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Anki
INF ML AI
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Independence & Conditional Independence

Two random variables, x and y are independent if the occurance of one does not give any information on the likelihood of the other event occuring. In other words, their probability distribution can be expressed as a product of two factors, one only involving x, and one only involving y:

$\forall x \varepsilon X, y \varepsilon Y, p(X = x, Y= y) = p(X = x)p(Y = y)$

Two random variables, x and y are conditionally independent if, given knowledge of the occurance of Z, knowledge of the occurance of X provides no information on the likeihood of the occurance of Y. This can be expressed as:

$\forall x \varepsilon X, y \varepsilon Y, z \varepsilon Z, p(X = x, Y= y | Z = z) = p(X = x | Z = z)p(Y = y | Z = z)$

Example

Independent events

x: the number on a rolled dice

y: the number on another rolled dice

Conditionally independent

x: a person's height

y: the person's vocabulary

z: the person's age

19.01.2017

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