Marginal Independence and Partial Set Partitions

Francisco Ponce Carrión and Seth Sullivant Department of Mathematics
North Carolina State University, Raleigh, NC, 27695 [email protected] [email protected]

Abstract.

We establish a bijection between marginal independence models on $n$ random variables and split closed order ideals in the poset of partial set partitions. We also establish that every discrete marginal independence model is toric in cdf coordinates. This generalizes results of Boege, Petrovic, and Sturmfels [1] and Drton and Richardson [2], and provides a unified framework for discussing marginal independence models.

1. Introduction

Marginal independence models are often studied by associating a combinatorial structure to describe the independence relations between random variables. For instance, Drton and Richardson [2] used bidirected graphs to develop a family of marginal independence models for discrete random variables, and Boege, Petrovic, and Sturmfels [1] used simplicial complexes. The marginal independence models of [2] fit more broadly into the class of graphical models [7, 9]. Each of these structures describes a very different family of models, with very little overlap between them. In both cases it had been shown that when we restrict to discrete random variables, we obtain toric models after a linear change of coordinates. However, there are collections of marginal independence statements that cannot be expressed either as graphs or simplicial complexes. This motivates the proposal of a more general combinatorial structure which we use in our treatment of marginal independence.

In this paper, we study marginal independence using the poset of partial set partitions, $P\Pi_{n}$ . In Section 2, we give a combinatorial description of marginal independence models on $n$ random variables by showing a correspondence between the models and certain order ideals of $P\Pi_{n}$ .

In Section 3, we then restrict to discrete random variables and show that the resulting models are toric and smooth after a linear change of coordinates. In [1, 2, 9], the proof that the mentioned family of marginal independence models are toric uses a linear change of coordinates into Möbius coordinates, by using the Möbius transformation on an appropriately defined poset. In this paper, we describe a change of coordinates using the cumulative distribution function, which is classically used to describe marginal independence. This description has the advantage of tying the results to the classical description of marginal independence. We then move from the affine description of our models to the projective description by showing a parametrization of the homogeneous ideal corresponding to the model.

Finally, in Section 4 we discuss how marginal independence models associated to graphs and simplicial complexes can fit in the framework proposed in this paper. We then compare both classes of models and give an explicit characterization of the models that can be described using both simplicial complexes and graphs. We also count the number of distinct models in these three classes for up to $4$ random variables.

2. Marginal Independence and the Poset of Partial Set Partitions

In this section we discuss general marginal independence models, for completely general types of random variables. Marginal independence is naturally described via constraints on the joint cumulative distribution function of a collection of random variables. We will also see a combinatorial structure which can be used to capture all marginal independence models. This structure is called the poset of partial set partitions, and we will show how certain order ideals of this poset correspond to marginal independence models. We achieve this by introducing a closure operation on order ideals which is based on translating certain implications of marginal independence statements to the language of this poset.

Definition 2.1.

Let $X_{1},\ldots,X_{n}$ be random variables in $\mathbb{R}$ . The joint cumulative distribution function (cdf) of $X_{1},\ldots,X_{n}$ , is the function defined on $(\mathbb{R}\cup\{\pm\infty\})^{n}$ defined by the rule

F(a_{1},\ldots,a_{n})={\rm P}(X_{1}\leq a_{1},\ldots,X_{n}\leq a_{n}).

Note that for a function $F$ to be a joint cumulative distribution function it must satisfy a few properties.

•

For all $i$ , $\lim_{x_{i}\rightarrow-\infty}F(x_{1},\ldots,x_{n})=0$
•

$\lim_{x_{1},\ldots,x_{n}\rightarrow\infty}F(x_{1},\ldots,x_{n})=1$ .
•

$F$ should be nondecreasing in each coordinate: That is, if $a\leq b$ coordinate-wise, then $F(a)\leq F(b)$ .
•

$F$ should be right continuous in each coordinate: That is, for each $i$ , and all $a_{1},\ldots,a_{n}$ ,

$\lim_{x_{i}\rightarrow a_{i}^{+}}F(a_{1},\ldots,a_{i-1},x_{i},a_{i+1},\ldots,a_{n})=F(a_{1},\ldots,a_{n})$
•

For all $a,b$ , $F(\max(a,b))+F(\min(a,b))\geq F(a)+F(b)$ .

These conditions are not exhaustive; that is, there are functions that satisfy all of these properties but are not cdfs of a collection of random variables. However, we do not need to know a precise characterization of functions which are joint cdfs to derive our results.

Note that the joint cdf contains all the marginal cdfs as well. For example, with three random variables, we have

F(a_{1},a_{2},\infty)={\rm P}(X_{1}\leq a_{1},X_{2}\leq a_{2},X_{3}\leq\infty)={\rm P}(X_{1}\leq a_{1},X_{2}\leq a_{2})

Marginal independence of collections of random variables can be expressed in terms of the joint cdf. Let $X=(X_{1},\ldots,X_{n})$ denote the random vector, and $X_{A}=(x_{i}:i\in A)$ be a subvector.

Definition 2.2.

Let $A,B\subseteq[n]$ , be disjoint subsets. We say that $X_{A}$ and $X_{B}$ are marginally independent (denoted $X_{A}\mbox{$\perp\kern-5.5pt\perp$}X_{B}$ , or just $A\mbox{$\perp\kern-5.5pt\perp$}B$ for short), if

F(x)=F(y)F(z)

for all $x$ such that $x_{i}=\infty$ for all $i\in[n]\setminus(A\cup B)$ and where

y_{i}=\begin{cases}x_{i}&\mbox{if }i\in A\\ \infty&\mbox{otherwise}\end{cases}\quad\quad\quad\mbox{ and }\quad\quad\quad z_{i}=\begin{cases}x_{i}&\mbox{if }i\in B\\ \infty&\mbox{otherwise}\end{cases}

Example 2.3.

Suppose $n=4$ , and consider the marginal independence statement $X_{1}\mbox{$\perp\kern-5.5pt\perp$}(X_{2},X_{4})$ (or $\{1\}\mbox{$\perp\kern-5.5pt\perp$}\{2,4\}$ , or $1\mbox{$\perp\kern-5.5pt\perp$}24$ ). This translates into the condition:

F(x_{1},x_{2},\infty,x_{4})=F(x_{1},\infty,\infty,\infty)F(\infty,x_{2},\infty,x_{4}),\qquad\forall\;x_{1},x_{2},x_{4}\in\mathbb{R}\cup\{\infty\}

Said another way, we have, for all $x_{1},x_{2},x_{4}$ ,

{\rm P}(X_{1}\leq x_{1},X_{2}\leq x_{2},X_{4}\leq x_{4})={\rm P}(X_{1}\leq x_{1}){\rm P}(X_{2}\leq x_{2},X_{4}\leq x_{4}).

More generally, we can define marginal independence for a collection of sets.

Definition 2.4.

Let $A_{1},\ldots,A_{k}\subseteq[n]$ with $A_{i}\cap A_{j}=\emptyset$ for all $i\neq j$ . Then $X_{A_{1}}\mbox{$\perp\kern-5.5pt\perp$}\cdots\mbox{$\perp\kern-5.5pt\perp$}X_{A_{k}}$ if, for all $x$ with $x_{i}=\infty$ for all $i\in[n]\setminus(A_{1}\cup\cdots\cup A_{k})$ , we have

F(x)=F(y^{(1)})\cdots F(y^{(k)})

where

y^{(j)}_{i}=\begin{cases}x_{i}&\mbox{if }i\in A_{j}\\ \infty&\mbox{otherwise}.\end{cases}

Now that we have the notion of generalized marginal independence, we can discuss marginal independence models. A marginal independence model is a collection of marginal independence statements and the set of probability distributions that are compatible with those statements. We will describe a completely general combinatorial structure for representing all such marginal independence models. This depends on the structure of a certain poset called the poset of partial set partitions.

Definition 2.5.

A partial set partition of $[n]$ is an unordered list of nonempty sets $\pi=\pi_{1}|\cdots|\pi_{k}$ such that each $\pi_{i}\subseteq[n]$ , and $\pi_{i}\cap\pi_{j}=\emptyset$ for all $i\neq j$ . The set of all partial set partitions of $[n]$ is denoted $P\Pi_{n}$ . A set $\pi_{i}$ in the partial set partition $\pi$ is called a block. The ground set, $|\pi|$ of $\pi$ is defined as $|\pi|=\cup_{i=1}^{k}\pi_{i}$ . The set of partial set partitions where each $\pi$ has at least $2$ blocks is denoted $P\Pi_{n,2}$ .

Note the elements of $P\Pi_{n,2}$ correspond to marginal independence statements. Specifically $\pi=\pi_{1}|\cdots|\pi_{k}$ corresponds to the independence statement $X_{\pi_{1}}\mbox{$\perp\kern-5.5pt\perp$}\cdots\mbox{$\perp\kern-5.5pt\perp$}X_{\pi_{k}}$ .

We want to turn $P\Pi_{n}$ into a poset that is compatible with marginal independence implications. We will explore some implications between marginal independence statements that will lead naturally to a poset structure on $P\Pi_{n}$ .

Lemma 2.6.

If a cdf $F$ satisfies $A_{1}\mbox{$\perp\kern-5.5pt\perp$}\ldots\mbox{$\perp\kern-5.5pt\perp$}A_{k}$ then,

(1)

$F$ satisfies $\cup_{s\in S}A_{s}\mbox{$\perp\kern-5.5pt\perp$}\cup_{t\in T}A_{t}$ for all $S,T\subseteq[k]$ such that $S\cap T=\emptyset$ .
(2)

$F$ satisfies $B_{1}\mbox{$\perp\kern-5.5pt\perp$}\ldots\mbox{$\perp\kern-5.5pt\perp$}B_{k}$ where $B_{i}\subseteq A_{i}$ for all $i$ .

Proof.

