t**k
发帖数: 260 | 1 Given a joint distribution P(A,B,C), we can compute various marginal
distributions. Now suppose:
P1(A,B,C) = P(A) P(B) P(C)
P2(A,B,C) = P(A,B) P(C)
P3(A,B,C) = P(A,B,C)
Is it true that d(P1,P3)>=d(P2,P3) where d is the total variation distance?
In other words, is it provable that P(A,B)P(C) is a better approximation of
P(A,B,C) than P(A)P(B)P(C) in terms of the total variation distance? | K*****2
发帖数: 9308 | 2 我假定你的问题是用P(A,B)P(C)的形式，或者P(A)P(B)P(C)的形式，去近似给定的一个
P(A,B,C)。
那估计你算distance的时候，是要求inf的吧，比如inf d(P1,P3), inf对P1求。
那既然P(A)P(B)永远都是P(A,B)的一个特例，取inf肯定是P(A,B)P(C)产生的distance
更小啊。 | f**********d
发帖数: 4960 | 3 应该没有。
total variation distance基于sigma field的任意取最大差的事件。直觉上感觉是很
难保证。
of
【在 t**k 的大作中提到】
: Given a joint distribution P(A,B,C), we can compute various marginal
: distributions. Now suppose:
: P1(A,B,C) = P(A) P(B) P(C)
: P2(A,B,C) = P(A,B) P(C)
: P3(A,B,C) = P(A,B,C)
: Is it true that d(P1,P3)>=d(P2,P3) where d is the total variation distance?
: In other words, is it provable that P(A,B)P(C) is a better approximation of
: P(A,B,C) than P(A)P(B)P(C) in terms of the total variation distance?
| t**k
发帖数: 260 | 4 Thanks for the replies. I just find a counter-example, so it is not true.
http://stats.stackexchange.com/a/65975/28478
of
【在 t**k 的大作中提到】
: Given a joint distribution P(A,B,C), we can compute various marginal
: distributions. Now suppose:
: P1(A,B,C) = P(A) P(B) P(C)
: P2(A,B,C) = P(A,B) P(C)
: P3(A,B,C) = P(A,B,C)
: Is it true that d(P1,P3)>=d(P2,P3) where d is the total variation distance?
: In other words, is it provable that P(A,B)P(C) is a better approximation of
: P(A,B,C) than P(A)P(B)P(C) in terms of the total variation distance?
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