In probability theory, the mixing time of a Markov chain is the time until the Markov chain is "close" to its steady state distribution.
More precisely, a fundamental result about Markov chains is that a finite state irreducible aperiodic chain has a unique stationary distribution π and, regardless of the initial state, the time-t distribution of the chain converges to π as t tends to infinity. Mixing time refers to any of several variant formalizations of the idea: how large must t be until the time-t distribution is approximately π ? One variant, variation distance mixing time, is defined as the smallest t such that
for all subsets A of states and all initial states. This is the sense in which David Bayer and Persi Diaconis proved that the number of riffle shuffles needed to mix an ordinary 52 card deck is 7. Mathematical theory focuses on how mixing times change as a function of the size of the structure underlying the chain. For an n-card deck, the number of riffle shuffles needed grows as 1.5 log (n) / log (2). The most developed theory concerns randomized algorithms for #P-Complete algorithmic counting problems such as the number of graph colorings of a given n vertex graph. Such problems can, for sufficiently large number of colors, be answered using the Markov chain Monte Carlo method and showing that the mixing time grows only as n log (n). This example and the shuffling example possess the rapid mixing property, that the mixing time grows at most polynomially fast in log (number of states of the chain). Tools for proving rapid mixing include arguments based on conductance and the method of coupling. In broader uses of the Markov chain Monte Carlo method, rigorous justification of simulation results would require a theoretical bound on mixing time, and many interesting practical cases have resisted such theoretical analysis.
Famous quotes containing the words chain, mixing and/or time:
“Loyalty to petrified opinions never yet broke a chain or freed a human soul in this world—and never will.”
—Mark Twain [Samuel Langhorne Clemens] (1835–1910)
“It was not till the middle of the second dance, when, from some pauses in the movement wherein they all seemed to look up, I fancied I could distinguish an elevation of spirit different from that which is the cause or the effect of simple jollity.—In a word, I thought I beheld Religion mixing in the dance.”
—Laurence Sterne (1713–1768)
“The little toy dog is covered with dust,
But sturdy and stanch he stands;
And the little toy soldier is red with rust,
And the musket moulds in his hands.
Time was when the little toy dog was new,
And the soldier was passing fair;
And that was the time when our Little Boy Blue
Kissed them and put them there.”
—Eugene Field (1850–1895)