$

I keep the defining relation \(\mathsf D\) above the derived relation \(\mathsf C\). This exposes whether cancellation used \(x\ne0\) and whether the conclusion is canonical.

$$ \pi_i p_{i_0i_1}\cdots p_{i_{n-1}i_n}=\pi_{i_n}p_{i_ni_{n-1}}\cdots p_{i_1i_0} $$

One explicit computation

An explicit case prevents the notation from becoming ceremonial. Every subscript and superscript in the display contributes to the value.

$$ \sum_i\pi_ip_{ij}=\sum_i\pi_jp_{ji}=\pi_j\sum_ip_{ji}=\pi_j $$

Why the identity matters

The aligned summary deliberately puts the datum and conclusion on different rows. Mathematically, this is the distinction between specifying an object and proving a property of it.

$$ \begin{aligned} \mathsf{D}\;&:\quad \pi_ip_{ij}=\pi_jp_{ji},\\[5pt] \mathsf{C}\;&:\quad \pi_i p_{i_0i_1}\cdots p_{i_{n-1}i_n}=\pi_{i_n}p_{i_ni_{n-1}}\cdots p_{i_1i_0}. \end{aligned} $$

Where it can fail

A stationary distribution need not be unique without irreducibility, and convergence to it can fail without aperiodicity.

$$ \boxed{\begin{gathered} \text{compact conclusion}\\[-2pt] \pi_i p_{i_0i_1}\cdots p_{i_{n-1}i_n}=\pi_{i_n}p_{i_ni_{n-1}}\cdots p_{i_1i_0} \end{gathered}} $$

With the dependency made explicit, the same pattern can be recognised safely in nearby problems. A changed hypothesis should now be easy to spot.

This article was posted on Fri 26 April 2019. Facts and circumstances may have changed since publication.
Please contact me before jumping to conclusions if something seems wrong or unclear.