Questions on the law of total multivariate probability
0
I am wondering whether these formulations are established:
$P(Y|X,A)=int_Z P(Y|Z,A)P(Z|X,A) dZ$
$P(Y|X,A)=int_Z P(Y|Z)P(Z|X,A) dZ$
The $Y$, $A$, $Z$ are all random variables.
And if so, how to prove and comprehend them?
thanks so much!
probability
asked Dec 26 at 10:50
YRS
42
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0
I am wondering whether these formulations are established:
$P(Y|X,A)=int_Z P(Y|Z,A)P(Z|X,A) dZ$
$P(Y|X,A)=int_Z P(Y|Z)P(Z|X,A) dZ$
The $Y$, $A$, $Z$ are all random variables.
And if so, how to prove and comprehend them?
thanks so much!
probability
asked Dec 26 at 10:50
YRS
42
New contributor
YRS is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
1
What exactly is meant with the notations like $p(Ymid X,A)$ and $P(Ymid Z,A)$? Also: is there any difference between e.g. $p(Ymid X,A)$ and $P(Ymid X,A)$? In the integrals on RHS variable $Z$ seems to be overloaded with meanings.
– drhab
Dec 26 at 11:13
The $p(Y|X,A)$ and $P(Y|X,A)$ are the same. I am sorry about making this typo.
– YRS
Dec 26 at 12:17
BTW, could you explain the meaning of variable $Z$ overloaded?
– YRS
Dec 26 at 12:19
I see $Z$ as part of integral sign ($int_Z$, why?) and also as intergrating variable ($dZ$). Do the notations stand for conditional PDF'S maybe? Is $Z $ a random variable or a normal variable? Questions like this cross my mind.
– drhab
Dec 26 at 12:24
How to integrate over events (I mean $Z $)? For that it must be a variable.
– drhab
Dec 26 at 12:32
|
show 4 more comments
0
0
0
I am wondering whether these formulations are established:
$P(Y|X,A)=int_Z P(Y|Z,A)P(Z|X,A) dZ$
$P(Y|X,A)=int_Z P(Y|Z)P(Z|X,A) dZ$
The $Y$, $A$, $Z$ are all random variables.
And if so, how to prove and comprehend them?
thanks so much!
probability
asked Dec 26 at 10:50
YRS
42
New contributor
YRS is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
I am wondering whether these formulations are established:
$P(Y|X,A)=int_Z P(Y|Z,A)P(Z|X,A) dZ$
$P(Y|X,A)=int_Z P(Y|Z)P(Z|X,A) dZ$
The $Y$, $A$, $Z$ are all random variables.
And if so, how to prove and comprehend them?
thanks so much!
probability
probability
asked Dec 26 at 10:50
YRS
42
New contributor
YRS is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
asked Dec 26 at 10:50
YRS
42
New contributor
YRS is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
edited Dec 26 at 15:01
asked Dec 26 at 10:50
YRS
42
New contributor
YRS is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
asked Dec 26 at 10:50
YRS
42
asked Dec 26 at 10:50
YRS
42
42
New contributor
YRS is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
New contributor
YRS is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
YRS is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
1
What exactly is meant with the notations like $p(Ymid X,A)$ and $P(Ymid Z,A)$? Also: is there any difference between e.g. $p(Ymid X,A)$ and $P(Ymid X,A)$? In the integrals on RHS variable $Z$ seems to be overloaded with meanings.
– drhab
Dec 26 at 11:13
The $p(Y|X,A)$ and $P(Y|X,A)$ are the same. I am sorry about making this typo.
– YRS
Dec 26 at 12:17
BTW, could you explain the meaning of variable $Z$ overloaded?
– YRS
Dec 26 at 12:19
I see $Z$ as part of integral sign ($int_Z$, why?) and also as intergrating variable ($dZ$). Do the notations stand for conditional PDF'S maybe? Is $Z $ a random variable or a normal variable? Questions like this cross my mind.
– drhab
Dec 26 at 12:24
How to integrate over events (I mean $Z $)? For that it must be a variable.
