Marginalization

Creator
Creator
Seonglae Cho
Created
Created
2023 Mar 7 2:19
Editor
Edited
Edited
2025 Apr 28 12:43

Law of total Probability, Sum rule of probability

p(A,B)=C,Dp(A,B,C,D)p(A,B) = \sum_{C,D}p(A,B,C,D)
Marginal probability from conditional probability means considering all cases by integral
P(A)=P(A,B)dB=P(AB)P(B)dBP(A) = \int P(A, B) \, dB = \int P(A \mid B) P(B) \, dBP(A)=P(A,Bi)=P(ABi)P(Bi)P(A) = \sum{P(A, B_i)} = \sum{P(A|B_i)}P(B_i)

Product rule which involves
Bayes Theorem

P(X,Y)=P(X)P(YX)=P(Y)P(XY)=P(XY)P(X, Y) = P(X)P(Y|X) = P(Y)P(X|Y) = P(X \cap Y)
Joint probability of X and Y is simply understood as choosing X first and choose Y given X. Confusing part is that regardless of order, reverse probability is expressed as same form.
After that we can decide
Pairwise Independence
or not
P(X)=P(XY),P(Y)=P(YX)P(X) = P(X|Y), P(Y) = P(Y|X)
 
 
 
 

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