Finite Factored Sets: Polynomials and Probability

In this post, given a finite factored set , we will show how to associate each with a characteristic polynomial, . We will discuss how to factor these characteristic polynomials, and use these characteristic polynomials to build up to the fundamental theorem of finite factored sets, which associates conditional orthogonality with conditional independence in probability distributions.

5.1. Characteristic Polynomials

Definition 28. Given a finite factored set , let denote the ring of polynomials with coefficients in and variables in .

Definition 29. Given a finite factored set , a , and an , we write for the evaluation of at , computed by replacing each with .

Definition 30. Given a finite factored set , and a polynomial , denotes the set of all variables that appear in . is called the support of .

Definition 31. Given a finite factored set and an , let be given by . is called the characteristic polynomial of (in ).

We will be building up to an understanding of how to factor into irreducibles. For that, we will first need to give some basic notation for manipulating polynomials in .

Definition 32. Given a finite factored set , an , and a , let be given by .

Definition 33. Given a finite factored set , an , and a , let be given by .

Definition 34. Given a finite factored set , an , and a , let be given by .

Proposition 26. Let be a finite factored set, and let . Then .

Proof. We start by showing that for all , .

Let be arbitrary. By Proposition 3, if , there must be some such that . Then, note that . If were also in , then would be in both and , contradicting the fact that these two sets are disjoint. Therefore .

Thus has exactly one element for each element of , so we have that .

Proposition 27. Let be a finite factored set, and let be subsets of . Let be disjoint subsets of . Let , and let . Then .

Proof. For , let . We will start by showing that , given by , is a well-defined function and a bijection.

First, observe that it follows immediately from the definition that for all , if we have that , , and . Combining these, we get that .

For all , there exists some such that , and some such that , and this gives us that . Thus, is well-defined.

To see that is surjective, observe that for all , there exists an such that , and there exist and such that , and we have .

To see that is injective, observe that for , for all , . Further, and are disjoint. Thus, for all and , .

This means that for all and , if , then and . However, every monomial in or is just equal to the product of all variables in its support. Thus and . Thus is injective, and thus a bijection between and .

Now, we have that

Proposition 28. Let be a finite factored set, and let be a nonempty subset of . If divides , then , for some and .

Proof. Let be a finite factored set, and let be a nonempty subset of . Let satisfy . We thus must have .

If there were some , then the degree of in would be at least 2, contradicting the definition of and Corollary 1. Thus, .

There can be no combining like terms, then, in the product . The monomial terms in are in bijective correspondence to the pairs of monomial terms in and monomial terms in .

In particular, this means that since all the coefficients in are equal to , all the coefficients in must be equal to some , and all of the coefficients in must be equal to .

Further, for all , if is nonempty, must be empty, since otherwise would contain a term with two factors in , which clearly never happens according to the definition of .

Since is nonempty, for each there must be some . Thus at least one of and must be nonempty, so exactly one of and must be nonempty.

Let be the set of all such that is nonempty.

For every , every term of has exactly one factor in . Thus, every term in has exactly one factor in . These cover all variables in the support of , so each term in must have total degree .

For each , divides a term in . Since has no common support with , must also divide a term in . Thus must be a term in . Conversely, every term in divides a term in , and thus must be in . Thus every term in is of the form for some . Thus .

5.2. Factoring Characteristic Polynomials

We will now show how to factor characteristic polynomials into irreducibles.

Definition 35. Given a finite factored set , and a nonempty subset , let denote the set of all such that:

  1. is nonempty,

  2. , and

  3. there is no nonempty strict subset such that .

Proposition 29. Let be a finite factored set, and let be a nonempty subset of . Then .

Proof. Let be a finite factored set, and let be a nonempty subset of . It suffices to show that the sets in are pairwise disjoint and cover .

We start by showing that the set of all satisfying is closed under intersection. Indeed, if and , then .

Next, observe that . Thus, for all , we can consider . Since is an intersection of a finite nonempty collection of sets satisfying , we have that . Further, , so is nonempty.

Assume for the purpose of contradiction that there is some nonempty strict subset such that . If , then we have a contradiction by the definition of . If , then note that , so , and is a nonempty strict subset of that contains , contradicting the definition of .

Thus for all , and since , this means that the sets in cover .

Next, we need to show that the sets in are pairwise disjoint. Let be arbitrary distinct elements. We have that , and is a subset of and , and thus a strict subset of at least one of them. Thus is empty.

Thus .

The following two propositions constitute a factorization of into irreducibles.

Proposition 30. Let be a finite factored set, and let be a nonempty subset of . Then .

Proof. Let be a finite factored set, and let be a nonempty subset of . Let , and let . For , let .

We will show by induction on that for all .

If , the result is trivial, as .

For , observe that and are disjoint, and that , thus by Proposition 27, we have . Thus, by induction, we get .

In the case where , this gives that .

Proposition 31. Let be a finite factored set, and let be a nonempty subset of . Then is irreducible for all .

