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Thread: joint probability with conditional

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    joint probability with conditional




    Hello,
    What is P(A|B, C)?
    I have a dataset and I want to estimate the mutual information of A|B and C.
    The formula for mutual information is
    Sum_A|B Sum_C p(a|b, c) log( p(a|b, c) / p(a|b) p(b) ) da db
    I'm looking for a formula for p(a|b, c)

    Many thanks in advance
    Xavier

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    Re: joint probability with conditional


    To be more specific, I was looking for a formula for P(D, C) with D = A|B.
    The idea being that I want to know how much additional information C brings about A when we already know B.
    But I figured that if "we know B for A", "we know B, period" (assuming it makes any sense).
    So I'm really looking for the mutual information of A and C knowing B.

    P(D, C) = P(A, C | B)

    p(a, c | b) = p(a, c, b) / p(b)

    I(A, C | B) = Sum_B Sum_A Sum_C p(a, c | b) log( p(a, c | b) / (p(a | b) p(c | b)) )
    = Sum_B Sum_A Sum_C (p(a, c, b) / p(b)) log( p(a, c, b) p(b) / (p(a, b) p(c, b)) )

    For which a simple (biased) estimator would be
    Iest(A, C | B) = Sum_B Sum_A Sum_C (#(a, c, b) / #b) log( #(a, c, b) #b / (#(a, b) #(c, b)) )
    where #(x1, ..., xn) is the number of joint occurrences of (x1, ..., xn).

    Is this correct?

    Thanks,
    X.

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