Probability and Statistics - Geomathematics: Theoretical Foundations, Applications and Future Developments - page 49

where O O /(1 + O ), and W B \ C is the

“weight of evidence” for occurrence of the event D given B and C . Suppose that
the probability p refers to occurrence of a mineral deposit D under a small area on
the map (circular or square unit area). Suppose further that B represents a binary
indicator pattern, and that C is the study area within which D and B have been
determined. Under the assumption of equipossibility (or equiprobability), the prior
probability is equal to total number of deposits in the study area divided by total
number of unit cells in the study area. Theoretically, C is selected from an infinitely
large universe (parent population) with constant probabilities for the relationship
between D and B . In practical applications, only one study area is selected per
problem and C can be deleted from the preceding equation. Then Bayes' theorem
can be written in the form:
ln OD B

¼ p /(1 p ) are the odds corresponding to p ¼

¼

ln OD B c

W B þ W B þ

ln OD

ðÞ; ¼

ln OD

ðÞ

for presence or absence of B , respectively. If the area of the unit cell underlain by
B is small in comparison with the total area underlain by B , the odds O are
approximately equal to the probability p . The weights satisfy:

pB c

ln pB

ð \ D Þ ð \ D Þ

W B ¼ W B ¼

Þ ; D c pB c D c pB ð \ ð \ Þ

As an example of this type of application of Bayes' theorem, suppose that a
study area C , which is a million times as large as the unit cell, contains ten deposits;
20 % of C is underlain by rock type B , which contains eight deposits. The prior
probability p ( D ) then is equal to 0.000 01; the posterior probability for a unit
cell on B is p ( D | B )

¼ 0.000 04, and the posterior probability for a unit cell not on

B is p ( D | B c ) 10 6 . The weights of evidence are W B + ¼ 2.5 ¼ 0.982 and

W B ¼ 1.056, respectively. In this example, the two posterior probabilities can

be calculated without use of Bayes' theorem. However, the weights themselves
provide useful information as will be seen in Chap. 5 .

2.2.2 Probability Generating Functions

A random variable X is either discrete or continuous. Some geological frequency
distributions are best modeled as compound frequency distributions that require the
use of more advanced methods of mathematical statistics including use of proba-
bility generating functions. Our treatment of this subject is kept brief for reasons of
space. The reader is referred to textbooks of mathematical statistics (e.g., Feller
1968 ) for further explanations.