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A have found an alternative definition of independency for a given conditional probability $P(A|B)$, they are independent, iff all columns of the probability table are equal.

What does equal mean in this case? For instance

$$P(A|B) = \begin{pmatrix}0.3 & 0.7\\0.7 & 0.3\end{pmatrix}$$

Where $A$ is associated with the columns and $B$ is associated with the rows, for instance $P(A=0|B=1) = 0.7$ read at the bottom on the left.

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up vote 1 down vote accepted

It means that $P(A=i\mid B=j)=P(A=i\mid B=k)$ for every $i$, $j$ and $k$.

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+1 Easy to comprehend notation – Mahoni Jun 9 '12 at 15:06

Independence means P(A and B)=P(A|B)P(B)=P(B|A)P(A)=P(A)P(B). I think this is just a way of showing it in a table when A and B consists of {0, 1} with certain probabilities associated with the 4 joint occurrences {0, 0}, {0, 1}, {1. 0}, {1, 1}.

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