Stochastic Modeling

Download Mathematical Aspects of Mixing Times in Markov Chains by Ravi Montenegro, Prasad Tetali PDF

By Ravi Montenegro, Prasad Tetali

Presents an creation to the analytical facets of the speculation of finite Markov chain blending occasions and explains its advancements. This ebook appears to be like at numerous theorems and derives them in easy methods, illustrated with examples. It comprises spectral, logarithmic Sobolev concepts, the evolving set method, and problems with nonreversibility.

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EZ g(EZ). 2 284 Evolving Set Methods It is fairly easy to translate these to mixing time bounds. 6 it is appropriate to let f (z) = 1−z z for L bounds. 5) τ2 ( ) ≤ 2x(1 − x)(1 − C√z(1−z) (x))  π∗       1   1+ 2 /4 dx     4π∗ x(1 − x)(1 − C√ (x)) z(1−z) 1+3π∗ 1 with the first integral requiring x 1 − C√z(1−z) 1+x to be convex. 2 r By making the change of variables x = 1+r and applying a few pessimistic approximations one obtains a result more strongly resembling spectral profile bounds:   1 1   log √  √  1 − C z(1−z) π∗         1/ 2 dr τ2 ( ) ≤ √ 2r(1 − C z(1−z) (r))  π∗        4/ 2  dr     √ r(1 − C (r)) 4π∗ z(1−z) For total variation distance related results are in terms of Cz(1−z) (r), and Cz log(1/z) (r) for relative entropy.

PΛ(S, y) = ΛK(S, Proof. PΛ(S, y) = z∈S ˆ y) = ΛK(S, S y Q(S, y) π(z) P(z, y) = π(S) π(S) ˆ S ) π(y) = π(y) K(S, π(S ) π(S) The final equality Q(S, y)/π(y). is because S K(S, S ) = S y y K(S, S Q(S, y) π(S) ) = P rob(y ∈ S ) = With duality it becomes easy to write the n step transitions in terms ˆ of the walk K. ˆ n . 4. Let E {x} then ˆ n πS (y) , Pn (x, y) = E n where πS (y) = set S by π. 1S (y)π(y) π(S) denotes the probability distribution induced on Proof. ˆ n )({x}, y) = E ˆ n πS (y) Pn (x, y) = (Pn Λ)({x}, y) = (ΛK n The final equality is because Λ(S, y) = πS (y).

3. 15. λP ≥ 1 λˆ , MA P ρP ≥ 1 ρˆ , MA P ΛP (r) ≥ 1 Λ ˆ (r). MA P The log-Sobolev and spectral profile mixing time bounds of P are ˆ thus at worst a factor M A times larger than those of P. ˆ along with the If the distribution π = π ˆ then a Nash inequality for P, 1 relation EP (f, f ) ≥ A EPˆ (f, f ), immediately yields a Nash inequality for P. It is not immediately clear how to compare Nash inequality bounds if π = π ˆ . 5) gives for P. EP (f, f ) to EPˆ (f, f ) and Varπ (f ) to Varπˆ (f ) in the original proofs of the mixing times.

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