Marginalizing over product of same probability distribution
$begingroup$
Let's say I have a probability distribution defined as a product of probability distributions:
begin{equation}tag{1}label{1}
p(boldsymbolmu)=
prod_{r=1}^{R}prod_{c=1}^{C}p(mu_{r-1,c})p(mu_{r,c})p(mu_{r+1,c})p(mu_{r,c+1})p(mu_{r,c-1})
end{equation}
where $boldsymbolmu$ is a matrix with dimensions $R, C$.
Let's say I want to marginalize out all variables except two particular ones:
$$
sum_{boldsymbolmu notin {mu_{alpha, beta}, mu_{alpha, beta+1} }} p(boldsymbolmu) = p(mu_{alpha, beta}, mu_{alpha, beta+1})
$$
I want to perform this marginalization on (ref{1}). How do I do that? Let's for example say that $R = 1$ and $C = 2$ and we want $p(mu_{2,1}, mu_{2,2})$. Trying to marginalize, we get:
$$
p(mu_{2,1}, mu_{2,2}) = sum_{mu_{0,1}} sum_{mu_{1,1}} sum_{mu_{1,0}} sum_{mu_{1,2}} sum_{mu_{0,2}} p(mu_{0,1})p(mu_{1,1})p(mu_{2,1})p(mu_{1,0})p(mu_{1,2})p(mu_{0,2})p(mu_{1,2})p(mu_{2,2})p(mu_{1,1})p(mu_{1,2})
$$
Three of the variables are easy to eliminate since they only show up once in the product. We're then left with:
$$
sum_{mu_{1,1}} sum_{mu_{1,2}} p(mu_{1,1}) p(mu_{1,1}) p(mu_{1,2}) p(mu_{1,2}) p(mu_{1,2}) p(mu_{2,1}) p(mu_{2,2})
$$
where there are many duplicates of the distributions we want to marginalize out. Is it correct to simply write
$$
p(mu_{2,1}, mu_{2,2}) = p(mu_{2,1}) p(mu_{2,2})?
$$
And why?
probability-theory
asked Dec 5 '18 at 15:45
SandiSandi
255112
$endgroup$
add a comment |
$begingroup$
Let's say I have a probability distribution defined as a product of probability distributions:
begin{equation}tag{1}label{1}
p(boldsymbolmu)=
prod_{r=1}^{R}prod_{c=1}^{C}p(mu_{r-1,c})p(mu_{r,c})p(mu_{r+1,c})p(mu_{r,c+1})p(mu_{r,c-1})
end{equation}
where $boldsymbolmu$ is a matrix with dimensions $R, C$.
Let's say I want to marginalize out all variables except two particular ones:
$$
sum_{boldsymbolmu notin {mu_{alpha, beta}, mu_{alpha, beta+1} }} p(boldsymbolmu) = p(mu_{alpha, beta}, mu_{alpha, beta+1})
$$
I want to perform this marginalization on (ref{1}). How do I do that? Let's for example say that $R = 1$ and $C = 2$ and we want $p(mu_{2,1}, mu_{2,2})$. Trying to marginalize, we get:
$$
p(mu_{2,1}, mu_{2,2}) = sum_{mu_{0,1}} sum_{mu_{1,1}} sum_{mu_{1,0}} sum_{mu_{1,2}} sum_{mu_{0,2}} p(mu_{0,1})p(mu_{1,1})p(mu_{2,1})p(mu_{1,0})p(mu_{1,2})p(mu_{0,2})p(mu_{1,2})p(mu_{2,2})p(mu_{1,1})p(mu_{1,2})
$$
Three of the variables are easy to eliminate since they only show up once in the product. We're then left with:
$$
sum_{mu_{1,1}} sum_{mu_{1,2}} p(mu_{1,1}) p(mu_{1,1}) p(mu_{1,2}) p(mu_{1,2}) p(mu_{1,2}) p(mu_{2,1}) p(mu_{2,2})
$$
where there are many duplicates of the distributions we want to marginalize out. Is it correct to simply write
$$
p(mu_{2,1}, mu_{2,2}) = p(mu_{2,1}) p(mu_{2,2})?
$$
And why?
probability-theory
asked Dec 5 '18 at 15:45
SandiSandi
255112
$endgroup$
add a comment |
$begingroup$
Let's say I have a probability distribution defined as a product of probability distributions:
begin{equation}tag{1}label{1}
p(boldsymbolmu)=
prod_{r=1}^{R}prod_{c=1}^{C}p(mu_{r-1,c})p(mu_{r,c})p(mu_{r+1,c})p(mu_{r,c+1})p(mu_{r,c-1})
end{equation}
where $boldsymbolmu$ is a matrix with dimensions $R, C$.