First, we will show that for $S,T\subseteq[k]$ such that $S\cap T=\emptyset$ , if $F$ satisfies $A_{1}\mbox{$\perp\kern-5.5pt\perp$}\ldots\mbox{$\perp\kern-5.5pt\perp$}A_{k}$ , then it satisfies $\cup_{s\in S}A_{s}\mbox{$\perp\kern-5.5pt\perp$}\cup_{t\in T}A_{t}$ . Let $x$ be such that $x_{i}=\infty$ if $i\in[n]\setminus\cup_{s\in S}A_{s}$ , in particular, $x_{i}=\infty$ if $i\in[n]\setminus(A_{1}\cup\cdots\cup A_{k})$ as well. Thus,

F(x)=F(y^{(1)})\cdots F(y^{(k)})

where

y^{(j)}_{i}=\begin{cases}x_{i}&\mbox{if }i\in A_{j}\mbox{ and }j\in S\\ \infty&\mbox{otherwise}.\end{cases}

Hence,

F(x)=\prod_{s\in S}F\left(y^{(s)}\right).

We get an analogous result for $\cup_{t\in T}A_{t}$ .

Now, let $x$ be such that $x_{i}=\infty$ if $i\in[n]\setminus\cup_{t\in S\cup T}A_{t}$ . Then, it holds that

F(x)=F(y^{(1)})\cdots F(y^{(k)})

where

y^{(j)}_{i}=\begin{cases}x_{i}&\mbox{if }i\in A_{j}\mbox{ and }j\in S\cup T\\ \infty&\mbox{otherwise}.\end{cases}

Hence,

	$\displaystyle F(x)$	$\displaystyle=\prod_{s\in S}F\left(y^{(s)}\right)\prod_{t\in T}F\left(y^{(t)}\right)$
		$\displaystyle=F(w)F(z)$

where

w_{i}=\begin{cases}x_{i}&\mbox{if }i\in\cup_{s\in S}A_{s}\\ \infty&\mbox{otherwise}\end{cases}\quad\quad\quad\mbox{ and }\quad\quad\quad z_{i}=\begin{cases}x_{i}&\mbox{if }i\in\cup_{t\in T}A_{t}\\ \infty&\mbox{otherwise}\end{cases}

Now we will show that if $F$ satisfies $A_{1}\mbox{$\perp\kern-5.5pt\perp$}\ldots\mbox{$\perp\kern-5.5pt\perp$}A_{k}$ , then it satisfies $B_{1}\mbox{$\perp\kern-5.5pt\perp$}\ldots\mbox{$\perp\kern-5.5pt\perp$}B_{k}$ whenever $B_{i}\subseteq A_{i}$ for all $i$ . Assume $F$ satisfies $A_{1}\mbox{$\perp\kern-5.5pt\perp$}\ldots\mbox{$\perp\kern-5.5pt\perp$}A_{k}$ . Let $x$ be such that $x_{i}=\infty$ for all $i\in[n]\setminus(B_{1}\cup\ldots\cup B_{k})$ , then it follows that

F(x)=F(y^{(1)})\cdots F(y^{(k)})

where

y^{(j)}_{i}=\begin{cases}x_{i}&\mbox{if }i\in B_{j}\subseteq A_{j}\\ \infty&\mbox{otherwise}.\end{cases}

which proves $F$ also satisfies $B_{1}\mbox{$\perp\kern-5.5pt\perp$}\ldots\mbox{$\perp\kern-5.5pt\perp$}B_{k}$ . ∎

Proposition 2.7.

Suppose that the joint cdf $F$ satisfies the marginal independence statement $A_{1}\mbox{$\perp\kern-5.5pt\perp$}\cdots\mbox{$\perp\kern-5.5pt\perp$}A_{k}$ . Let $B_{1}\mbox{$\perp\kern-5.5pt\perp$}\cdots\mbox{$\perp\kern-5.5pt\perp$}B_{l}$ be another marginal independence statement such that for each $i\in[l]$ , there is a set $I_{i}\subseteq[k]$ such that $I_{i}\cap I_{j}=\emptyset$ for all $i\neq j$ , and $B_{i}\subseteq\cup_{t\in I_{i}}A_{t}$ . Then $F$ also satisfies $B_{1}\mbox{$\perp\kern-5.5pt\perp$}\cdots\mbox{$\perp\kern-5.5pt\perp$}B_{l}$ .

Proof.

Assume $F$ satisfies the marginal independence statement $A_{1}\mbox{$\perp\kern-5.5pt\perp$}\cdots\mbox{$\perp\kern-5.5pt\perp$}A_{k}$ . Let $B_{1}\mbox{$\perp\kern-5.5pt\perp$}\cdots\mbox{$\perp\kern-5.5pt\perp$}B_{l}$ be another marginal independence statement such that for each $i\in[l]$ , there is a set $I_{i}\subseteq[k]$ such that $I_{i}\cap I_{j}=\emptyset$ for all $i\neq j$ , and $B_{i}\subseteq\cup_{t\in I_{i}}A_{t}$ . By Lemma 2.6 (1), we know that $F$ satisfies $\cup_{t\in I_{1}}A_{t}\mbox{$\perp\kern-5.5pt\perp$}\cdots\mbox{$\perp\kern-5.5pt\perp$}\cup_{t\in I_{l}}A_{t}$ . Finally, $F$ satisfies $B_{1}\mbox{$\perp\kern-5.5pt\perp$}\cdots\mbox{$\perp\kern-5.5pt\perp$}B_{l}$ by Lemma 2.6 (2), since $B_{i}\subseteq\cup_{t\in I_{i}}A_{t}$ for all $i\in[l]$ . ∎

This yields a partial order on $P\Pi_{n}$ by the same rule as in Proposition 2.7.

Definition 2.8.

Let $\pi=\pi_{1}|\cdots|\pi_{k}$ and $\tau=\tau_{1}|\cdots|\tau_{l}$ be two partial set partitions. Define a partial order on $P\Pi_{n}$ by declaring $\pi>\tau$ if for each $i\in[l]$ , there is a set $I_{i}\in[k]$ such that $I_{i}\cap I_{J}=\emptyset$ for all $i\neq j$ , and $\tau_{i}\subseteq\cup_{t\in I_{i}}\pi_{t}$ .

Note that the covering relations in $P\Pi_{n}$ are of two types. We have that $\pi$ covers $\tau$ if either

(1)

$\pi$ can be obtained from $\tau$ by adding a single element to one of the blocks of $\tau$ , or
(2)

$\pi$ can be obtained from $\tau$ by splitting a block of $\tau$ into two blocks.

Proposition 2.9.

In $P\Pi_{n}$ , $\pi<\tau$ if and only if $S=|\pi|\subseteq|\tau|$ and $\pi\succ_{S}\tau\cap S$ , where $\succ_{S}$ is the standard order in the lattice of set partitions of $S$ and $\tau\cap S=\tau_{1}\cap S|\cdots|\tau_{l}\cap S$ .

Proof.

This follows directly from the covering relations. ∎

Remark.

Note that $P\Pi_{n}$ is not, in general, a lattice. Indeed, the elements $1|2$ and $3|4$ have two least upper bounds, namely $13|24$ and $14|23$ . This shows a key difference with the poset of set partitions $\Pi_{n}$ , which is well known to be a lattice [8].

On the other hand, $P\Pi_{n}$ is a graded poset, with

\rho(\pi_{1}|\cdots|\pi_{k})=\#(\cup_{i}\pi_{i})+k-1.

This formula follows by counting the number of covering relations to get from the bottom element $\emptyset$ to a partial set partition $\pi$ .

Remark.

Note that $P\Pi_{n}$ contains the Boolean lattice $B_{n}$ as a subposet. The lattice of set partitions $\Pi_{n}$ is not a subposet of $P\Pi_{n}$ , but rather its opposite $\Pi_{n}^{opp}$ is a subposet. In fact, $P\Pi_{n}$ contains the opposite of the lattice of set partitions of any $S\subseteq[n]$ .

For studying marginal independence models, it is more natural to consider the poset $P\Pi_{n,2}$ of partial set partitions where each partition has at least two blocks. We will see that marginal independence models correspond to certain types of order ideals in $P\Pi_{n,2}$ .

Definition 2.10.

Let $\pi^{(1)},\ldots,\pi^{(k)}\subseteq P\Pi_{n,2}$ , then

\mathcal{C}=\left\{\pi_{1}^{(1)}|\cdots|\pi_{i_{1}}^{(1)},\ldots,\pi_{1}^{(k)}|\cdots|\pi_{i_{k}}^{(k)}\right\}

or, equivalently,

\mathcal{C}=\left\{X_{\pi_{1}^{(1)}}\mbox{$\perp\kern-5.5pt\perp$}\ldots\mbox{$\perp\kern-5.5pt\perp$}X_{\pi_{i_{1}}^{(1)}},\ldots,X_{\pi_{1}^{(k)}}\mbox{$\perp\kern-5.5pt\perp$}\ldots\mbox{$\perp\kern-5.5pt\perp$}X_{\pi_{i_{k}}^{(k)}}\right\}

is a list of independence statements on random variables $X_{1},\ldots,X_{n}$ . A marginal independence model, $\mathcal{M}_{\mathcal{C}}$ , on these random variables is the set of distributions $P$ that satisfy all the statements in $\mathcal{C}$ .

Definition 2.11.

Let $\pi=\pi_{1}|\cdots|\pi_{k}\in P\Pi_{n,2}$ and $\tau=\tau_{1}|\tau_{2}\in P\Pi_{n,2}$ . We say $\tau$ splits $\pi$ if $\pi_{i}=\tau_{1}\cup\tau_{2}$ for some $i$ . We denote the splitting of $\pi$ by $\tau$ with $\pi^{\tau}=\pi_{1}|\cdots|\pi_{i-1}|\tau_{1}|\tau_{2}|\pi_{i+1}|\cdots|\pi_{k}$ .

Definition 2.12.

Let $\mathcal{C}$ be an order ideal of $P\Pi_{n,2}$ . We say $\mathcal{C}$ is split closed if $\pi^{\tau}\in\mathcal{C}$ for all $\pi,\tau\in\mathcal{C}$ such that $\tau$ splits $\pi$ .

Definition 2.13.

The split closure $\overline{\mathcal{C}}$ of $\mathcal{C}$ is the smallest split closed order ideal containing $\mathcal{C}$ .

Remark.

This allows us to make a precise connection between collections of marginal independence statements $\mathcal{C}$ and certain order ideals in $P\Pi_{n,2}$ , namely, their split closure $\overline{\mathcal{C}}$ .

