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Advanced Algorithms (6311) Gautam Das

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Advanced Algorithms (6311) Gautam Das Notes: 04/28/2009 Ranganath M R. Outline Tail Inequalities Markov distribution Chebyshev s Inequality Chernoff Bounds Tail ... – PowerPoint PPT presentation

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Title: Advanced Algorithms (6311) Gautam Das


1
Advanced Algorithms (6311)Gautam Das
  • Notes 04/28/2009
  • Ranganath M R.

2
Outline
  • Tail Inequalities
  • Markov distribution
  • Chebyshevs Inequality
  • Chernoff Bounds

3
Tail Inequalities
  • Markov Distribution says that the probability of
    X being greater that t is
  • p(Xgtt) lt µ/t

4
  • Chebyshevs inequality
  • P(X-µ gt t.s) 1/t2

5
Chernoff bounds
  • This is a specific distribution where we can
    obtain much sharper tail inequalities
    (exponentially sharp). If the trials are repeated
    more, the chances of getting very accurate
    results are more. Lets see how this is possible.
  • Example imagine we have n coins (X1Xn ).
  • and let the probabilities of each coins (say
    heads) be (p1pn).
  • Now the Randon variable X ?ni1 xi
  • and µ EX ?ni1 pi in general.

6
  • Some special cases
  • All coins are equally unbiased i.e. pi ½.
  • µ n pi n1/2 n/2, s vn/2, Example for n
    100, s v100/2 5

7
  • Chernoff bounds is given by
  • If EX gt µ
  • P(X- µ µ) e /(1 )1 µ
    -----------------eqn 1
  • If µ gt EX
  • P(µ - X µ) e µ2 /2
  • Here the µ is the power of right hand side
    expression. Hence If more trials are taken, µ
    n/2 increaese, hence we get accurate results as
    the expression (e /(1 )1 ) would be less
    than 1.
  • Example problem to illustrate this.
  • Probability of a team winning is 1/3 .
  • What is the probability that the team will win 50
    out of the 100 games

8
  • µ n pi 100 1/3 100/3
  • s (no of games to win - µ)/ µ
  • (50 100/3)/(100/3) ½.
  • Now to calculating probability of winning we need
    to substitute all these in eqn 1.
  • e ½ /(3/2 3/2 )100/3 0.027 (approx).
  • Here if we increase no of games(in general no of
    trials), the µ increases, and the expression e
    /(1 )1 evaluates to less than 1. hence we
    get more accurate results, when more trails are
    done.

9
Derivation
  • Let X ?ni1 xi
  • Let Y etX
  • P(X- µ µ) P(X (1 ) µ )
  • P(Y et (1 ) µ) EY/ et
    (1 ) µ
  • Now EY EetX e tX1 tX2 tXn
  • EetX1 EetX2 EetXn

10
  • Now lets consider EetXi
  • Xi is either 0 or 1
  • Xi will be 0 with probability 1-Pi
  • and 1 with probability Pi.

11
  • P(et) (1 - Pi)(1) if x 1 then y et
    if x 0 then y 1
  • to be continued in next class
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