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Variation distance?

  1. Jan 9, 2010 #1
    I was doing some reading and I came across this:

    http://en.wikipedia.org/wiki/Total_variation_distance_of_probability_measures

    So apparently for the finite case,

    [tex]\max_{x} ( \left| P(x) - Q(x) \right|)\quad \mbox{ is equivalent to}\quad \frac{1}{2} \sum_x {\left| P(x)-Q(x)\right|}[/tex]


    but isn't this is a counterexample?
    Code (Text):

    x         1        2        3        4
    P(x)    0.25     0.25     0.25      0.25

    Q(x)    0.10     0.20     0.35      0.35

    |P-Q|   0.15     0.05     0.10      0.10
     
    sum(|P-Q|)/2= 0.2

    max(|P-Q|)=0.15



    So I was thinking, maybe they meant 'equivalent' in a different sense? Could somebody please explain?
     
  2. jcsd
  3. Jan 9, 2010 #2
    I believe they might mean that those two metrics define the same topology on the set of probability measures.
     
  4. Jan 10, 2010 #3

    HallsofIvy

    User Avatar
    Science Advisor

    Rochfor1 is correct. "Equivalent" simply means that they will give the same results in any probability measures, not that they are equal.
     
  5. Jan 10, 2010 #4
    Thanks guys.
     
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