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# 375 Class Note for STAT 416 at PSU

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## About this Document

COURSE
PROF.
No professor available
TYPE
Class Notes
PAGES
8
WORDS
KARMA
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This 8 page Class Notes was uploaded by an elite notetaker on Friday February 6, 2015. The Class Notes belongs to a course at Pennsylvania State University taught by a professor in Fall. Since its upload, it has received 27 views.

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Date Created: 02/06/15
Time Reversible Markov Chain 0 Consider a stationary ergodic irreducible Markov chain 0 Let the limiting probabilities be 7n 0 The original MC 39 7Xn 27 Xn la X717 39 39 39 0 Trace the MC backwards XnXn1Xn2 o Xnz39 0 l is also a Markov chain 0 Why Given the present the past and future are in dependent past gt future future gt past Given the present the future and past are in dependent Hence the reverse process is also a MC o What are the transition probabilities of the reversed MC Qij PiXmjiXm1i PXmj7Xm1i PXm1 z PiXmjPXm1 Z i Xm PXm1 z 7Tij39 7Tz39 0 Time reversible MC A Markov chain is time reversible if QM 13 that is the reverse MC has the same tran sition probability matrix as the original MC QM PM is equivalent to 791 1317 0 Proposition Suppose an ergodic irreducible MC have transition probabilities PM If we can nd nonnegatz39ve num bers 511 summing to one 51 l and satisfying equation xZPZijP for all 239 j then the MC is time reversible and x is the limiting probability 7T1 Proof Since xiPZj CUijZ39 sum over 239 ijP JIjZPjZ39 L j Hence for any j we have 512739 E Jl z39Pz39j Z39 In addition 2 51 1 By Theorem 41 51 is the limiting probability 7n 0 Interpretation for 79PM MPH Pseeing a transition from j to PseeingjPtransit toi j Wijz39 o 7rijZ 7nsz means the probability of seeing a tran sition from j to 239 is the same as seeing a transition from 239 to j o A transition from j to 2 for the original MC is a tran sition from 239 to j for the reversed MC 0 Example Consider a random walk with states 0 1 M and PM B7111 05 1 P 7 1Z 1 1 1301 060 1 1300 PMM OdM 1 PALM 1 0 0 0L1 0 2 0 3 0 4 0L5 0 9 1 0L1 1 012 1 0L3 10L4 10L5 1 0L0 o This MC is time reversible In between every 2 transitions from 239 to 21 there has to be a transition from Z 1 to z In between every 2 transitions from 21 to 239 there has to be a transition from 239 to Z 1 The transitions from 2 to 2 1 and from 2 1 to 2 are sandwiched Hence 7Tthi 7T 1P i If1j z 1 gt 1132 P 0 hence Wijz39 0 Limiting probabilities W00 7T11 051 711061 7T21 052 7120 7TH11 244 0717quot397M 1 Solve in terms of 7m 060 7T1 7T0 1 051 051 051050 7Q 7T1 7r0 1 052 1 Otglt1 Otl 01239710123972m010 Ingeneral 7TZ39 WWI0 fOI39Z 12 Subject to 7n 1 we have 1 7T0 jil M 1 12 01239 il gia Odi 1Odi 2 39 39 39Oto mea m4 u mgt for z 12M 7U 7T0 0 Example Consider an arbitrarily connected graph A link between vertices 239 and j has weight wij or w wij z j E 1 2 A Markov chain is de ned by the transition probabilities Pi 2125 Show that this Markov chain is time reversible l 7 32l 7 6 H i 3 7 l 7 For instance P12 7 31 7 2 P14 7 Proof 39 i 2k wz39k i Cons1der 51 7 lekwlk z 7 12 M It is easy to check that 1 51 7 kaz39k wzj wzj xZP i Zle wlk 2k wz39k Xsz wlk39 Similarly w 39 wzj 7 P 3 Xsz wlk Xsz Wk Z Z By the proposition the MC is time reversible with x being the limiting probabilities

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