Let's say I want to marginalize out all variables except two particular ones:
$$
sum_{boldsymbolmu notin {mu_{alpha, beta}, mu_{alpha, beta+1} }} p(boldsymbolmu) = p(mu_{alpha, beta}, mu_{alpha, beta+1})
$$
I want to perform this marginalization on (ref{1}). How do I do that? Let's for example say that $R = 1$ and $C = 2$ and we want $p(mu_{2,1}, mu_{2,2})$. Trying to marginalize, we get:
$$
p(mu_{2,1}, mu_{2,2}) = sum_{mu_{0,1}} sum_{mu_{1,1}} sum_{mu_{1,0}} sum_{mu_{1,2}} sum_{mu_{0,2}} p(mu_{0,1})p(mu_{1,1})p(mu_{2,1})p(mu_{1,0})p(mu_{1,2})p(mu_{0,2})p(mu_{1,2})p(mu_{2,2})p(mu_{1,1})p(mu_{1,2})
$$
Three of the variables are easy to eliminate since they only show up once in the product. We're then left with:
$$
sum_{mu_{1,1}} sum_{mu_{1,2}} p(mu_{1,1}) p(mu_{1,1}) p(mu_{1,2}) p(mu_{1,2}) p(mu_{1,2}) p(mu_{2,1}) p(mu_{2,2})
$$
where there are many duplicates of the distributions we want to marginalize out. Is it correct to simply write
$$
p(mu_{2,1}, mu_{2,2}) = p(mu_{2,1}) p(mu_{2,2})?
$$
And why?
probability-theory
asked Dec 5 '18 at 15:45
SandiSandi
255112
$endgroup$
Let's say I have a probability distribution defined as a product of probability distributions:
begin{equation}tag{1}label{1}
p(boldsymbolmu)=
prod_{r=1}^{R}prod_{c=1}^{C}p(mu_{r-1,c})p(mu_{r,c})p(mu_{r+1,c})p(mu_{r,c+1})p(mu_{r,c-1})
end{equation}
where $boldsymbolmu$ is a matrix with dimensions $R, C$.
Let's say I want to marginalize out all variables except two particular ones:
$$
sum_{boldsymbolmu notin {mu_{alpha, beta}, mu_{alpha, beta+1} }} p(boldsymbolmu) = p(mu_{alpha, beta}, mu_{alpha, beta+1})
$$
I want to perform this marginalization on (ref{1}). How do I do that? Let's for example say that $R = 1$ and $C = 2$ and we want $p(mu_{2,1}, mu_{2,2})$. Trying to marginalize, we get:
$$
p(mu_{2,1}, mu_{2,2}) = sum_{mu_{0,1}} sum_{mu_{1,1}} sum_{mu_{1,0}} sum_{mu_{1,2}} sum_{mu_{0,2}} p(mu_{0,1})p(mu_{1,1})p(mu_{2,1})p(mu_{1,0})p(mu_{1,2})p(mu_{0,2})p(mu_{1,2})p(mu_{2,2})p(mu_{1,1})p(mu_{1,2})
$$
Three of the variables are easy to eliminate since they only show up once in the product. We're then left with:
$$
sum_{mu_{1,1}} sum_{mu_{1,2}} p(mu_{1,1}) p(mu_{1,1}) p(mu_{1,2}) p(mu_{1,2}) p(mu_{1,2}) p(mu_{2,1}) p(mu_{2,2})
$$
where there are many duplicates of the distributions we want to marginalize out. Is it correct to simply write
$$
p(mu_{2,1}, mu_{2,2}) = p(mu_{2,1}) p(mu_{2,2})?
$$
And why?
probability-theory
probability-theory
asked Dec 5 '18 at 15:45
SandiSandi
255112
asked Dec 5 '18 at 15:45
SandiSandi
255112
asked Dec 5 '18 at 15:45
SandiSandi
255112
asked Dec 5 '18 at 15:45
SandiSandi
255112
asked Dec 5 '18 at 15:45
SandiSandi
255112
255112
add a comment |
add a comment |
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