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Probalistc Models
Part 2 : Fundamental concepts of probability
Indepencies
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Cards (5)
Two event \alpha and \beta said to be independant in a joint distribution P, denoted as (A \bot B) (is
symmetrix
), if
P
(
α
∣
β
)
=
P(\alpha | \beta) =
P
(
α
∣
β
)
=
P
(
α
)
P(\alpha)
P
(
α
)
or
P
(
β
∣
α
)
=
P(\beta | \alpha) =
P
(
β
∣
α
)
=
P
(
β
)
P(\beta)
P
(
β
)
or
P
(
α
∩
β
)
=
P(\alpha \cap \beta) =
P
(
α
∩
β
)
=
P
(
α
)
P
(
β
)
P(\alpha) P(\beta)
P
(
α
)
P
(
β
)
Two random variables X,Y are
independant
if for all values
Number of entries in joint distributino grows
exponentially
with number of variables.
Two events \alpha, \beta , \gamma :
P
(
α
∣
β
∩
γ
)
=
P(\alpha | \beta \cap \gamma) =
P
(
α
∣
β
∩
γ
)
=
P
(
α
∣
γ
)
,
P
(
β
∣
α
∩
γ
)
=
P(\alpha | \gamma), P(\beta | \alpha \cap \gamma) =
P
(
α
∣
γ
)
,
P
(
β
∣
α
∩
γ
)
=
P
(
β
∣
γ
)
,
P
(
α
∩
β
∣
γ
)
=
P(\beta| \gamma), P(\alpha \cap \beta | \gamma) =
P
(
β
∣
γ
)
,
P
(
α
∩
β
∣
γ
)
=
P
(
α
∣
γ
)
P
(
β
∣
γ
)
P(\alpha | \gamma)P(\beta|\gamma)
P
(
α
∣
γ
)
P
(
β
∣
γ
)
P
(
X
,
Y
,
Z
)
=
P(X,Y,Z) =
P
(
X
,
Y
,
Z
)
=
P
(
X
∣
Y
,
Z
)
P
(
Y
,
Z
)
=
P(X|Y,Z) P(Y,Z) =
P
(
X
∣
Y
,
Z
)
P
(
Y
,
Z
)
=
P
(
X
∣
Y
,
Z
)
P
(
Y
∣
Z
)
P
(
Z
)
=
P(X|Y,Z) P(Y|Z) P(Z) =
P
(
X
∣
Y
,
Z
)
P
(
Y
∣
Z
)
P
(
Z
)
=
P
(
X
∣
Z
)
P
(
Y
∣
Z
)
P
(
Z
)
P(X|Z) P(Y|Z) P(Z)
P
(
X
∣
Z
)
P
(
Y
∣
Z
)
P
(
Z
)
