Patrick's Theorem

Preliminaries

     Patrick's theorem is a theorem on a posteriori probabilities that will allow for multiple

classes active at the same time. As defined in Bayes Theorem, the category space Γ consists

of M mutually exclusive and exhaustive categories γ₁*, γ₂*, ....γ_M*. In addition, define a

feature space Χ with x є Χ. There exists category-feature relationships p(x|γ_i* ), i = 1 to M

as defined in Bayes Framework created by Patrick but not present in Bayes Theorem.  

     A departure from Bayes now begins with the definition of a class space Ώ:

                                             Ώ: ώ_1, ώ_{2, ....,} ώ_M1 .

The category space Γ and class space  Ώ must be considered separate, although there will be

components from the respective spaces which are the same. For example, the two spaces will

have certain features in common. As another example, a class in Ώ is a logical union of all

categories in Γ which "contain" the class. Even more important are certain properties of categories

in Γ which exist for Patrick's theorem but not for Bayes theorem. For Patrick's theorem, categories

γ₁*, γ₂*, ....γ_M*  are mutually exclusive but there can exist relationships among the

category conditional probability density functions. These relationships constitute knowledge in terms

of significant and insignificant features of each category. Significance or insignificance is

defied in terms of probability or relative frequency  This relative frequency itself becomes part   

of the knowledge base. It is used to provide knowledge which relates some of the category-

conditional probability density functions. The models relating certain category conditional probability

density functions may be considered in another space, say Μ, distinct from the class space Ώ,

category space Γ, feature space Ғ, or cross-product space Ώ x Ғ and  Γ x Ғ. The space Μ,

called the Model Space, contains the "glue" relating classes, only classes, and complex classes.

This "glue can be functional relationships, logical relationships, logical relationships involving

probability weights, and so on.

     Likelihood (Patrick)

                                                     p (x|ώ_i^*)                                                  p(Ώ_ξ*)

                p (x|ώ_i)  =  p (x|ώ_i^*)      --------                +          Σ p(x | Ώ_ξ*)   -------

                                                     p (x|ώ_i)                        ώ_i є  Ώ_ξ*            p (x|ώ_i)

     Theorem on A Posteriori Class Probabilities (Patrick)

Two Class Example

     It is helpful to consider a two class problem where the two classes are denoted ώ₁  and  ώ₂ .

But these classes are "intermediate concepts," not themselves in the decision space. What are in

the decision space are three categories defined:

                                                  ώ₁^*  =  (ώ_1,ώ_{2) :}        means class ώ₁   but not class ώ₂

                                                                   ώ₂^*  =  (ώ_2,ώ_{1) :}        means class ώ₂   but not class ώ₁

                                                                   Ώ₁₂^*  ₌   (ώ₁,ώ₂):       means class ώ₁ and class ώ₂.

The concepts ώ₁^*  and  ώ₂^* are called only classes, while the concept  Ώ₁₂^*  is called a complex

class. The superscript ^* denotes a category.

     Categories ώ₁^* , ώ₂^* , and  Ώ₁₂^* are mutually exclusive as in Bayes Theorem. We will show

how to learn the concept  Ώ₁₂^*   by discovery without a teacher. During the learning process

there are statistical dependencies among the categories, a property not present in Bayes

Theorem.