Proposition 2.14.

Let $\mathcal{C}$ be a a split closed order ideal. If $\pi,\mu\in\mathcal{C}$ are such that $S=|\pi|=|\mu|$ for some $S\subseteq[n]$ , then $\pi\wedge\mu\in I$ where $\wedge$ is the common refinement in the lattice of set partitions of $S$ .

Proof.

Let $\pi,\mu\in I$ be such that $S=|\pi|=|\mu|$ for some $S\subseteq[n]$ . Observe that $\pi\wedge\mu=\tau$ where for each $i$ , $\tau_{i}=\pi_{r}\cap\mu_{t}$ for some $r,t$ . Now consider the partition $\gamma^{i}=\mu\cap\pi_{i}=\mu_{1}\cap\pi_{i}|\cdots|\mu_{l}\cap\pi_{i}$ , by definition, either $\gamma^{i}=\pi_{i}\subseteq\mu_{j}$ for some $j$ or $\gamma^{i}\leq\mu$ . Observe further that we can write $\tau=\gamma^{1}|\cdots|\gamma^{k}$ . If for all $i$ , we have that $\gamma^{i}=\pi_{i}$ , then $\tau=\pi\in I$ . On the other hand, if $\gamma^{i}\leq\mu$ , clearly $\gamma^{i}\in I$ and since $I$ is a split closed order ideal, so will $\pi^{\gamma^{i}}$ . Thus, $\tau=\pi^{\gamma^{1}\cdots\gamma^{k}}\in I$ ∎

We finish this section stating a general theorem about marginal independence models that makes the correspondence between split closed order ideals and models precise. We prove this theorem in Section 3.

Theorem 2.15.

Marginal independence models $\mathcal{M}_{\mathcal{C}}$ on $n$ random variables are in bijective correspondence to split closed order ideals $\mathcal{C}\subseteq P\Pi_{n,2}$ .

In this level of generality, it is not clear how to prove the correspondence yet. One of the directions is straightforward, based on results we have already proven: for any marginal independence model, there exists a split closed order ideal by taking all implied marginal independence statements. For the other direction, it suffices to show the existence of probability distributions that satisfy exactly the independence statements in a given split closed order ideal, and no others. We will derive such result as a consequence of our study of marginal independence for discrete random variables in the following section.

3. Discrete marginal independence models

In this section, we will discuss marginal independence models restricting the class of random variables to have finitely many states to allow us to translate the independence statements into polynomial constraints. Past work has been done on certain classes of these models where the marginal independence statements are given by graphs or simplicial complexes [1, 2]. However, there are marginal independence models that cannot be represented with either of these structures, which is why we present a more general approach using split closed order ideals from the set of partial set partitions. In previous work [1, 2, 9], it has been shown these models are toric after a certain linear change of coordinates whose inverse could be found by applying Möbius inversion on a specific poset. In our treatment of marginal independence, we will use the cumulative distribution function as our coordinate system. This allows us to express marginal independence constraints in a very natural way while preserving the structure of the inverse map as a Möbius inversion on a poset isomorphic to the one used in [1].

We begin by defining what a single marginal independence statement looks like in terms of the probability tensor and then we will generalize it further to define our models. For this, we need to first discuss some notation that will be used.

Definition 3.1.

For finite random variables $X_{1},\ldots,X_{n}$ , with respective state spaces $[r_{i}]$ , $r_{i}\in\mathbb{N}$ , we use the shorthand $p_{i_{1}\ldots i_{n}}$ to denote $P(X_{1}=i_{1},\ldots X_{n}=i_{n})$ . Furthermore, when we discuss marginalizations of the probability tensor, we will allow $i_{k}=+$ to symbolize adding over all the states of the corresponding variable, i.e. if $i_{k}=+$ , then

p_{i_{1},\ldots,i_{k},\ldots,i_{n}}=p_{i_{1},\ldots,+,\ldots,i_{n}}=\sum_{j_{k}=1}^{r_{k}}p_{i_{1},\ldots,j_{k},\ldots,i_{n}}.

For an index vector $i=i_{1}\ldots i_{n}$ , we define the support of $i$ as $\mathrm{supp}(i)=\{l\in[n]\;|\;i_{l}\neq+\}$ .

Definition 3.2.

Let $X_{1},\ldots,X_{n}$ be finite random variables, we say that for disjoint $A,B\subseteq[n]$ , $X_{A}\mbox{$\perp\kern-5.5pt\perp$}X_{B}$ holds if and only if for a probability distribution $P$ indexed by $p_{i}=p_{i_{1}\ldots i_{n}}$ we have that

p_{i_{1}\ldots i_{n}}p_{j_{1}\ldots j_{n}}-p_{k_{1}\ldots k_{n}}p_{l_{1}\ldots l_{n}}=0,

for all $i,j,k,l$ , with $\mathrm{supp}(i)=\mathrm{supp}(j)=\mathrm{supp}(k)=\mathrm{supp}(l)=A\cup B$ , $k_{A}=i_{A}$ , $k_{B}=i_{B}$ , $l_{A}=j_{A}$ , and $l_{B}=j_{B}$ .

Equivalently, these constraints can be obtained from all the $2\times 2$ minors of the matrix resulting from marginalizing the probability tensor $P$ by summing out the indices not in $A\cup B$ and whose columns are indexed by the states of $X_{A}$ and the rows are indexed by the states of $X_{B}$ , where $X_{A}$ and $X_{B}$ are seen as random vectors.

Definition 3.3.

Let $P$ be a probability distribution on random variables $X_{1},\ldots,X_{n}$ with respective state spaces $[r_{1}],\ldots,[r_{n}]$ , with $r_{i}\in\mathbb{N}$ for $i\in[n]$ . Then, we define $\mathcal{R}=[r_{1}]\times\cdots\times[r_{n}]$ .

Definition 3.4.

We define the ring of polynomials in probability coordinates, $\mathbb{R}[P]$ , for the probability tensor of format $\mathcal{R}=[r_{1}]\times\ldots\times[r_{n}]$ as

\mathbb{R}[P]=\mathbb{R}[p_{i}:\;i\in\mathcal{R}].

Definition 3.5.

The marginal independence ideal ( $MI_{\mathcal{C}}$ ) associated to the split closed order ideal $\mathcal{C}$ on discrete random variables $X_{1},\ldots,X_{n}$ is

\displaystyle MI_{\mathcal{C}}=\left\langle 2\times 2\mbox{ minors of all flattenings of }P_{\pi}:\;\pi\in\mathcal{C}\right\rangle,

where $\pi=\pi_{1}|\cdots|\pi_{k}$ and $P_{\pi}$ is the $k$ -way tensor which is obtained by first marginalizing the tensor $P$ by summing out the indices not in $|\pi|$ , and then flattening the resulting tensor according to the set partition $\pi$ .

Alternatively, $MI_{\mathcal{C}}$ is the ideal resulting from imposing rank 1 conditions on the tensors $P_{\pi}$ , for all $\pi\in\mathcal{C}$ . This definition is a slight modification of the one in [1] to fit our class of models.

Example 3.6.

Let $X_{1}$ be a binary random variable and let $X_{2},X_{3}$ be ternary randcom variables and consider the marginal independence statement $X_{2}\mbox{$\perp\kern-5.5pt\perp$}X_{3}$ . Then the equations that define $MI_{2\mbox{$\perp\kern-5.5pt\perp$}3}$ are

	$\displaystyle p_{+i_{2}i_{3}}p_{+j_{2}j_{3}}-p_{+i_{2}j_{3}}p_{+j_{2}i_{3}}=$	$\displaystyle(p_{1i_{2}i_{3}}+p_{2i_{2}i_{3}})(p_{1j_{2}j_{3}}+p_{2j_{2}j_{3}})$
		$\displaystyle-(p_{1i_{2}j_{3}}+p_{2i_{2}j_{3}})(p_{1j_{2}i_{3}}+p_{2j_{2}i_{3}})=0,$

for all $i_{2},i_{3},j_{2},j_{3}\in[3]$ . We can also obtain these equations from the $2\times 2$ minors of the matrix

\begin{pmatrix}p_{+11}&p_{+12}&p_{+13}\\ p_{+21}&p_{+22}&p_{+23}\\ p_{+31}&p_{+32}&p_{+33}\end{pmatrix}.

Example 3.7.

Let $n=4$ and consider the model given by binary random variables such that $X_{1,2}\mbox{$\perp\kern-5.5pt\perp$}X_{3}$ . Then, the equations that define $MI_{12\mbox{$\perp\kern-5.5pt\perp$}3}$ are

	$\displaystyle p_{i_{1}i_{2}i_{3}+}p_{j_{1}j_{2}j_{3}+}-p_{i_{1}i_{2}j_{3}+}p_{j_{1}j_{2}i_{3}+}=$	$\displaystyle(p_{i_{1}i_{2}i_{3}1}+p_{i_{1}i_{2}i_{3}2})(p_{j_{1}j_{2}j_{3}1}+p_{j_{1}j_{2}j_{3}2})$
		$\displaystyle-(p_{i_{1}i_{2}j_{3}1}+p_{i_{1}i_{2}j_{3}2})(p_{j_{1}j_{2}i_{3}1}+p_{j_{1}j_{2}i_{3}2})=0,$

for all $i_{1},i_{2},i_{3},j_{1},j_{2},j_{3}\in[2]$ . These equations can again be obtained from the $2\times 2$ minors of the matrix

\begin{pmatrix}p_{111+}&p_{112+}\\ p_{121+}&p_{122+}\\ p_{211+}&p_{212+}\\ p_{221+}&p_{222+}\\ \end{pmatrix}.

The ideal $MI_{\mathcal{C}}$ tends not to be prime when $\mathcal{C}$ has more than one maximal element, which represents a challenge when studying these models. We get around this issue by using the cumulative distribution function as our new system of coordinates, since marginal independence constraints can be expressed in a simpler way using the cdf.

Definition 3.8 (cdf coordinates).

