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Variation distance?

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


    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


    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


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    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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