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included partition conditional independence idea

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edited Aug 17, 2018 at 17:45

πr8

Say I define a probability measure over the symmetric group $S_n$ as follows:

I specify $n$ positive `potential' functions $G_i : \{1, 2, \cdots, N\} \to (0,\infty)$
I then set

$$\mathbb{P}(\sigma) = \frac{1}{Z} \prod_{i=1}^N G_i(\sigma (i))$$

for permutations $\sigma \in S_n$, where $Z$ is a normalisation constant.

Is there a name for measures of this form?

It's certainly not completely generic - there are only $\Theta(n^2)$ of these measures over $S_n$, so clearly not every distribution over permutations can be written in this form.
It seems like it should be able to express this condition in terms of some sort of conditional independence relations, but I haven't been able to work out how to do so.
- A partial answer to this is that if you have a partition $\{1,2,\cdots,N\} = A \sqcup B$, then conditional on knowing that $\sigma(A) = A', \sigma(B) = B'$, you have that $\{\sigma(x)\}_{x \in A}$ and $\{\sigma(x)\}_{x \in B}$ are independent. So there's a sort of partition/refinement-based conditional independence structure, I suppose?

If there isn't a specific name for this family of measures, but they do have some useful properties which would otherwise be useful, I'll also be happy to accept answers which indicate this.

Say I define a probability measure over the symmetric group $S_n$ as follows:

I specify $n$ positive `potential' functions $G_i : \{1, 2, \cdots, N\} \to (0,\infty)$
I then set

$$\mathbb{P}(\sigma) = \frac{1}{Z} \prod_{i=1}^N G_i(\sigma (i))$$

for permutations $\sigma \in S_n$, where $Z$ is a normalisation constant.

Is there a name for measures of this form?

It's certainly not completely generic - there are only $\Theta(n^2)$ of these measures over $S_n$, so clearly not every distribution over permutations can be written in this form.
It seems like it should be able to express this condition in terms of some sort of conditional independence relations, but I haven't been able to work out how to do so.

If there isn't a specific name for this family of measures, but they do have some useful properties which would otherwise be useful, I'll also be happy to accept answers which indicate this.

Say I define a probability measure over the symmetric group $S_n$ as follows:

I specify $n$ positive `potential' functions $G_i : \{1, 2, \cdots, N\} \to (0,\infty)$
I then set

$$\mathbb{P}(\sigma) = \frac{1}{Z} \prod_{i=1}^N G_i(\sigma (i))$$

for permutations $\sigma \in S_n$, where $Z$ is a normalisation constant.

Is there a name for measures of this form?

It's certainly not completely generic - there are only $\Theta(n^2)$ of these measures over $S_n$, so clearly not every distribution over permutations can be written in this form.
It seems like it should be able to express this condition in terms of some sort of conditional independence relations, but I haven't been able to work out how to do so.
- A partial answer to this is that if you have a partition $\{1,2,\cdots,N\} = A \sqcup B$, then conditional on knowing that $\sigma(A) = A', \sigma(B) = B'$, you have that $\{\sigma(x)\}_{x \in A}$ and $\{\sigma(x)\}_{x \in B}$ are independent. So there's a sort of partition/refinement-based conditional independence structure, I suppose?

If there isn't a specific name for this family of measures, but they do have some useful properties which would otherwise be useful, I'll also be happy to accept answers which indicate this.

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asked Aug 17, 2018 at 8:50

πr8

Probability Distribution on permutations with factor structure

Say I define a probability measure over the symmetric group $S_n$ as follows:

I specify $n$ positive `potential' functions $G_i : \{1, 2, \cdots, N\} \to (0,\infty)$
I then set

$$\mathbb{P}(\sigma) = \frac{1}{Z} \prod_{i=1}^N G_i(\sigma (i))$$

for permutations $\sigma \in S_n$, where $Z$ is a normalisation constant.

Is there a name for measures of this form?

It's certainly not completely generic - there are only $\Theta(n^2)$ of these measures over $S_n$, so clearly not every distribution over permutations can be written in this form.
It seems like it should be able to express this condition in terms of some sort of conditional independence relations, but I haven't been able to work out how to do so.

If there isn't a specific name for this family of measures, but they do have some useful properties which would otherwise be useful, I'll also be happy to accept answers which indicate this.