Proof. Let be a finite factored set, let be a nonempty subset of , and let .

Assume for the purpose of contradiction that , and that both and have nonempty support. By Proposition 28, we have that , for some , and .

We will first need to show that and are nonempty and disjoint. They must be nonempty, because and have nonempty support. Assume for the purpose of contradiction that . Let be an element of , and note that for , we have . Thus must be degree at least 2 in , which contradicts the fact that every variable clearly has degree at most 1 in .

Next, we need to show that . We already know that

Let be an element of . Given an arbitrary , we have that if and only if if and only if for some if and only if .

We now have that and are disjoint and that . Thus, by Proposition 27, we have that . Thus , so .

Let be arbitrary, and let . Note that , so there is some such that . Thus for all . However, we also have that for all , so . Since , , so . Since and were arbitrary elements of , we have that . Since is a nonempty strict subset of , this contradicts the fact that .

Thus, is irreducible for all .

5.3. Characteristic Polynomials and Orthogonality

We can now give an alternate characterization of conditional orthogonality in terms of divisibility of characteristic polynomials.

Lemma 3. Let be a finite factored set, and let be partitions of . The following are equivalent.

  1. .

  2. divides for all , , and .

  3. for all , , and .

Proof. Clearly condition 3 implies condition 2. We will first show that condition 1 implies condition 3, and then show that condition 2 implies condition 1.

Let , and let satisfy . Consider an arbitrary , , and . We want to show that .

Let . Clearly . We thus have that , so . We also have that , so . These two together give that .

Since , we have that . Thus, by Proposition 27, we have that . Similarly, since , we have that .

Since , and , we have . We also have that

Thus .

By Proposition 27, this gives that .

Finally, since , we have that .

Thus, and are both equal to .

Thus, condition 1 implies condition 3. It remains to show that condition 2 implies condition 1.

Fix , and , and let divide for all , , and . Assume for the purpose of contradiction that it is not the case that . Thus, there exists some such that . Let and satisfy .

Let be such that and , and let . Thus, is an irreducible factor of .

Either divides for all or divides for all , since otherwise there would exist an and a such that divides neither nor , but does divide their product, contradicting the fact that is irreducible, and thus prime.

Assume without loss of generality that divides for all . Fix an . Let us first restrict attention to the case where is nonempty.

Let . By Proposition 28, and for some and . We will show that , and .

Let be an element of . Then for all , if and only if if and only if if and only if . Thus .

For all , we have and , so , so . Similarly, for all , , so , so . Thus .

Since and both have all coefficients equal to , we have . Thus,

Similarly, since all the coefficients of are and all the coefficients of are , all the coefficients of are , so . Thus, .

We thus have that .

In the case where is empty, we also have , since both sides are .

By Proposition 27, . Thus, , so .

Since for all , we have that . However, this contradicts the fact that , and .

Thus, condition 2 implies condition 1.

5.4. Probability Distributions on Finite Factored Sets

The primary purpose of all this discussion of characteristic polynomials has been to build up to thinking about the relationship between orthogonality and probabilistic independence. We will now discuss probability distributions on finite factored sets.

Recall the definition of a probability distribution.

Definition 36. Given a finite set , a probability distribution on is a function such that

  1. for all ,

  2. ,

  3. , and

  4. whenever satisfy .

A probability distribution on a finite factored set is a probability distribution on its underlying set that also satisfies another condition, which represents the probability distribution coming from a product of distributions on the underlying factors.

Definition 37. Given a finite factored set , a probability distribution on is a probability distribution on such that for all , we have .

Proposition 32. Given a finite factored set , a probability distribution on is a probability distribution on if and only if for all .

Proof. If for all , in particular this means that for all .

Conversely, if for all , then for all .

5.5. The Fundamental Theorem of Finite Factored Sets

We are now ready to state and prove the fundatmental theorem of finite factored sets.

Theorem 3. Let be a finite factored set, and let be partitions of . Then if and only if for all probability distributions on and all , , and , we have .

Proof. We already have by Lemma 3 that if , then for all , , and , . Thus for any probability distribution on , we have

Conversely, assume that for all probability distributions on , and all , , and , we have .

If is empty, then is the unique partition of , and we have . Thus, we can restrict our attention to the case where is nonempty.

Fix an arbitrary , , and . Let . We will first show that for all .

Given an arbitrary , we can define by , and we will show that is a distribution on .

is well-defined because is a nonempty sum of products of positive real numbers, and thus positive. Further, since is a sum of products of positive real numbers, for all . Since , we also have . Clearly . Finally, for all with , we have

Therefore is a distribution on . We still need to show that is a distribution on .

Observe that for all and , since , we have that , and since , we have that . Thus, we have that

Thus, for all ,

Thus is a distribution on .

It follows that P. We therefore have that

Thus, is a polynomial that is zero on an open subset of inputs, so is the zero polynomial. Thus