We define the linear change of coordinates $\varphi:\mathbb{R}^{\mathcal{R}}\rightarrow\mathbb{R}^{\mathcal{R}}$ , $P\mapsto Q$ by transforming the tensor $P$ so that the entries of $Q$ are cumulative distributions as follows:

	$\displaystyle q_{j_{1}\ldots j_{n}}$	$\displaystyle=P\left\{X_{1}\leq j_{1},\ldots,X_{n}\leq j_{n}\right\}$
		$\displaystyle=\sum_{i_{1}\leq j_{1},\ldots,i_{n}\leq j_{n}}p_{i_{1}\ldots i_{n}}$

Remark.

In the case of binary random variables, cdf coordinates become much simpler. We have that for any $q_{i_{1}\ldots i_{n}}$ , either $i_{l}=1$ or $i_{l}=2$ , thus we can represent each cdf coordinate by indexing it with its support.

We can discuss this transformation, and in particular, its inverse, in terms of the poset of the index vectors of the tensors. Let $\mathcal{P}=[r_{1}]\times\ldots\times[r_{n}]$ be the usual direct product poset, where two index vectors $j\leq i$ if $j_{k}\leq i_{k}$ for all $k\in[n]$ . Then, we can state the equation for cdf coordinates in Definition 3.8 as follows, for all $i\in\mathcal{P}$ ,

\displaystyle q_{i}:=\sum_{j\leq i}p_{j}.

Using the Möbius inversion formula we have that for all $i\in\mathcal{P}$ ,

	$\displaystyle p_{i}$	$\displaystyle=\sum_{j\leq i}\mu(j,i)q_{j}$
		$\displaystyle=\sum_{j\leq i}\prod_{k=1}^{n}\mu(j_{k},i_{k})q_{j},$

where

\displaystyle\mu(j_{k},i_{k})=\begin{cases}1&\mbox{ if }i_{k}=j_{k}\\ -1&\mbox{ if }i_{k}=j_{k}+1\\ 0&\mbox{ otherwise.}\\ \end{cases}

Example 3.9.

In the case when $r_{1}=r_{2}=3$ , we have that the poset $\mathcal{P}$ would be as in Figure 1.

Figure 1. Index poset

\mathcal{P}

from Example 3.9.

Then, we have that

\displaystyle q_{22}=\sum_{(i_{1}i_{2})\leq(22)}p_{i_{1}i_{2}}.

Hence, using Möbius inversion on this poset yields

	$\displaystyle p_{22}$	$\displaystyle=\sum_{(i_{1}i_{2})\leq(22)}\mu(i_{1}i_{2},22)q_{i_{1}i_{2}}$
		$\displaystyle=q_{22}-q_{12}-q_{21}+q_{11}.$

Marginal independence is usually stated in terms of the cumulative probability distribution, which is why this is a natural transformation to consider as it simplifies the equations for marginal independence.

Our goal now is to to explore the ideals of discrete marginal independence models in the cdf coordinates. To do this, we need to explore some properties of split closed order ideals.

Definition 3.10.

Let $\mathcal{C}$ be a split closed order ideal of $P\Pi_{n,2}$ , then

(1)

We say that a nonempty set $D\subseteq[n]$ is disconnected (with respect to $\mathcal{C}$ ) if there exists $\pi\in\mathcal{C}$ such that $D=|\pi|$ .
(2)

A connected set is a set that is not disconnected.
(3)

A connected set $C$ is maximal with respect to some disconnected set $D$ if for any set $C^{\prime}\subseteq D$ such that $C\subset C^{\prime}$ , then $C^{\prime}$ is disconnected.
(4)

We define the set $\mathrm{Con}(\mathcal{C})=\left\{C\subseteq[n]:\;C\mbox{ is connected}\right\}$

Proposition 3.11.

For any disconnected set $D$ with respect to $\mathcal{C}$ , there exists a unique maximal $\pi\in\mathcal{C}$ such that

D=|\pi|=\cup_{i=1}^{k}\pi_{i},

where all $\pi_{i}$ are maximal connected sets.

Proof.

Since $D$ is disconnected, there exists $\gamma\in\mathcal{C}$ such that $D=|\gamma|$ . If some block $\gamma_{k}$ is disconnected, then there is some $\tau\in\mathcal{C}$ such that $\gamma_{k}=|\tau|$ . Then, $D=|\gamma^{\tau}|$ is the new decomposition, additionally, $\gamma^{\tau}\in\mathcal{C}$ and $\gamma^{\tau}>\gamma$ . We can repeat this process on $\gamma^{\tau}$ if any of its blocks are disconnected. This process terminates since every chain of a finite poset is finite.

To show uniqueness of $\pi$ , let us consider a different maximal decomposition $\mu$ . Since $D=|\pi|=|\mu|$ , and by assumption, $\pi\neq\mu$ , then, $\mu,\pi<\pi\wedge\mu\in\mathcal{C}$ thus contradicting maximality of $\pi$ and $\mu$ . ∎

Let $i\in\mathcal{R}$ . Define the support of $i$ to be $\mathrm{supp}(i)=\{l\in[n]\mid i_{l}\neq r_{l}\}$ . Note that for cdf coordinates this will agree with our previous notion of support, since if $i_{l}=r_{l}$ in $q_{i}$ , it means we are assuming $X_{l}\leq\infty$ in our cdf calculations, a condition denoted by $+$ in $p_{i}$ .

Corollary 3.12.

Let $\mathcal{C}$ be a split closed order ideal. A probability tensor $P$ lies in the model $\mathcal{M}_{\mathcal{C}}$ if and only if its cdf coordinates satisfy the following condition. For any $D\subseteq[n]$ that is disconnected by $\mathcal{C}$ , and $i,j^{(l)}\in\mathcal{R}$ be index vectors where

•

$D=\pi_{1}\cup\ldots\cup\pi_{k}$ is the decomposition of $D$ into maximal connected sets.
•

$D=\mathrm{supp}(i)$ , $\pi_{l}=\mathrm{supp}(j^{(l)})$ , and
•

$i_{\mathrm{supp}(j^{(l)})}=j^{(l)}_{\mathrm{supp}(j^{(l)})}$

we have

(1)

q_{i}=q_{j^{(1)}}\ldots q_{j^{(k)}}.

Proof.

We know $P$ lies in the model $\mathcal{M}_{\mathcal{C}}$ if and only if its corresponding cdf satisfies all of the marginal independence constraints. We know that the cdf must satisfy that for all $\pi\in\mathcal{C}$ ,

q_{i}=q_{j^{(1)}}\ldots q_{j^{(k)}}

where

•

$|\pi|=\mathrm{supp}(i)$ , $\pi_{l}=\mathrm{supp}(j^{(l)})$ ,
•

$i_{\mathrm{supp}(j^{(l)})}=j^{(l)}_{\mathrm{supp}(j^{(l)})}$ .

If we have a non maximal $\pi$ , then we can decompose it further using Proposition 3.11. Hence, the non maximal $\pi$ yield equations that are implied by maximal set partitions in the sense of Proposition 3.11. ∎

Remark.

Observe that in the case of binary random variables, we get that for any disconnected set $D\subseteq[n]$ with respect to $\mathcal{C}$ with maximal connected sets $\pi_{1},\ldots,\pi_{k}$ ,

q_{D}=q_{\pi_{1}}\ldots q_{\pi_{k}}.

Definition 3.13.

We define the ring of polynomials in cdf coordinates $\mathbb{R}[Q]$ for the cdf tensor $Q$ of format $\mathcal{R}=[r_{1}]\times\ldots\times[r_{n}]$ as

\mathbb{R}[Q]=\mathbb{R}[q_{i}:\;i\in\mathcal{R}].

Definition 3.14.

Let $\mathcal{C}$ be a split closed order ideal on $X_{1},\ldots,X_{n}$ and $i\in\mathcal{R}$ such that $\mathrm{supp}(i)$ is disconnected with respect to $\mathcal{C}$ . Then, the associated maximal equation $f_{i}$ is given by

f_{i}=q_{i}-q_{j^{(1)}}\ldots q_{j^{(k)}}

where

•

$\pi_{l}=\mathrm{supp}(j^{(l)})$ ,
•

$i_{\mathrm{supp}(j^{(l)})}=j^{(l)}_{\mathrm{supp}(j^{(l)})}$ , and
•

$\mathrm{supp}(i)=\pi_{1}\cup\ldots\cup\pi_{k}$ is the decomposition of $\mathrm{supp}(i)$ into maximal connected sets.

Definition 3.15.

The inhomogeneous marginal independence ideal ( $J_{\mathcal{C}}$ ) associated to the split closed order ideal $\mathcal{C}$ on discrete random variables $X_{1},\ldots,X_{n}$ is

J_{\mathcal{C}}=\left\langle f_{i}\mbox{ maximal}:\;i\in\mathcal{R},\;\mathrm{supp}(i)\mbox{ is disconnected}\right\rangle.

Example 3.16.

In Example 3.7, there are 16 cdf coordinates indexed by subsets of $[4]$ . The equations that define the ideal $J_{\mathcal{C}}$ without indexing them with their support are

\displaystyle f_{1112}=q_{1112}-q_{1122}q_{2212}=0,

\displaystyle f_{1212}=q_{1212}-q_{1222}q_{2212}=0,\;\;

\displaystyle f_{2112}=q_{2112}-q_{1212}q_{1122}=0.

However, since all random variables are binary, we can instead index them by their support as follows:

\displaystyle f_{123}=q_{123}-q_{12}q_{3}=0,

\displaystyle f_{13}=q_{13}-q_{1}q_{3}=0,\;\;

\displaystyle f_{23}=q_{23}-q_{2}q_{3}=0.

This ideal is prime and therefore toric in cdf coordinates.

Definition 3.17.

The marginal independence ideal in cdf coordinates ( $MJ_{\mathcal{C}}$ ) associated to the split closed order ideal $\mathcal{C}$ on discrete random variables $X_{1},\ldots,X_{n}$ is

\displaystyle MJ_{\mathcal{C}}=\left\langle 2\times 2\mbox{ minors of all flattenings of }Q_{\pi}:\;\pi\in\mathcal{C}\right\rangle,

where $\pi=\pi_{1}|\cdots|\pi_{k}$ and $Q_{\pi}$ is the $k$ -way tensor which is obtained by first marginalizing the tensor $Q$ by making all the indices not in $|\pi|$ equal to their maximum value (this corresponds to summing out indices in Definition 3.5, as we are in cdf coordinates), and then flattening the resulting tensor according to the set partition $\pi$ .

