The fundamental theorem of finite factored sets tells us that (conditional) orthogonality data can be inferred from probabilistic data. Thus, if we can infer temporal data from orthogonality data, we will be able to combine these to infer temporal data purely from probabilistic data. In this section, we will discuss the problem of inferring temporal data from orthogonality data, mostly by going through a couple of examples.

6.1. Factored Set Models

We'll begin with a sample space, $\Omega$.

Naively, one might except that temporal inference in this paradigm involves inferring a factorization of $\Omega$. What we'll actually be doing, however, is inferring a factored set model of $\Omega$. This will allow for the possibility that some situations are distinct without being distinct in $\Omega$—that there can be latent structure not represented in $\Omega$.

Definition 38 (model). Given a set $\Omega$, a model of $\Omega$ is a pair $M=(F,f)$, where $F$ is a finite factored set and $f:\text{set}\left(F\right)\to \Omega$ is a function from the set of $F$ to $\Omega$.

Definition 39. Let $S$ and $\Omega$ be sets, and let $f:S\to \Omega$ be a function from $S$ to $\Omega$. 

Given a $\omega \in \Omega$, we let $f^{-1}\left(\omega \right)=\{s\in S\mid f(s)=\omega \}$. 

Given an $E\subseteq \Omega$, we let $f^{-1}\left(E\right)=\{s\in S\mid f(s)\in E\}$. 

Given an $X\in \text{Part}\left(\Omega \right)$, we let $f^{-1}\left(X\right)\in \text{Part}\left(S\right)$ be given by $f^{-1}\left(X\right)=\left\{f^{-1}\right(x)|x\in X,f^{-1}(x)\ne \{\left\}\right\}$.

Definition 40 (orthogonality database). Given a set $\Omega$, an orthogonality database on $\Omega$ is a pair $D=(O,N)$, where $O$ and $N$ are both subsets of $\text{Part}\left(\Omega \right)\times \text{Part}\left(\Omega \right)\times \text{Part}\left(\Omega \right)$.

Definition 41. Given an orthogonality database $D=(O,N)$ on a set $\Omega$, and partitions $X,Y,Z\in \text{Part}\left(\Omega \right)$, we write $X{\perp }_{D}Y|Z$ if $(X,Y,Z)\in O$, and we write $X{\rightleftharpoons }_{D}Y|Z$ if $(X,Y,Z)\in N$. 

Definition 42. Given a set $\Omega$, a model $M=(F,f)$ of $\Omega$, and an orthogonality database $D=(O,N)$ on $\Omega$, we say $M$ models $D$ if for all $X,Y,Z\in \text{Part}\left(\Omega \right)$,

1. if $X{\perp }_{D}Y|Z$ then $f^{-1}\left(X\right){\perp }^F{f}^{-1}\left(Y\right)|f^{-1}\left(Z\right)$, and
2. if $X{\rightleftharpoons }_{D}Y|Z$ then $\neg \left(f^{-1}\left(X\right){\perp }^F{f}^{-1}\left(Y\right)|f^{-1}\left(Z\right)\right)$.

Definition 43. An orthogonality database $D$ on a set $\Omega$ is called consistent if there exists a model $M$ of $\Omega$ such that $M$ models $D$. 

Definition 44. An orthogonality database $D$ on a set $\Omega$ is called complete if for all $X,Y,Z\in \text{Part}\left(\Omega \right)$, either $X{\perp }_{D}Y|Z$ or $X{\rightleftharpoons }_{D}Y|Z$.

Definition 45. Given a set $\Omega$, an orthogonality database $D$ on $\Omega$, and $X,Y\in \text{Part}\left(\Omega \right)$, we say $X{<}_{D}Y$ if for all models $(F,f)$ of $\Omega$ that model $D$, we have $f^{-1}\left(X\right){<}^F{f}^{-1}\left(Y\right)$. 

6.2. Examples

Example 1. Let $\Omega =\{00,01,10,11\}$ be the set of all bit strings of length $2$. For $i\in \{0,1\}$, let $x_i=\{i0,i1\}$ be the event that the first bit is $i$, and let $y_i=\{0i,1i\}$ be the event that the second bit is i. Let $X=\{x_0,x_1\}$ and let $Y=\{y_0,y_1\}$. 

