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Mathematics for Computer Science MIT 6.042J/18.062J Stationary Distributions Albert R Meyer, December 11, 2013 stationary.1 Distribution Over Nodes 1/4 1/4 1/4 1/4 B 1/2 1/2 1/4 1 1/3 0 1/4 1/4 1/4 1 2/3 00 (pB, pO, pG) Suppose you start at B: (1, 0, 0) 1/2 1/2 What are p’B, p’O, p’G after 1 step? Albert R Meyer, December 11, 2013 stationary.2 Distribution Over Nodes 1/4 1/4 1/4 1/4 B 1/4 1/2 1 1/2 1/2 1/2 1/3 0 1/4 1/4 1/4 1 2/3 00 Dist after 1 step: (p’B,p’O,p’G), only get places from B, so Albert R Meyer, December 11, 2013 stationary.3 Distribution Over Nodes 1/4 B 1/2 1/2 1/4 1/4 1 1/3 2/3 1/4 Dist after 1 step: Dist after 2 steps: (p’’B, p’’O, p’’G) Albert R Meyer, December 11, 2013 stationary.6 Distribution Over Nodes 1/4 B 1/2 1/2 1/4 1/4 1 1/3 2/3 1/4 Dist after 1 step: p’’O = Pr[B to O|at B]•p’B + Pr[O to O|at O]•p’o + Pr[G to O|at G]•p’G Albert R Meyer, December 11, 2013 stationary.7 Distribution Over Nodes 1/4 B 1/2 1/2 1/4 1/4 1 1/3 2/3 1/4 Dist after 1 step: 1/4 p’’O = Pr[B to O|at B]•p’1/2 B 1/4 + Pr{O to 1/3 O|at O}•p’ o 1/4 + Pr{G to0O|at G}•p’ G Albert R Meyer, December 11, 2013 = 5/24 stationary.8 Distribution Over Nodes 1/4 B 1/2 5/24 1/4 1/2 1 1/3 2/3 7/24 distribution after 2 steps: ’’ (p B, ’’ p O, Albert R Meyer, ’’ p G) December 11, 2013 stationary.9 Distribution Over Nodes 1/4 B 1/2 pt B ptO 1/4 1 1/3 2/3 pt G distribution after t steps? Albert R Meyer, December 11, 2013 stationary.10 Linear Algebra the edge probability matrix December 11, 2013 stationary.11 Linear Algebra the edge probability matrix December 11, 2013 stationary.12 Linear Algebra the edge probability matrix December 11, 2013 stationary.13 Linear Algebra the edge probability matrix December 11, 2013 stationary.14 Linear Algebra December 11, 2013 stationary.15 Distribution Over Nodes ptO 1/4 B 1/2 pt B 1/4 1 1/3 2/3 pt G distribution after t steps? …and as t ? Albert R Meyer, December 11, 2013 stationary.16 Stationary Distribution 1/4 B 1/2 pB pO 1/4 1 1/3 2/3 pG distribution (pB, pO, pG) is stationary if next-step distribution is the same. What is a stationary dist. here? Albert R Meyer, December 11, 2013 stationary.17 Finding Stationary Dist. 1/4 1/2 pB pO 1/4 1 1/3 2/3 pG pB = pB’ = (1/2)pB + 1pG pO = pO’ = (1/4)pB + (1/3)pO pG = pG’ = (1/4)pB + (2/3)pO pB + pO + pG = 1 Albert R Meyer, December 11, 2013 stationary.18 Finding Stationary Dist. 1/4 1/2 pB pO 1/4 1/3 2/3 pG solving for (pB, pO, pG): Albert R Meyer, December 11, 2013 stationary.19 Linear Algebra Find stationary dist vector s by solving: sM = s |s | = 1 December 11, 2013 stationary.20 Stationary Difficulties 1 0 p 10 does not converge to stable distribution q uncountably many stable distributions December 11, 2013 stationary.21 Questions on Stationary Dist Yes stationary dist exist? (if graph finite) • Is it unique? Sometimes • Does a random walk approach it from any Sometimes starting distribution? – How quickly? Varies • Albert R Meyer, December 11, 2013 stationary.22