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Is marginalization a sum of conditioned observations?

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0

$begingroup$

I have the following probability model:

the joint probability p(A,B,D) is to be evaluated. I think there are 2 ways how to think about the problem:

1. Marginalization over C of p(A,B,C,D) ignoring the conditional probabilities within the model
   $$p(A,B,D) = sum_Cp(A,B,C,D) = p(A,B,C,D) + p(A,B,overlineC,D)$$
   

Assuming the $C = [C, overlineC]$, that is, it has only 2 values.

2. Marginalization over C of p(A,B,C,D) taking the conditional links p(C|A,B) and p(D|C) into account

It can be shown, that according to the graph, the $p(A,B,C,D)$ can be marginalized as

$$p(A,B,D) = sum_Cp(A,B,C,D)
\ = sum_Cp(A)p(B)p(C
\ = p(A)p(B)sum_CC)
\ = p(A)p(B)left(p(C|A,B)p(D|C) + p(overlineC|A,B)p(D|overlineC)right)$$

---

Question

Is the marginalization of the joint distribution a summation over conditioned observations ? (that is, the marginalization over C is the same as summation of conditioned cases on C)

probability conditional-probability marginal-distribution

share|cite|improve this question

asked Mar 29 at 3:37

Martin GMartin G

618

$endgroup$

add a comment | 

0

$begingroup$

I have the following probability model:

the joint probability p(A,B,D) is to be evaluated. I think there are 2 ways how to think about the problem:

1. Marginalization over C of p(A,B,C,D) ignoring the conditional probabilities within the model
   $$p(A,B,D) = sum_Cp(A,B,C,D) = p(A,B,C,D) + p(A,B,overlineC,D)$$
   

Assuming the $C = [C, overlineC]$, that is, it has only 2 values.

2. Marginalization over C of p(A,B,C,D) taking the conditional links p(C|A,B) and p(D|C) into account

It can be shown, that according to the graph, the $p(A,B,C,D)$ can be marginalized as

$$p(A,B,D) = sum_Cp(A,B,C,D)
\ = sum_Cp(A)p(B)p(C
\ = p(A)p(B)sum_CC)
\ = p(A)p(B)left(p(C|A,B)p(D|C) + p(overlineC|A,B)p(D|overlineC)right)$$

---

Question

Is the marginalization of the joint distribution a summation over conditioned observations ? (that is, the marginalization over C is the same as summation of conditioned cases on C)

probability conditional-probability marginal-distribution

share|cite|improve this question

asked Mar 29 at 3:37

Martin GMartin G

618

$endgroup$

add a comment | 

$begingroup$

I have the following probability model:

the joint probability p(A,B,D) is to be evaluated. I think there are 2 ways how to think about the problem:

1. Marginalization over C of p(A,B,C,D) ignoring the conditional probabilities within the model
   $$p(A,B,D) = sum_Cp(A,B,C,D) = p(A,B,C,D) + p(A,B,overlineC,D)$$
   

Assuming the $C = [C, overlineC]$, that is, it has only 2 values.

2. Marginalization over C of p(A,B,C,D) taking the conditional links p(C|A,B) and p(D|C) into account

It can be shown, that according to the graph, the $p(A,B,C,D)$ can be marginalized as

$$p(A,B,D) = sum_Cp(A,B,C,D)
\ = sum_Cp(A)p(B)p(C
\ = p(A)p(B)sum_CC)
\ = p(A)p(B)left(p(C|A,B)p(D|C) + p(overlineC|A,B)p(D|overlineC)right)$$

---

Question

Is the marginalization of the joint distribution a summation over conditioned observations ? (that is, the marginalization over C is the same as summation of conditioned cases on C)

probability conditional-probability marginal-distribution

share|cite|improve this question

asked Mar 29 at 3:37

Martin GMartin G

618

$endgroup$

share|cite|improve this question

asked Mar 29 at 3:37

Martin GMartin G

618

share|cite|improve this question

asked Mar 29 at 3:37

Martin GMartin G

618

asked Mar 29 at 3:37

Martin GMartin G

618

asked Mar 29 at 3:37

Martin GMartin G

618

1 Answer
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oldest

votes

1

$begingroup$

Question

Is the marginalization of the joint distribution a summation over conditioned observations ? (that is, the marginalization over C is the same as summation of conditioned cases on C)

Yes, it is an application of the Law of Total Probability.