Let $v_0=\{00,11\}$ be the event that the two bits are equal, let $v_1=\{01,10\}$ be the event that the two bits are unequal, and let $V=\{v_0,v_1\}$. 

Let $D=(O,N)$, where $O=\left\{\right(X,V,\left\{\Omega \right\}\left)\right\}$ and $N=\left\{\right(V,V,\left\{\Omega \right\}\left)\right\}$.

Proposition 33.  In Example 1, $D$ is consistent.

Proof. First observe that $F=(\Omega ,\{X,V\left\}\right)$ is a factored set, and so $M=(F,f)$ is a model of $\Omega$, where $f$ is the identity on $\Omega$. It suffices to show that $M$ models $D$.

Indeed $h^F\left(X\right)=\left\{X\right\}$, and $h^F\left(V\right)=\left\{V\right\}$, so $X{\perp }^{F}V$, so $f^{-1}\left(X\right){\perp }^F{f}^{-1}\left(V\right)|f^{-1}\left(\right\{\Omega \left\}\right)$. 

Further, it is not the case that $V{\perp }^{F}V$, since $V\ne {\text{Ind}}_{\Omega }$. Thus it is not the case that $f^{-1}\left(V\right){\perp }^F{f}^{-1}\left(V\right)|f^{-1}\left(\right\{\Omega \left\}\right)$.

Thus $M$ satisfies all of the conditions to model $D$, so $D$ is consistent. $\square$

Proposition 34. In Example 1, $X{<}_{D}Y$.

Proof. Let $(F,f)$ be any model of $\Omega$ that models $D$. Let $F=(S,B)$. For any $A\in \text{Part}\left(\Omega \right)$, let $H_A=h^F\left(f^{-1}\right(A\left)\right)$. Our goal is to show that $H_X$ is a strict subset of $H_Y$.

First observe that $X{\le }_{\Omega }Y{\vee }_{\Omega }V$, so for any $s,t\in S$, if $s{\sim }_{f^{-1}\left(Y\right)}t$ and $s{\sim }_{f^{-1}\left(V\right)}t$, then $f\left(s\right){\sim }_{Y}f\left(t\right)$ and $f\left(s\right){\sim }_{V}f\left(t\right)$, so $f\left(s\right){\sim }_{X}f\left(t\right)$, so $s{\sim }_{f^{-1}\left(X\right)}t$. Thus $f^{-1}\left(X\right){\le }_S{f}^{-1}\left(Y\right){\vee }_S{f}^{-1}\left(V\right)$.

It follows that $H_X\subseteq h^F\left(f^{-1}\right(Y\left){\vee }_S{f}^{-1}\right(V\left)\right)=H_Y\cap H_V$. However, since $X{\perp }_{D}V|\left\{\Omega \right\}$, we have that $H_X\cap H_V=\left\{\right\}$, so $H_X\subseteq H_Y$.

By swapping $X$ and $V$ in the argument above, we also get that $H_V\subseteq H_Y$. Since $V{\rightleftharpoons }_{D}V|\left\{\Omega \right\}$, we have that $H_V\ne \left\{\right\}$. Thus $H_V$ contains some element $b$. Observe that $b\notin H_X$, but $b\in H_Y$. Thus $H_X$ is a strict subset of $H_Y$, so $f^{-1}\left(X\right){<}^F{f}^{-1}\left(Y\right)$.

Since $(F,f)$ was an arbitrary model of $\Omega$ that models $D$, this implies that $X{<}_{D}Y$. $\square$

Example 2. Let $\Omega =\{000,001,010,011,100,101,110,111\}$ be the set of all bit strings of length $3$. For $i\in \{0,1\}$, let $x_i=\{i00,i01,i10,i11\}$ be the event that the first bit is $i$, let $y_i=\{0i0,0i1,1i0,1i1\}$ be the event that the second bit is $i$, and let $z_i=\{00i,01i,10i,11i\}$ be the event that the third bit is $i$. Let $X=\{x_0,x_{}$