Alternatively, $MJ_{\mathcal{C}}$ is the ideal resulting from imposing rank 1 conditions on the tensors $Q_{\pi}$ , for all $\pi\in\mathcal{C}$ .

Example 3.18.

Let us consider the same model as in Example 3.7. Then, $MJ_{12\mbox{$\perp\kern-5.5pt\perp$}3}$ is going to be given by imposing rank $1$ conditions on the tensors $Q_{12\mbox{$\perp\kern-5.5pt\perp$}3}$ , $Q_{1\mbox{$\perp\kern-5.5pt\perp$}3}$ and $Q_{2\mbox{$\perp\kern-5.5pt\perp$}3}$ , all of which are matrices since they are all partial set partitions with 2 blocks. Following the construction in Definition 3.17, we obtain

Q_{12\mbox{$\perp\kern-5.5pt\perp$}3}=\begin{pmatrix}q_{1112}&q_{1122}\\ q_{1212}&q_{1222}\\ q_{2112}&q_{2122}\\ q_{2212}&q_{2222}\\ \end{pmatrix}.

Furthermore, since the random variables are binary, we can reindex the cdf coordinates by their support as follows:

Q_{12\mbox{$\perp\kern-5.5pt\perp$}3}=\begin{pmatrix}q_{123}&q_{12}\\ q_{13}&q_{1}\\ q_{23}&q_{2}\\ q_{3}&q_{\emptyset}\\ \end{pmatrix}.

Observe that all of the rank 1 conditions in $Q_{1\mbox{$\perp\kern-5.5pt\perp$}3}$ and $Q_{2\mbox{$\perp\kern-5.5pt\perp$}3}$ are found in the $2\times 2$ minors of $Q_{12\mbox{$\perp\kern-5.5pt\perp$}3}$ . Thus, $MJ_{12\mbox{$\perp\kern-5.5pt\perp$}3}$ is given by the $2\times 2$ minors of $Q_{12\mbox{$\perp\kern-5.5pt\perp$}3}$

This ideal and its generators generally have a more complicated structure than $J_{\mathcal{C}}$ . However, once we add a constraint that is imposed by the probability simplex, both ideals have the same generators, which allows us to work with the simpler ideal $J_{\mathcal{C}}$ to study these models. To prove this result, we first need to prove the following:

Lemma 3.19.

Let $A=(a_{ij})_{(i,j)\in[m]\times[n]}\in\mathbb{C}^{m\times n}$ be such that $a_{kl}=1$ for some $(k,l)\in[m]\times[n]$ . Then, $\mathrm{rank}(A)=1$ if and only if the $2\times 2$ minors involving $a_{kl}$ vanish.

Proof.

The forward direction is direct. For the converse, without loss of generality (since rank is invariant under row/column operations), assume $a_{11}=1$ and the $2\times 2$ minors involving $a_{11}$ vanish. Let $a_{i}$ denote the $i$ -th row of $A$ . To show $A$ is rank 1, it suffices to show $a_{i}$ is a scalar multiple of $a_{1}$ , for any $i\in[m]$ . We claim that scalar is $a_{i1}$ . Let $i\in[m]$ , note

\displaystyle a_{i}=a_{i1}a_{1}\iff a_{ij}=a_{i1}a_{1j}\;\;\forall j\in[n],

which is true as these correspond to the vanishing minors involving $a_{11}$ . ∎

Proposition 3.20.

For a split closed order ideal $\mathcal{C}$ , we have that $MJ_{\mathcal{C}}+\langle q_{r_{1}\cdots r_{n}}-1\rangle=J_{\mathcal{C}}+\langle q_{r_{1}\cdots r_{n}}-1\rangle$ .

Proof.

First observe that for any $\pi=\pi_{1}|\cdots|\pi_{k}\in\mathcal{C}$ , there exist partial set partitions $\tau_{1},\ldots,\tau_{k-1}\in\mathcal{C}$ such that

\pi=((\tau_{1}^{\tau_{2}})^{\tau_{3}}\cdots)^{\tau_{k}}

these are given by

\tau_{i}=\pi_{i}|\bigcup_{j=i+1}^{k}\pi_{j}.

In other words, we can get any partial set partition by taking partitions with 2 blocks and applying the splitting operation to them. On the algebraic side, this means the maximal equations $f_{i}$ that generate $J_{\mathcal{C}}$ can be obtained if we have all the equations corresponding to set partitions with 2 blocks in $\mathcal{C}$ .

Note that the matrices $Q_{\pi}$ with $\pi=\pi_{1}|\pi_{2}\in\mathcal{C}$ always have $q_{r_{1}\cdots r_{n}}$ appearing. Furthermore, the $2\times 2$ minors that involve $q_{r_{1}\cdots r_{n}}$ are of the form

\displaystyle g_{i}=q_{r_{1}\cdots r_{n}}q_{i}-q_{j}q_{k}\in MJ_{\mathcal{C}}

where $i,j,k\in\mathcal{R}$ are such that

•

$\mathrm{supp}(i)=|\pi|$ ,
•

$\mathrm{supp}(j)=\pi_{1}$ ,
•

$\mathrm{supp}(k)=\pi_{2}$ ,
•

$i_{\mathrm{supp}(j)}=j_{\mathrm{supp}(j)}$ , and
•

$i_{\mathrm{supp}(k)}=k_{\mathrm{supp}(k)}$

Once we add the constraint $q_{r_{1}\cdots r_{n}}-1$ , we observe that

\displaystyle\hat{g_{i}}=q_{i}-q_{j}q_{k}\in MJ_{\mathcal{C}}+\langle q_{r_{1}\cdots r_{n}}-1\rangle.

Since any set partition in $\mathcal{C}$ can be obtained by splitting set partitions with 2 blocks, we can keep splitting $q_{j}$ and $q_{k}$ until we get the maximal equation $f_{i}\in J_{\mathcal{C}}$ . Thus showing $MJ_{\mathcal{C}}+\langle q_{r_{1}\cdots r_{n}}-1\rangle\supseteq J_{\mathcal{C}}+\langle q_{r_{1}\cdots r_{n}}-1\rangle$ .

For the other direction, observe that for any $\pi\in\mathcal{C}$ , we know that any flattening of the tensor $Q_{\pi}$ contains $q_{r_{1}\cdots r_{n}}$ . Additionally, under the constraint $q_{r_{1}\cdots r_{n}}-1=0$ , any such matrix contains a 1. Thus, a flattening of $Q_{\pi}$ is rank 1 if and only if the minors involving the entry of $1$ vanish, by Lemma 3.19. However, these $2\times 2$ minors are all equations of the same form as $\hat{g_{i}}$ above, all of which are in $J_{\mathcal{C}}$ by construction. Hence showing the tensor $Q_{\pi}$ is rank 1 if and only if its entries lie in $V(J_{\mathcal{C}}+\langle q_{r_{1}\cdots r_{n}}-1\rangle)$ , which in turn shows $J_{\mathcal{C}}+\langle q_{r_{1}\cdots r_{n}}-1\rangle$ generates $MJ_{\mathcal{C}}+\langle q_{r_{1}\cdots r_{n}}-1\rangle$ . ∎

Now we will exploit the combinatorial structure of the equations in $J_{\mathcal{C}}$ to study these models. Observe that Corollary 3.12 gives us a unique factorization for all cdf coordinates and, in particular, it tells us that the cdf coordinates that appear in the defining equations of the variety are precisely those with disconnected support.

Theorem 3.21.

Every discrete marginal independence model $\mathcal{M}_{\mathcal{C}}$ is toric in cdf coordinates.

Proof.

By Corollary 3.12, we know that the ideal $J_{\mathcal{C}}$ that generates the model is given by equations of the form

f_{i}=q_{i}-q_{j^{(1)}}\ldots q_{j^{(k)}}

where

•

$D$ is disconnected.
•

$D=\mathrm{supp}(i)$ , $\pi_{l}=\mathrm{supp}(j^{(l)})$ ,
•

$i_{\mathrm{supp}(j^{(l)})}=j^{(l)}_{\mathrm{supp}(j^{(l)})}$ and
•

$D=\pi_{1}\cup\ldots\cup\pi_{k}$ is the decomposition of $D$ into maximal connected sets.

Now, let us consider the term order $\prec$ on $\mathbb{R}[Q]$ defined by a lexicographic order with the condition that if $|\mathrm{supp}(i)|<|\mathrm{supp}(j)|$ , then $q_{i}\prec q_{j}$ . Therefore, for each generator $f_{i}$ of $J_{\mathcal{C}}$ , we have that $\mbox{lt}_{\prec}(f_{i})=q_{i}$ .

This is a Gröbner basis for $J_{\mathcal{C}}$ since the leading terms of any pair of equations are coprime, so any $S$ -pair reduces to $0$ modulo the basis of difference of binomials. Thus, the initial ideal of $J_{\mathcal{C}}$ , ${\rm in}_{\prec}(J_{\mathcal{C}})$ is generated by linear terms, so it is prime. This implies that $J_{\mathcal{C}}$ is a prime ideal generated by differences of binomials, hence it is toric. ∎

The explicit description of the generators of $J_{\mathcal{C}}$ also allows us to show that its associated affine variety is smooth.

Example 3.22.

Consider the split closed order ideal $\mathcal{C}$ on 3 binary random variables generated by the independence statement $X_{1}\mbox{$\perp\kern-5.5pt\perp$}X_{2,3}$ . Then, by Corollary 3.12,

J_{\mathcal{C}}=\left\langle q_{12}-q_{1}q_{2},q_{13}-q_{1}q_{3},q_{123}-q_{1}q_{23}\right\rangle

Hence, the Jacobian of $J_{\mathcal{C}}$ is

\mbox{Jac}(J_{\mathcal{C}})=\begin{bmatrix}0&-q_{2}&-q_{1}&0&1&0&0&0\\ 0&-q_{3}&0&-q_{1}&0&1&0&0\\ 0&-q_{23}&0&0&0&0&-q_{1}&1\\ \end{bmatrix}.

