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- Markov_chain_mixing_time abstract "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 thatfor all subsets A of states and all initial states. This is the sense in which Dave Bayer and Persi Diaconis (1992) 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) (Jerrum 1995). 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.".
- Markov_chain_mixing_time wikiPageExternalLink MARKOV.
- Markov_chain_mixing_time wikiPageExternalLink book.html.
- Markov_chain_mixing_time wikiPageID "6101938".
- Markov_chain_mixing_time wikiPageRevisionID "604971370".
- Markov_chain_mixing_time author1Link "Dave Bayer".
- Markov_chain_mixing_time author2Link "Persi Diaconis".
- Markov_chain_mixing_time first "Dave".
- Markov_chain_mixing_time first "Persi".
- Markov_chain_mixing_time hasPhotoCollection Markov_chain_mixing_time.
- Markov_chain_mixing_time last "Bayer".
- Markov_chain_mixing_time last "Diaconis".
- Markov_chain_mixing_time year "1992".
- Markov_chain_mixing_time subject Category:Markov_processes.
- Markov_chain_mixing_time type Abstraction100002137.
- Markov_chain_mixing_time type Act100030358.
- Markov_chain_mixing_time type Activity100407535.
- Markov_chain_mixing_time type Event100029378.
- Markov_chain_mixing_time type MarkovProcesses.
- Markov_chain_mixing_time type Procedure101023820.
- Markov_chain_mixing_time type PsychologicalFeature100023100.
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- Markov_chain_mixing_time comment "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.".
- Markov_chain_mixing_time label "Markov chain mixing time".
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