Which convention for transition matrices in Markov chains is correct

[songoku]

TL;DR

Transition matrix is matrix that shows the probability of going into future state from a certain current state

The note I get from the teacher states that for transition matrix, the column part will be current state and the row part will be future state (let this be matrix A) so the sum of each column must be equal to 1. But I read from another source, the row part is the current state and the column part is the future state (let this be matrix B) so the sum of row is equal to 1. Matrix B is transpose of matrix A but when I try to multiply each of them with other matrix (matrix of the current value of observation), I get different results

https://www.dartmouth.edu/~chance/teaching_aids/books_articles/probability_book/Chapter11.pdf

That link states that the row part is the current state and the column part is the future state

https://www.math.ucdavis.edu/~dadde...tions/MarkovChain/MarkovChain_9_18/node1.html

The second link states that the column part will be current state and the row part will be future state

So which one is correct, matrix A or matrix B? Or maybe I am missing something?

Thanks

 

[mfb]

It's a matter of convention. Do you prefer to multiply the transition matrix with a column vector on the right or with a row vector on the left?
It's unfortunate that there are places where people didn't agree on a single convention, but as long as you keep the convention consistent within your work it will give the right result.

 

[songoku]

It's a matter of convention. Do you prefer to multiply the transition matrix with a column vector on the right or with a row vector on the left?

Oh I see. I prefer to multiply the transition matrix with a column vector on the right side of transition matrix so the one I use should be matrix A, correct?

Thanks

 

[mfb]

Right.

 

[songoku]

Thank you very much mfb

 

[John Finn]

This brings up a related issue. One can iterate a Markov chain $p(i,t+1)=\sum_j T_{i,j) p(j,t)$ from $t=0$ to $t=N$, i.e. in vector form $p(t+1)=T p(t)$ and then make the measurement $c=(q(N),p(N))$, where $(\cdot , \cdot)$ is the $l^2$ inner product. Or you could advance the measurement vector q(N) \emph{backwards} by the transpose $T^T$ of the transition matrix $T$, and then take the inner product at $t=0$. This is a basic adjoint method. Have such adjoint methods been used in Markov processes?