Observe that $J_{\mathcal{C}}$ is smooth everywhere since its Jacobian has full rank independent of the point of evaluation.

Theorem 3.23.

For a split closed order ideal $\mathcal{C}$ , the affine variety that defines the model $V(J_{\mathcal{C}})$ is smooth.

Proof.

It suffices to show that the Jacobian of $J_{\mathcal{C}}$ is full rank. Observe that $J_{\mathcal{C}}$ is generated by equations of the form

(2)

f_{i}=q_{i}-q_{j^{(1)}}\ldots q_{j^{(k)}}

where

•

$D=\mathrm{supp}(i)$ , $\pi_{l}=\mathrm{supp}(j^{(l)})$ ,
•

$i_{\mathrm{supp}(j^{(l)})}=j^{(l)}_{\mathrm{supp}(j^{(l)})}$ and
•

$D=\pi_{1}\cup\ldots\cup\pi_{k}$ is the decomposition of $D$ into maximal connected sets.

for all disconnected $D$ . Hence, the equations that appear on the generating set are precisely those indexed by $i$ such that $\mathrm{supp}(i)$ is disconnected. In addition, observe that we have that for all index vectors $i$ and $j$ with disconnected support,

\frac{\partial}{\partial q_{j}}f_{i}=\begin{cases}1&\mbox{if $j=i$}\\ 0&\mbox{else}\end{cases}.

This follows because of the uniqueness of the decomposition of $q_{i}$ . Hence, the columns of the Jacobian contain some permutation of the columns of the identity matrix of size $\#\{i\in[r_{1}]\times\cdots\times[r_{n}]:\mathrm{supp}(i)\mbox{ is disconnected}\}$ , which shows that $\mbox{Jac}(J_{\mathcal{C}})$ has full rank everywhere. ∎

We are now ready to prove a result about existence of distributions in the binary case.

Proposition 3.24.

For each split closed order ideal $\mathcal{C}\subseteq P\Pi_{n,2}$ , there exists a probability distribution $P$ in the binary model $\mathcal{M}_{\mathcal{C}}$ such that $P\in\Delta_{2^{n}}\setminus\partial(\Delta_{2^{n}})$ and $P$ satisfies all the marginal independence statements in $\mathcal{C}$ and no other marginal independence statements.

Proof.

We want to show there is a point in the interior of the probability simplex that satisfies exactly the statements in $\mathcal{C}$ and no others.

Since we are using binary random variables, the description of our generators becomes much simpler,

J_{\mathcal{C}}=\langle f_{D}|\;D\mbox{ disconnected}\rangle.

Take $\pi\notin\mathcal{C}$ to be minimal, this corresponds to an equation $g=q_{\pi}-q_{\pi_{1}}\cdots q_{\pi_{k}}$ where $\pi$ is connected. Let

I=J_{\mathcal{C}}+\langle q_{\pi}-q_{\pi_{1}}\cdots q_{\pi_{k}}\rangle

Observe that ${\rm in}_{\prec}(I)={\rm in}_{\prec}(J_{\mathcal{C}})+\langle q_{\pi}\rangle$ . So that we still get a Gröbner basis for $I$ composed of linear terms and therefore, $\dim(I)<\dim(J_{\mathcal{C}})$ . Hence, we have a strict inclusion, $V(I)\subset V(J_{\mathcal{C}})$ , showing that $V(J_{\mathcal{C}})\setminus V(I)\neq\emptyset$ . Since the uniform distribution $U\in V(J_{\mathcal{C}})$ , it intersects the interior of the probability simplex (in cdf coordinates). Finally, since we additionally know $V(J_{\mathcal{C}})$ is smooth, we conclude that $\dim(V(J_{\mathcal{C}}))=\dim(V(J_{\mathcal{C}})\cap\Delta_{2^{n}}))$ . ∎

With Proposition 3.24 in hand, we are ready to prove Theorem 2.15.

Proof of Theorem 2.15.

Given a model $\mathcal{M}_{\mathcal{C}}$ , we obtain a unique split closed order ideal given by the closure of $\mathcal{C}$ . Now let us start with a split closed order ideal $\mathcal{C}\in P\Pi_{n,2}$ . It suffices to show that for any $\pi\notin\mathcal{C}$ , there exists a distribution $P$ that satisfies the statements in $\mathcal{C}$ and does not satisfy $\pi$ , this follows directly from Proposition 3.24. ∎

We have now proven the bijective correspondence between split closed order ideals and marginal independence models, which provides us with a combinatorial description of this class of models.

So far, we have been describing our models and varieties in affine space. However, Proposition 3.11 and Corollary 3.12 allow us to give an explicit homogeneous parametrization of the ideals defining our models.

Corollary 3.25.

Let $\mathbb{R}[\Theta]:=\mathbb{R}\left[t,\theta_{l}^{(C)}:\;C\in\mathrm{Con}(\mathcal{C}),\;\mathrm{supp}(l)=C,\;l\in\prod_{i=1}^{n}[r_{i}]\right]$ . A homogeneous parametrization of the model in cdf coordinates is given by the homomorphism between polynomial rings $\phi_{\mathcal{C}}:\mathbb{R}[Q]\rightarrow\mathbb{R}[\Theta]$ defined by

\phi_{\mathcal{C}}(q_{i_{1}\ldots i_{n}})=t\prod_{j=1}^{k}\theta_{i_{\pi_{j}}}^{(\pi_{j})},

where $\mathrm{supp}(i)=\pi_{1}\cup\ldots\cup\pi_{k}$ is the decomposition of $\mathrm{supp}(i)$ into maximal connected sets.

Remark.

In the binary case, we can lose the subscript on our parameters $\theta$ as a consequence of the reindexing of binary cdf coordinates with their support.

Definition 3.26.

The homogeneous marginal independence ideal ( $I_{\mathcal{C}}$ ) associated to the split closed order ideal $\mathcal{C}$ on discrete random variables $X_{1},\ldots,X_{n}$ is given by

I_{\mathcal{C}}=\ker\phi_{\mathcal{C}},

where $\phi_{\mathcal{C}}$ is the parametrization given in Corollary 3.25.

Proposition 3.27.

For a split order model ideal $\mathcal{C}$ , we have

I_{\mathcal{C}}=\overline{J_{\mathcal{C}}},

where $\overline{J_{\mathcal{C}}}$ is the homogenization of $J_{\mathcal{C}}$ .

Proof.

This follows from construction of the parametrization in Corollary 3.25 and the definition of $I_{\mathcal{C}}$ . ∎

In general, toric varieties are defined by a monomial map with an associated integer matrix $A_{\mathcal{C}}$ . This parametrization allows us to compute this integer matrix and the associated polytope $\mathfrak{P}_{\mathcal{C}}$ given by the convex hull of the columns of $A_{\mathcal{C}}$ .

Example 3.28.

Consider the model on 4 binary variables given by $X_{1}\mbox{$\perp\kern-5.5pt\perp$}X_{2}\mbox{$\perp\kern-5.5pt\perp$}X_{3,4}$ . Then, we have 16 cdf coordinates, and the equations that define $J_{\mathcal{C}}$ are:

	$\displaystyle q_{12}-q_{1}q_{2}$	$\displaystyle=0,$	$\displaystyle q_{13}-q_{1}q_{3}$	$\displaystyle=0,$	$\displaystyle q_{14}-q_{1}q_{4}$	$\displaystyle=0,$	$\displaystyle q_{23}-q_{2}q_{3}$	$\displaystyle=0,$	$\displaystyle q_{24}-q_{2}q_{4}$	$\displaystyle=0,$
	$\displaystyle q_{234}-q_{2}q_{34}$	$\displaystyle=0,$	$\displaystyle q_{123}-q_{1}q_{2}q_{3}$	$\displaystyle=0,$	$\displaystyle q_{124}-q_{1}q_{2}q_{4}$	$\displaystyle=0,$	$\displaystyle q_{1234}-q_{1}q_{2}q_{34}$	$\displaystyle=0$

From this description, we get the generators for $J_{\mathcal{C}}$ and consequently all coordinates with disconnected support. Hence, the parametrization described in Corollary 3.25 yields the following integer matrix:

A_{\mathcal{C}}=\hbox{}\vbox{\kern 0.86108pt\hbox{$\kern 0.0pt\kern 2.5pt\kern-5.0pt\left(\kern 0.0pt\kern-2.5pt\kern-6.66669pt\vbox{\kern-0.86108pt\vbox{\vbox{ \halign{\kern\arraycolsep\hfil\@arstrut$\kbcolstyle#$\hfil\kern\arraycolsep& \kern\arraycolsep\hfil$\@kbrowstyle#$\ifkbalignright\relax\else\hfil\fi\kern\arraycolsep&& \kern\arraycolsep\hfil$\@kbrowstyle#$\ifkbalignright\relax\else\hfil\fi\kern\arraycolsep\cr 5.0pt\hfil\@arstrut$\scriptstyle$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle q$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle q_{1}$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle q_{2}$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle q_{3}$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle q_{4}$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle q_{12}$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle q_{13}$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle q_{14}$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle q_{23}$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle q_{24}$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle q_{34}$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle q_{123}$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle q_{124}$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle q_{134}$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle q_{234}$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle q_{1234}\\t$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle 1$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle 1$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle 1$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle 1$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle 1$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle 1$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle 1$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle 1$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle 1$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle 1$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle 1$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle 1$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle 1$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle 1$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle 1$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle 1\\\theta^{(1)}$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle 0$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle 1$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle 0$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle 0$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle 0$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle 1$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle 1$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle 1$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle 0$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle 0$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle 0$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle 1$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle 1$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle 1$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle 0$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle 1\\\theta^{(2)}$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle 0$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle 0$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle 1$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle 0$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle 0$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle 1$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle 0$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle 0$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle 1$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle 1$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle 0$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle 1$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle 1$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle 0$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle 1$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle 1\\\theta^{(3)}$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle 0$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle 0$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle 0$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle 1$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle 0$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle 0$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle 1$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle 0$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle 1$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle 0$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle 0$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle 1$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle 0$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle 0$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle 0$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle 0\\\theta^{(4)}$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle 0$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle 0$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle 0$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle 0$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle 1$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle 0$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle 0$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle 1$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle 0$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle 1$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle 0$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle 0$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle 1$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle 0$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle 0$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle 0\\\theta^{(34)}$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle 0$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle 0$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle 0$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle 0$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle 0$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle 0$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle 0$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle 0$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle 0$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle 0$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle 1$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle 0$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle 0$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle 1$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle 1$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle 1$\hfil\kern 5.0pt\crcr}}}}\right)$}}

