Kemeny's constant
In probability theory, Kemeny’s constant is the expected number of time steps required for a Markov chain to transition from a starting state i to a random destination state sampled from the Markov chain's stationary distribution. Surprisingly, this quantity does not depend on which starting state i is chosen.[1] It is in that sense a constant, although it is different for different Markov chains. When first published by John Kemeny in 1960 a prize was offered for an intuitive explanation as to why quantity was constant.[2][3]
Definition
For a finite ergodic Markov chain[4] with transition matrix P and invariant distribution π, write mij for the mean first passage time from state i to state j (denoting the mean recurrence time for the case i = j). Then
is a constant and not dependent on i.[5]
Prize
Kemeny wrote, (for i the starting state of the Markov chain) “A prize is offered for the first person to give an intuitively plausible reason for the above sum to be independent of i.”[2] Grinstead and Snell offer an explanation by Peter Doyle as an exercise, with solution “he got it!”[6][7]
In the course of a walk with Snell along Minnehaha Avenue in Minneapolis in the fall of 1983, Peter Doyle suggested the following explanation for the constancy of Kemeny's constant. Choose a target state according to the fixed vector w. Start from state i and wait until the time T that the target state occurs for the first time. Let Ki be the expected value of T. Observe that
and hence
By the maximum principle, Ki is a constant. Should Peter have been given the prize?
References
- ^ Attention: This template ({{cite doi}}) is deprecated. To cite the publication identified by doi:10.1080/00207179.2011.568005, please use {{cite journal}} (if it was published in a bona fide academic journal, otherwise {{cite report}} with
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instead. - ^ a b Kemeny, J. G.; Snell, J. L. (1960). Finite Markov Chains. Princeton, NJ: D. Van Nostrand. (Corollary 4.3.6)
- ^ Attention: This template ({{cite doi}}) is deprecated. To cite the publication identified by doi:10.1007/s10915-010-9382-1, please use {{cite journal}} (if it was published in a bona fide academic journal, otherwise {{cite report}} with
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instead. - ^ Attention: This template ({{cite jstor}}) is deprecated. To cite the publication identified by jstor:3072398, please use {{cite journal}} with
|jstor=3072398
instead. - ^ Hunter, Jeffrey J. (2012). "The Role of Kemeny's Constant in Properties of Markov Chains". arXiv:1208.4716 [math.PR].
- ^ Grinstead, Charles M.; Snell, J. Laurie. Introduction to Probability (PDF).
- ^ "Two exercises on Kemeny's constant" (PDF). Retrieved 1 March 2013.