# Measure the distance between two probability transition matrices

I have a probability transition matrix $$P$$ that contains some values very close to zero. I want to sparsify this matrix by taking the k largest values for each row and setting the others to zero. For instance, taking $$k=2$$

$$P = \left( \begin{array}{cccc} 0.9 & 0.0999 & 0.0001 \\ 0.199 & 0.8 & 0.001 \\ 0.199 & 0.001 & 0.8 \end{array} \right)$$

and

$$P_{\text{sparse}} = \left( \begin{array}{cccc} 0.9 & 0.1 & 0.0 \\ 0.2 & 0.8 & 0.0 \\ 0.2 & 0.0 & 0.8 \end{array} \right) .$$

What is the best way to measure the distance between $$P$$ and $$P_{\text{sparse}}$$?

• Euclidean distance between $$P$$ and $$P_{\text{sparse}}$$?
• KL divergence between the underlying stationary state distribution?
• ...

I've look already at other questions (for instance https://mathoverflow.net/questions/225749/calculate-the-kl-divergence-between-two-transition-matrices) and articles (https://ieeexplore.ieee.org/document/1291741), but could not find a clear answer until now.

• How about absolute difference? Jul 23 '19 at 6:55
• I thought of it also, but I don't know if it makes sense when comparing these stochastic matrices. Jul 23 '19 at 12:18
• can you put the question in a maximum likelihood framework? then the metric might become more obvious Dec 27 '20 at 14:16