Furthermore, we know that the ideal $I_{\mathcal{C}}$ is going to be the kernel of the monomial map given by $A_{\mathcal{C}}$ , which, in this case we compute using Macaulay2 [3] and find that there are 46 degree 2 generators of the ideal. They include homogenized equations corresponding to the generators of $J_{\mathcal{C}}$

\begin{matrix}q_{1}q_{2}-qq_{12}\;&q_{1}q_{3}-qq_{13}\;&q_{1}q_{4}-qq_{14}\;&q_{2}q_{3}-qq_{23}\;&q_{2}q_{4}-qq_{24}\;&q_{1}q_{34}-qq_{134}\\ q_{2}q_{34}-qq_{234}\;&q_{1}q_{23}-qq_{123}\;&q_{2}q_{13}-qq_{123}\;&q_{3}q_{12}-qq_{123}\;&q_{1}q_{24}-qq_{124}\;&q_{2}q_{14}-qq_{124}\\ q_{4}q_{12}-qq_{124}&q_{1}q_{234}-qq_{1234}\;&q_{2}q_{134}-qq_{1234}\;&q_{12}q_{34}-qq_{1234}.\end{matrix}

However, we find more equations in the minimal generators of the homogeneous ideal, such as

\begin{matrix}&q_{134}q_{234}-q_{34}q_{1234}\;&q_{124}q_{234}-q_{24}q_{1234}\;&q_{123}q_{234}-q_{23}q_{1234}\;&q_{14}q_{234}-q_{4}q_{1234}\\ \end{matrix}

which are not necessary in the de-homogenized cdf coordinates since they are implied by the original equations (when $q=1$ ).

4. Graphical and simplicial marginal independence models

Two particularly interesting classes of marginal independence models are discussed in [1] and in [2]. They are graphical models and what we will refer to as simplicial models. We aim to contrast these different classes of models and show how they fit in the framework presented above.

4.1. Graphical models

Graphical models have been studied extensively because of their many applications and interesting structure that comes with a graphical representation of the independence relations. Different types of graphs are used to express different relationships between variables: directed acyclic graphs are been used to study causality in Bayesian networks, undirected graphs on the other hand, are used to study conditional independence models. Following the work of Drton and Richardson in [2], we use bidirected graphs as a framework to discuss marginal independence. Hence, for this section, $G=(V,E)$ will be a bidirected graph with vertex set $V=[n]$ and a set of edges $E$ .

Definition 4.1.

Let $G=(V,E)$ be a bidirected graph, for a vertex $v\in V$ we define the spouse of $v$ as the set of all vertices that share an edge with $v$ and $v$ itself, i.e., $\mathrm{Sp}(v):=\{w\in V:\;(w,v)\in E\}\cup\{v\}$ . For a subset $A\subseteq V$ , we define $\mathrm{Sp}(A)=\cup_{v\in A}\mathrm{Sp}(v)$ .

There are several different ways in which we can read different kinds of independent statements from a graph. The rules for reading a graph are called Markov properties, we are particularly interested in two sets of Markov properties: the global Markov property and the connected set Markov property.

Definition 4.2.

Let $G=\left([n],E\right)$ be a bidirected graph. A probability distribution is said to satisfy the global Markov property if for all $A,B,C\subseteq[n]$ pairwise disjoint such that $[n]\setminus A\cup B\cup C$ separates $A$ and $B$ , we have that

X_{A}\mbox{$\perp\kern-5.5pt\perp$}X_{B}|X_{C}.

Definition 4.3.

A probability distribution $P$ is said to satisfy the connected set Markov property associated to a bidirected graph $G$ if

X_{C}\mbox{$\perp\kern-5.5pt\perp$}X_{V\setminus\mathrm{Sp}(C)}

where $C$ is a connected subgraph of $G$ .

The global Markov property contains all of the marginal statements given by the connected set Markov property since for any connected set $C$ , we have that $C$ is separated from $V\setminus\mathrm{Sp}(C)$ by $\mathrm{Sp}(C)\setminus C=[n]\setminus(C\cup[n]\setminus\mathrm{Sp}(C))$ (in this case, the third set would be empty). Even though the global Markov property has conditional statements that are not given by the connected set property, we know that a probability distribution $P$ satisfies the global Markov property if and only if it satisfies the connected set property.

Additionally, since $v\in V$ is connected, the connected set Markov rules imply that if $u,v\in V$ do not have an edge between them, then $X_{v}\mbox{$\perp\kern-5.5pt\perp$}X_{u}$ . This is known as the pairwise Markov property, hence, both the global and connected set Markov properties imply all the statements in the pairwise Markov property.

Example 4.4.

Consider the chain with 4 vertices. The connected set Markov property yields $X_{1}\mbox{$\perp\kern-5.5pt\perp$}X_{3,4}$ and $X_{4}\mbox{$\perp\kern-5.5pt\perp$}X_{1,2}$ and the global Markov property gives us $X_{1}\mbox{$\perp\kern-5.5pt\perp$}X_{4}|X_{2,3}$ as well.

Definition 4.5 (Graphical model).

The marginal independence graphical model associated to $G$ , $\mathcal{M}_{G}$ , is given by the set of distributions $P=(p_{i_{1},\ldots,i_{n}})$ and discrete random variables $X_{1},\ldots,X_{n}$ that satisfy the marginal independence statements given by the connected set Markov property of $G$ .

Example 4.6.

Let $G$ be the graph on 3 vertices with the edge $(1,2)$ and binary random variables. Then, the marginal independence statements in this model are given by $X_{1,2}\mbox{$\perp\kern-5.5pt\perp$}X_{3}$ and the equations defining $J_{\mathcal{C}}$ are

\displaystyle q_{123}-q_{12}q_{3}=0,

\displaystyle q_{13}-q_{1}q_{3}=0,\;\;

\displaystyle q_{23}-q_{2}q_{3}=0.

These are the same defining equations for the model as the ones in Example 3.7. However, they represent slightly different models since here we have 3 random variables instead of 4.

In general, given a graph $G$ , the collection $\mathcal{C}$ of the marginal independence statements is defined by all the connected set Markov statements associated to the graph. This means that we can apply the results from Section 3 to this special class of models. Furthermore, the notion of connectedness relative to a collection of statements $\mathcal{C}$ agrees with the notion of connectedness of subgraphs of the corresponding graph $G$ .

Proposition 4.7.

Let $G$ be a bidirected graph and $\mathcal{C}$ the split closed order ideal defined by the connected set Markov statements. Then a set $D\subseteq[n]$ is disconnected with respect to $\mathcal{C}$ if and only if the corresponding subgraph with vertices in $D$ is disconnected.

Proof.

If $D$ is disconnected with respect to $\mathcal{C}$ , then there exists $\pi\in\mathcal{C}$ such that $D=|\pi|$ . Without loss of generality, we may assume that $\pi_{1}$ is connected with respect to $\mathcal{C}$ and that it only has two blocks, so, $\pi_{2}=V\setminus\mathrm{Sp}(\pi_{1})$ . Thus, the subgraph induced by $D$ is disconnected.

Conversely, if the subgraph of $D$ is disconnected, then we can certainly find a connected component $\pi$ of $D$ , and take $\pi|(V\setminus\mathrm{Sp}(\pi))$ so that $D$ is disconnected with respect to our collection $\mathcal{C}$ . ∎

Proposition 4.7 ensures that in this particular class of models, we can restate the results of Section 3 in terms of the graph theoretic sense of connectedness, this agrees with the results presented in [2] in the binary case and allows us to extend them to the general case.

Example 4.8.

Let $\mathcal{M}_{\mathcal{C}}$ be the model on 3 binary variables given by the marginal independence statements $X_{1}\mbox{$\perp\kern-5.5pt\perp$}X_{2}$ , $X_{2}\mbox{$\perp\kern-5.5pt\perp$}X_{3}$ , $X_{1}\mbox{$\perp\kern-5.5pt\perp$}X_{3}$ . The only graphical model on 3 binary variables that contains these marginal independence statements is given by the graph on 3 vertices that has no edges. However, because of the connected set Markov rule, this graphical model also has the statement $X_{1}\mbox{$\perp\kern-5.5pt\perp$}X_{2}\mbox{$\perp\kern-5.5pt\perp$}X_{3}$ , which cannot be inferred from $\mathcal{C}$ . We will show in the following section that this model is a simplicial complex model.

4.2. Simplicial complex models

This class of models was proposed in [1] and it concerns a very different class of models. We obtain these models from a simplicial complex instead of a graph, and the way we read statements from simplices is that all of the random variables indexed by a face are completely independent. Furthermore, this can be achieved by imposing rank 1 constraints on the probability tensor indexed by any particular face, which consequently makes the factorization in cdf coordinates and the homogeneous parametrization much simpler.

Definition 4.9 (Simplicial complex model).

Let $\Sigma\subseteq 2^{[n]}$ be a simplicial complex, then the model $\mathcal{M}_{\Sigma}$ is defined by the distributions $P=(p_{i_{1}\ldots i_{n}})$ and finite random variables such that the random variables $\{X_{i}:i\in\sigma\}$ are completely independent for any face $\sigma\in\Sigma$ .

Example 4.10.

Let $\Sigma$ be the simplicial complex on 3 binary variables defined by the 3 cycle, i.e., the marginal independence relations would be exactly the same as in Example 4.8. This shows that there are simplicial models that are not graphical models.

Remark.

We can discuss this class of models in terms of general marginal independence models by defining a collection of marginal independence statements generated by $\mathcal{C}=\left\{\sigma_{1}|\cdots|\sigma_{k}:\sigma\in\Sigma\right\}$ . Thus, we can apply all of the results from Section 3 to simplicial complex models, i.e. we get a toric model in cdf coordinates and a parametrization in terms of the connected sets. By construction, the disconnected sets of $\mathcal{C}$ are the faces of $\Sigma$ (except the vertices), and the connected sets are the non-faces and the vertices.