– drhab
Dec 26 at 12:32
|
show 4 more comments
1
What exactly is meant with the notations like $p(Ymid X,A)$ and $P(Ymid Z,A)$? Also: is there any difference between e.g. $p(Ymid X,A)$ and $P(Ymid X,A)$? In the integrals on RHS variable $Z$ seems to be overloaded with meanings.
– drhab
Dec 26 at 11:13
The $p(Y|X,A)$ and $P(Y|X,A)$ are the same. I am sorry about making this typo.
– YRS
Dec 26 at 12:17
BTW, could you explain the meaning of variable $Z$ overloaded?
– YRS
Dec 26 at 12:19
I see $Z$ as part of integral sign ($int_Z$, why?) and also as intergrating variable ($dZ$). Do the notations stand for conditional PDF'S maybe? Is $Z $ a random variable or a normal variable? Questions like this cross my mind.
– drhab
Dec 26 at 12:24
How to integrate over events (I mean $Z $)? For that it must be a variable.
– drhab
Dec 26 at 12:32
1
1
What exactly is meant with the notations like $p(Ymid X,A)$ and $P(Ymid Z,A)$? Also: is there any difference between e.g. $p(Ymid X,A)$ and $P(Ymid X,A)$? In the integrals on RHS variable $Z$ seems to be overloaded with meanings.
– drhab
Dec 26 at 11:13
What exactly is meant with the notations like $p(Ymid X,A)$ and $P(Ymid Z,A)$? Also: is there any difference between e.g. $p(Ymid X,A)$ and $P(Ymid X,A)$? In the integrals on RHS variable $Z$ seems to be overloaded with meanings.
– drhab
Dec 26 at 11:13
The $p(Y|X,A)$ and $P(Y|X,A)$ are the same. I am sorry about making this typo.
– YRS
Dec 26 at 12:17
The $p(Y|X,A)$ and $P(Y|X,A)$ are the same. I am sorry about making this typo.
– YRS
Dec 26 at 12:17
BTW, could you explain the meaning of variable $Z$ overloaded?
– YRS
Dec 26 at 12:19
BTW, could you explain the meaning of variable $Z$ overloaded?
– YRS
Dec 26 at 12:19
I see $Z$ as part of integral sign ($int_Z$, why?) and also as intergrating variable ($dZ$). Do the notations stand for conditional PDF'S maybe? Is $Z $ a random variable or a normal variable? Questions like this cross my mind.
– drhab
Dec 26 at 12:24
I see $Z$ as part of integral sign ($int_Z$, why?) and also as intergrating variable ($dZ$). Do the notations stand for conditional PDF'S maybe? Is $Z $ a random variable or a normal variable? Questions like this cross my mind.
– drhab
Dec 26 at 12:24
How to integrate over events (I mean $Z $)? For that it must be a variable.
– drhab
Dec 26 at 12:32
How to integrate over events (I mean $Z $)? For that it must be a variable.
– drhab
Dec 26 at 12:32
|
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1
What exactly is meant with the notations like $p(Ymid X,A)$ and $P(Ymid Z,A)$? Also: is there any difference between e.g. $p(Ymid X,A)$ and $P(Ymid X,A)$? In the integrals on RHS variable $Z$ seems to be overloaded with meanings.
– drhab
Dec 26 at 11:13
The $p(Y|X,A)$ and $P(Y|X,A)$ are the same. I am sorry about making this typo.
– YRS
Dec 26 at 12:17
BTW, could you explain the meaning of variable $Z$ overloaded?
– YRS
Dec 26 at 12:19
I see $Z$ as part of integral sign ($int_Z$, why?) and also as intergrating variable ($dZ$). Do the notations stand for conditional PDF'S maybe? Is $Z $ a random variable or a normal variable? Questions like this cross my mind.
– drhab
Dec 26 at 12:24
How to integrate over events (I mean $Z $)? For that it must be a variable.
– drhab
Dec 26 at 12:32