3 Chernoff Bound

Lecture 15: Chernoff bounds and Sequential detection

1 Chernoff Bounds 1.1 Bayesian Hypothesis Test A test using log-likelihood ratio statistic has the form, T (Y ) = logL(Y ) T τ. (1) Bound-1: The probability of error Pe is bounded as, Pe ≤ (π0 + π1e )eμT,0(s0)−s0τ , (2) where μT,0(s) = logE0[e ], and μ ′ T,0(s0) = τ . Bound-2: ∀ s ∈ [0, 1], Pe ≤ max(π0, π1e )eμT,0(s)−sτ . (3) Derivation of the above bound: Consider, Pe = π0P0(Γ1) + π1P1(Γ0), = ...

Equitable Coloring Extends Chernoff-Hoeffding Bounds

Lecture 03 : Chernoff Bounds and Intro to Spectral Graph Theory 3 1 . 1 Hoeffding ’ s Inequality

CS 229 r : Algorithms for Big Data Fall 2013 Lecture 4 — September 12 , 2013

2 Algorithm for Fp, p > 2 2 2.1 Alternate formulation of Chernoff bound . . . . . . . . . . . . . . . . . . . . . . . . . 3 2.2 Returning to proof of Theorem 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2.3 Digression on Perfect Hashing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2.4 Finishing proof of Theorem 1 . . . . . . . . . . . . . . . . . . . . . . . ...

Applied Stochastic Processes Problem Set 3

Compute the Chernoff bound on P [X ≥ a] where X is a random variable that satisfies the exponential law f X (x) = λe −λx u(x). Recall, from Equation 4.6-4 on page 214 in [3], that the Chernoff bound for a continuous random variable X is given by P [X ≥ a] ≤ argmin t>0 e −at θ X (t) , (1) where θ X (t) is the moment-generating function θ X (t) E[e tX ] = ∞ −∞ e tx f X (x)dx (2) as defined by Equ...