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, .

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

Definition 38 (model). Given a set , a model of  is a pair , where  is a finite factored set and  is a function from the set of  to .

Definition 39. Let  and  be sets, and let  be a function from  to 

Given a , we let 

Given an , we let 

Given an , we let  be given by .

Definition 40 (orthogonality database). Given a set , an orthogonality database on  is a pair , where  and  are both subsets of .

Definition 41. Given an orthogonality database  on a set , and partitions , we write  if , and we write  if 

Definition 42. Given a set , a model  of , and an orthogonality database  on , we say  models  if for all ,

  1. if  then , and
  2. if  then .

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

Definition 44. An orthogonality database  on a set  is called complete if for all , either  or .

Definition 45. Given a set , an orthogonality database  on , and , we say  if for all models  of  that model , we have 

 

6.2. Examples

Example 1. Let  be the set of all bit strings of length . For , let  be the event that the first bit is , and let  be the event that the second bit is i. Let  and let 

Let  be the event that the two bits are equal, let  be the event that the two bits are unequal, and let 

Let , where  and .

Proposition 33.  In Example 1,  is consistent.

Proof. First observe that  is a factored set, and so  is a model of , where  is the identity on . It suffices to show that  models .

Indeed , and , so , so 

Further, it is not the case that , since . Thus it is not the case that .

Thus  satisfies all of the conditions to model , so  is consistent. 

Proposition 34. In Example 1, .

Proof. Let  be any model of  that models . Let . For any , let . Our goal is to show that  is a strict subset of .

First observe that , so for any , if  and , then  and , so , so . Thus .

It follows that . However, since , we have that , so .

By swapping  and  in the argument above, we also get that . Since , we have that . Thus  contains some element . Observe that , but . Thus  is a strict subset of , so .

Since  was an arbitrary model of  that models , this implies that 

Example 2. Let  be the set of all bit strings of length . For , let  be the event that the first bit is , let  be the event that the second bit is , and let  be the event that the third bit is . Let , let