↳ Disclaimer: What We Know... Every irreducible, aperiodic Markov chain has a unique stationary distribution. Mixing Time ' n' Much handwaving today!! HIFI IT = π m (1) Goal is to get a high level intuition / picture for CS 70, Summer 2019 the concept of mixing time and applications m 0 . 8 0 . 2 0 π m = π 0 P m = π 0 0 0 . 3 0 . 7 . Emphasis is on heuristics rather than rigor 0 . 6 0 . 4 0 π m (2) Bonus Lecture, 8/9/19 π m (3) m Q: How long does it take to get close to the stationary distribution? 1 / 17 2 / 17 3 / 17 a Total Variation Distance Mixing Time: Definition Complete Graph With Loops 0%7*0 Just one of many ways to measure how close two I have an irreducible, aperiodic Markov chain. Mixing time analysis sometimes direct! Notation: µ ( n ) is the distribution at time n , and π distributions are. a ① n } { 1,2 , States over . , Take a random walk on a complete graph with . . is its (unique) stationary distribution . → Let P 1 , P 2 be two PMFs. Their TV distance is: loops . What is the stationary distribution? - - Elif I want to keep running my chain until: kn ftnh : . In ] / Phil symbtanwtnfn F- ( UM , Tl ) Estates i d small positive . . E E ← vertices Tv number What does the transition matrix look like? time in stationary . 9 ÷ dist at - TIME ' up thing lighted - The mixing time t mix ( " ) is the first time this .int . , di St happens. starting . . case worst = IT . Mixing Time? Dependence on " ? (Omitted fact: The TV distance between µ ( n ) 1 for E all U l l . . and π decreases as n increases.) prob , at with high least 4 / 17 5 / 17 6 / 17
I Random-To-Top Shuffling I Random-To-Top Shuffling II Random-To-Top Shuffling III con figs n ! S : All orderings of n cards in a deck. At each time, each card is labeled coupled or not. Time until all cards get coupled = time until Deck . number Transitions: Choose card randomly in the deck. 1 is fully random. Initially, all cards start uncoupled . Move it to the top. me all States here uniform Pick a random card C . is over Tl . 2nd number 1st 3rd Di ff erent strategy called coupling: T2 -13 couple couple couple - - In both decks, take card C and move it to the top. . assume 't const Geomfnnt ) ← Geomftn ) If C isn’t already coupled, mark it as coupled . . arena ; Hed " . What happens when we look at each deck collector coupon . - top - to A individually? LOOKS like random ! move T ECT .tTnT=nl0gN chosen , -1T , -1 I # . . . arbitrary " coupled ; start at v. a. r 0cnu④ . ④ once # labeled For all � : t mix ( " ) = " a IS it ?9eEuFo¥ has State we keep applying p , . " same the position both decks ! in at it dist Stays . . 7 / 17 8 / 17 9 / 17 Random Hypercube Walk I Random Hypercube Walk II Random Hypercube Walk III - length vertices n Take an n -dimensional hypercube. Each coordinate is labeled coupled or not. Use Coupon Collector again! bit strings Stationary distribution of hypercube walk: Initially all coordinates are uncoupled . In F- Cfn . In ] before all vertices as same Pick a random coordinate i . . . . same deg Try coupling again: . Flip a coin to set the i -th coordinate to 0 or 1. ④ Lazywatk vertex self loop Wp 's If i -th coordinate is uncoupled , set it to coupled. Every : transitions to up In Ef time . randomized ] all Nbr N log N = word iarbitraryvertexi → truly random : For all " , t mix ( " ) = bits all randomize 10 / 17 11 / 17 12 / 17
Conditions for Fast Mixing? Bottlenecks How To Measure Conductance? Complete graph K n ? We use a measurement called the conductance Measuring conductance = looking at all subsets =L ± to quantify the notion of a bottleneck. of states with vol ( A ) ≤ 2. E A How many subsets, potentially? 0¥ middle Path on n vertices? Start in The conductance of a set A ⊆ S is: . Milk ; EA ,jEA NZ mixing time ow - • - Φ( A ) = Alternative: Get lower bound on Φ( M ) using - second largest eigenvalue of transition matrix. Dumbbell? jeep 'll ← VOILA ) Φ( M ) ≥ nz 2 mixing time The conductance of the chain M is: min ECA ) Φ( M ) = Eigenvalues are much faster to compute! Ast VOKAKE mgaehffengnere pink dot chance f- . from getting hmltgefdte 13 / 17 14 / 17 15 / 17 Markov Chain Monte Carlo Markov Chain Monte Carlo Monte Carlo: randomized algorithm where the Key idea: Design a Markov chain so that its output is allowed to be incorrect stationary follows the distribution that you want Use cases: to sample from. Run the chain, wait for it to mix. I sampling from complicated distributions I counting combinatorial objects CE ) ! + Runtime depends on... mix I Bayesian inference I statistical physics I volume estimation, integration 16 / 17 17 / 17
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