Download Mathematical aspects of mixing times in Markov chains by Ravi Montenegro, Prasad Tetali PDF

By Ravi Montenegro, Prasad Tetali

ISBN-10: 1933019298

ISBN-13: 9781933019291

Presents an creation to the analytical elements of the idea of finite Markov chain blending occasions and explains its advancements. This booklet seems to be at a number of theorems and derives them in uncomplicated methods, illustrated with examples. It comprises spectral, logarithmic Sobolev strategies, the evolving set method, and problems with nonreversibility.

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Mathematical aspects of mixing times in Markov chains

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Y)|γxy | π ˆ (x)P(x, x=y:(a,b)∈γxy 40 Advanced Functional Techniques For variance we have Varπ (f ) = inf Eπ (f (x) − c)2 ≤ inf M Eπˆ (f (x) − c)2 = M Varπˆ (f ). c∈R c∈R For entropy, observe that Entπ (f 2 ) = f 2 (x) log π(x) x∈Ω = inf c>0 π(x) c>0 = M f 2 (x) log x∈Ω ≤ inf M f 2 (x) − f 2 (x) + Eπ f 2 Eπ f 2 π ˆ (x) f 2 (x) − f 2 (x) + c c f 2 (x) log x∈Ω Entπˆ (f 2 ) . f 2 (x) − f 2 (x) + c c The second equality follows from differentiating with respect to c to see that the minimum occurs at c = Eπ f 2 , while the inequality required the 2 fact that a log ab −a+b ≥ a (1− ab )−a+b = 0 and so f 2 log fc −f 2 +c ≥ 0.

K π(S) Heuristically, a step of the evolving set process consists of choosing a uniform value of u, and then Au is the set of vertices y that get at least a u-fraction of their size π(y) from the set A. The Doob transform produces another Markov chain because of a Martingale property. 2. If A ⊂ Ω then 1 π(A )K(A, A ) = A ⊂Ω 0 π(Au ) du = π(A) 46 Evolving Set Methods Proof. 1 0 π(Au )du = π(y)P rob(y ∈ Au ) = y∈Ω π(y) y∈Ω Q(A, y) = π(A) π(y) ˆ is a dual process of P. 3. If S ⊂ Ω, y ∈ Ω and Λ(S, y) = tion linkage, then ˆ PΛ(S, y) = ΛK(S, y) .

For c constant, E(f, f ) = E(f −c, f −c). Also, E(f −c, f − c) ≥ E((f − c)+ , (f − c)+ ) because ∀a, b ∈ R : (a − b)2 ≥ (a+ − b+ )2 . It follows that when 0 ≤ c < max f then E(f, f ) ≥ E((f − c)+ , (f − c)+ ) ≥ Var((f − c)+ ) inf u∈c+ 0 (f >c) E(u, u) Var(u) ≥ Var((f − c)+ ) Λ(π(f > c)) . The inequalities ∀a, b ≥ 0 : (a − b)2+ ≥ a2 − 2b a and (a − b)+ ≤ a show that Var((f − c)+ ) = E(f − c)2+ − (E(f − c)+ )2 ≥ Ef 2 − 2c Ef − (Ef )2 . Let c = Var(f )/4Ef and apply Markov’s inequality π(f > c) < (Ef )/c, E(f, f ) ≥ (Var(f ) − 2c Ef ) Λ(Ef /c) = 1 Var(f ) Λ 2 4(Ef )2 Var f A mixing time theorem then follows easily.

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Mathematical aspects of mixing times in Markov chains by Ravi Montenegro, Prasad Tetali


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