Next, we wish to compare graphical models and simplicial complex models. We need to be careful when comparing the two classes of models since the way we read the marginal independence statements from their corresponding structures is very different: in a simplicial complex, any face gives an marginal independence statement (in particular, edges represent marginal independence relations), whereas in graphs, any non-edge gives a marginal independence statement (and we get more from Markov rules). For this reason, when we are comparing them, it is useful to talk about the complementary graph $G^{\prime}$ of $G$ .

In general, we can associate a simplicial complex model to any graphical model such that the graphical model is a submodel of the simplicial complex model.

Proposition 4.11.

Let $G=(V,E)$ be a graph and $G^{\prime}$ be its complementary graph, $G^{\prime}=(V,E^{\prime})$ where $E^{\prime}$ is the usual complement of $E$ . Then, the associated simplicial complex, $\Sigma$ , given by

\Sigma(G)=\left\{F\subseteq V\;\big{|}\;e\nsubseteq F\;\forall e\in E\right\}

gives us that $\mathcal{M}_{G}\subseteq\mathcal{M}_{\Sigma}$ .

Proof.

We need to show the containment of the ideals $I_{\Sigma(G)}\subseteq I_{G}$ , which in turn requires us to show that any marginal independence statement from $\Sigma(G)$ can be obtained from $G$ as well. The edges (1 dimensional faces) of $\Sigma(G)$ are exactly the non-edges in $G$ and the faces of $\Sigma(G)$ are the completely disconnected subgraphs of $G$ . Hence, every marginal independence statement from $\Sigma(G)$ can be found in the model associated to $G$ . Thus proving that $\mathcal{M}_{G}$ is a submodel of $\mathcal{M}_{\Sigma(G)}$ . ∎

Remark.

Observe that the construction in Proposition 4.11 for a given $G$ implies that the $1$ -skeleton of $\Sigma(G)$ is the complimentary graph $G^{\prime}$ . The existence of a simplicial complex model that contains a given graphical model raises the question of when the simplicial complex models and graphical models are isomorphic. Consider the following example:

Example 4.12.

Let $\mathcal{M}_{G}$ be the model associated to the binary 4-cycle.

(a) 4-cycle graph

G

(b) Simplicial complex

\Sigma_{G}

Figure 2. Graph and simplicial complex in Example 4.12

This gives us the marginal independence relations $X_{1}\mbox{$\perp\kern-5.5pt\perp$}X_{3}$ and $X_{2}\mbox{$\perp\kern-5.5pt\perp$}X_{4}$ . As we can see in Figure 2, the simplicial complex $\Sigma_{G}$ associated to this graph gives us the same marginal independence statements as $G$ , i.e. $\mathcal{M}_{G}=\mathcal{M}_{\Sigma(G)}$ .

Example 4.13.

Let $\mathcal{M}_{G}$ be the model associated to the binary 4-chain.

(a) 4-chain graph

G

(b) Simplicial complex

\Sigma_{G}

Figure 3. Graph and simplicial complex in Example 4.13.

This graph gives us the following marginal independence statements: $X_{1}\mbox{$\perp\kern-5.5pt\perp$}X_{3.4}$ , $X_{4}\mbox{$\perp\kern-5.5pt\perp$}X_{1,2}$ . Its associated simplicial complex, $\Sigma_{G}$ , is again a 4 chain that implies the following: $X_{1}\mbox{$\perp\kern-5.5pt\perp$}X_{3}$ , $X_{1}\mbox{$\perp\kern-5.5pt\perp$}X_{4}$ and $X_{2}\mbox{$\perp\kern-5.5pt\perp$}X_{4}$ . We cannot derive the marginal independence statements given by $G$ from this . Thus, in this case the models are not isomorphic (even though the 1-skeleton of $\Sigma(G)$ and $G$ are isomorphic as graphs).

For a graphical model $\mathcal{M}_{G}$ to be isomorphic to its simplicial complex counterpart, we would need that the marginal independence statements on $G$ are only statements of complete marginal independence of individual variables, since those are the only type of marginal independence statements that we can obtain from the simplicial complex model.

Proposition 4.14.

For a graph $G$ , $\mathcal{M}_{G}=\mathcal{M}_{\Sigma(G)}$ if the complimentary graph $G^{\prime}$ of $G$ is a disjoint union of complete graphs.

Proof.

Let $G^{\prime}=G_{1}^{\prime}\cup\ldots\cup G_{k}^{\prime}$ where each $G_{i}^{\prime}$ , $i\in[k]$ , is a complete graph and they are disjoint. Then, observe that each $G_{i}^{\prime}$ corresponds to a totally disconnected subgraph of $G$ whose vertices are connected all vertices not in $G_{i}^{\prime}$ . Hence, each $G_{i}^{\prime}$ yields a complete marginal independence statement on the random variables indexed by the vertices of $G_{i}^{\prime}$ . Furthermore, by construction, these are all the statements that we are going to get from the graph $G$ . We know that those are precisely the statements that we get from $\Sigma(G)$ , thus showing that the models are isomorphic. ∎

Remark.

We can restate Proposition 4.14 without using the complimentary graph since that condition on $G^{\prime}$ is equivalent to $G$ being a multipartite graph. Additionally, we can see from this proposition that the number of nontrivial isomorphic models is going to be the number of partitions of $\#V$ .

Example 4.15.

Let $X_{1},\ldots,X_{4}$ be finite random variables, and consider the model given by the marginal independence statements $X_{1}\mbox{$\perp\kern-5.5pt\perp$}X_{2,3,4}$ and $X_{2}\mbox{$\perp\kern-5.5pt\perp$}X_{3}$ . This model is not simplicial and not graphical, but it still is toric.

4.3. Counting models and computational results.

We counted all models on 3 and 4 random variables and counted up to symmetry of relabeling the random variables and obtained the following:

Table 1. Number of models on

3

and

4

binary random variables

	Total	Up to symmetry	Total	Up to symmetry
	$n=3$	$n=3$	$n=4$	$n=4$
Graphical	8	4	64	11
Simplicial	9	5	114	20
Graphical and simplicial	5	3	18	5
General	12	6	496	53

For the simplicial and graphical models, we used data from OEIS [4, 5, 6]. For the general case, we do not yet have an efficient way of counting split closed order ideals, so we used exhaustive computation checking split closures of order ideals.

Additionally, we computed the homogeneous ideals for all models up to labeling symmetry for $n=4$ and found the highest degree of the generators, degree and dimension of the projective variety that comes from $I_{\mathcal{C}}$ and whether or not the models are simplicial or graphical.

Table 2. Computations for all models on 4 random variables.

Split order	Max degree	Degree	Dimension	Graphical	Simplicial
ideal	gens
$\emptyset$		1	15	x	x
$1\|2$	2	2	14	x	x
$1\|2,1\|3$	2	3	13		x
$1\|2,3\|4$	2	4	13	x	x
$1\|23$	2	4	12	x
$1\|2,1\|3,1\|4$	2	4	12		x
$1\|2,1\|3,2\|3$	3	5	12	x	x
$1\|2,1\|3,2\|4$	2	5	12		x
$1\|2,1\|3,1\|4,2\|3$	3	7	11		x
$1\|2,1\|3,2\|4,3\|4$	2	6	11		x
$1\|2,1\|34$	2	5	11
$1\|2,13\|4$	2	7	11
$1\|2\|3$	2	6	11		x
$1\|2,1\|3,1\|4,2\|3,2\|4$	3	9	10		x
$1\|2,1\|3,2\|34$	3	9	10
$1\|2,1\|3,23\|4$	2	9	10
$1\|2,1\|3\|4$	2	8	10		x
$1\|23,1\|24$	2	6	10
$1\|23,14\|2$	2	10	10	x
$1\|2,1\|3,1\|4,2\|3,2\|4,3\|4$	3	12	9		x
$1\|2,1\|3,1\|4,2\|34$	3	12	9
$1\|2,1\|3,2\|3\|4$	3	11	9		x
$1\|2,13\|4,23\|4$	3	11	9
$1\|2,13\|4,24\|3$	2	14	9
$1\|23,1\|24,1\|34$	2	7	9
$1\|23,23\|4$	2	10	9
$1\|2\|3,1\|24$	2	10	9
$1\|2,1\|3,1\|4,2\|3\|4$	3	16	8		x
$1\|2,1\|3,123\|4$	3	15	8
$1\|2,1\|3,123\|4,2\|3$	3	17	8
$1\|2,123\|4$	3	13	8	x
$1\|2,1\|3\|4,23\|4$	3	14	8
$1\|2,1\|34,2\|34$	3	14	8
$1\|23,14\|2,14\|3$	2	16	8
$1\|234$	2	8	8	x
$1\|2\|3,1\|24,1\|34$	2	12	8
$1\|2\|3,1\|2\|4$	2	12	8		x
$1\|2,1\|3,123\|4,2\|3\|4$	3	21	7
$1\|2,1\|3\|4,2\|3\|4$	3	18	7		x
$1\|2,1\|3\|4,123\|4$	3	17	7
$1\|23,14\|2,14\|3,23\|4$	2	19	7
$1\|2\|3,1\|234$	2	14	7
$1\|2\|3,1\|24,24\|3$	3	17	7
$1\|2\|3,1\|2\|4,1\|34$	2	15	7
$1\|2,1\|3\|4,123\|4,2\|3\|4$	3	23	6
$12\|34$	2	20	6	x
$1\|2\|3,1\|2\|4,1\|34,2\|34$	3	19	6
$1\|2\|3,1\|2\|4,1\|234$	2	18	6
$1\|2\|3,1\|2\|4,1\|3\|4$	3	20	6		x
$1\|2\|34$	2	20	5	x
$1\|2\|3,1\|2\|4,1\|3\|4,2\|3\|4$	3	23	5		x
$1\|2\|3,1\|2\|4,1\|234,1\|3\|4$	3	25	5
$1\|2\|3\|4$	2	24	4	x	x

References

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