When discussing Bayesian DAG, it is common to use $sum_C$ to mean "sum over the evaluations of $C$", which is typically just $C$ itself and its complement, $overline C$. So it is exactly as you had.

$$beginalignmathsf p(A,B,D)&=mathsf p(A,B,C,D)+mathsf p(A,B,overline C, D)\&=sum_C mathsf p(A,B,C,D)\&=mathsf p(A)mathsf p(B)sum_Cmathsf p(Cmid B,A)mathsf p(Dmid C)\&=mathsf p(A)mathsf p(B)left(mathsf p(Cmid B,A)mathsf p(Dmid C)+mathsf p(overline Cmid B,A)mathsf p(Dmid overline C)right)endalign$$

share|cite|improve this answer

answered Mar 29 at 4:02

Graham KempGraham Kemp

87.7k43578

$endgroup$

add a comment | 

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1

$begingroup$

Question

Is the marginalization of the joint distribution a summation over conditioned observations ? (that is, the marginalization over C is the same as summation of conditioned cases on C)

Yes, it is an application of the Law of Total Probability.

When discussing Bayesian DAG, it is common to use $sum_C$ to mean "sum over the evaluations of $C$", which is typically just $C$ itself and its complement, $overline C$. So it is exactly as you had.

$$beginalignmathsf p(A,B,D)&=mathsf p(A,B,C,D)+mathsf p(A,B,overline C, D)\&=sum_C mathsf p(A,B,C,D)\&=mathsf p(A)mathsf p(B)sum_Cmathsf p(Cmid B,A)mathsf p(Dmid C)\&=mathsf p(A)mathsf p(B)left(mathsf p(Cmid B,A)mathsf p(Dmid C)+mathsf p(overline Cmid B,A)mathsf p(Dmid overline C)right)endalign$$

share|cite|improve this answer

answered Mar 29 at 4:02

Graham KempGraham Kemp

87.7k43578

$endgroup$

add a comment | 

1

$begingroup$

Question

Is the marginalization of the joint distribution a summation over conditioned observations ? (that is, the marginalization over C is the same as summation of conditioned cases on C)

Yes, it is an application of the Law of Total Probability.

When discussing Bayesian DAG, it is common to use $sum_C$ to mean "sum over the evaluations of $C$", which is typically just $C$ itself and its complement, $overline C$. So it is exactly as you had.

$$beginalignmathsf p(A,B,D)&=mathsf p(A,B,C,D)+mathsf p(A,B,overline C, D)\&=sum_C mathsf p(A,B,C,D)\&=mathsf p(A)mathsf p(B)sum_Cmathsf p(Cmid B,A)mathsf p(Dmid C)\&=mathsf p(A)mathsf p(B)left(mathsf p(Cmid B,A)mathsf p(Dmid C)+mathsf p(overline Cmid B,A)mathsf p(Dmid overline C)right)endalign$$

share|cite|improve this answer

answered Mar 29 at 4:02

Graham KempGraham Kemp

87.7k43578

$endgroup$

add a comment | 

$begingroup$

Question

Is the marginalization of the joint distribution a summation over conditioned observations ? (that is, the marginalization over C is the same as summation of conditioned cases on C)

Yes, it is an application of the Law of Total Probability.

When discussing Bayesian DAG, it is common to use $sum_C$ to mean "sum over the evaluations of $C$", which is typically just $C$ itself and its complement, $overline C$. So it is exactly as you had.

$$beginalignmathsf p(A,B,D)&=mathsf p(A,B,C,D)+mathsf p(A,B,overline C, D)\&=sum_C mathsf p(A,B,C,D)\&=mathsf p(A)mathsf p(B)sum_Cmathsf p(Cmid B,A)mathsf p(Dmid C)\&=mathsf p(A)mathsf p(B)left(mathsf p(Cmid B,A)mathsf p(Dmid C)+mathsf p(overline Cmid B,A)mathsf p(Dmid overline C)right)endalign$$

share|cite|improve this answer

answered Mar 29 at 4:02

Graham KempGraham Kemp

87.7k43578

$endgroup$

share|cite|improve this answer

answered Mar 29 at 4:02

Graham KempGraham Kemp

87.7k43578

answered Mar 29 at 4:02

Graham KempGraham Kemp

87.7k43578

answered Mar 29 at 4:02

Graham KempGraham Kemp

87.7k43578