Independence

Definition

Two facts are independent if your knowledge of one does not influence your probability of the other.

List only those facts that are independent of every other fact. When facts are listed (as separate from one another), they a priori are considered to be distinguishable. Therefore, the act of listing facts separately is the first step toward establishing independence. Often this is sufficient, but not always. If you believe two or more facts might be mutually dependent, group them (thereby creating a single compound fact that is independent of the others.¹ Dependent facts invalidate the results of FactLogic.

1. Intuition

The notion of dependence is partially intuitive. Dependence is a relationship between two facts, and we often intuitively estimate the extent of the relationship.

2. Mathematical Test

From a mathematical point of view, Two facts are independent if your knowledge of one does not influence your subjective probability of the other.²

3. Extent of Dependence

The extent of dependence of two facts can vary from total dependence, to some dependence, and then to independence. A fact has a number of characteristics, and each characteristic has a value. For example, a characteristic is race, and Caucasian is a value of the characteristic, race. If two facts that share several characteristics also share values, the two facts are similar. Actually, a way to look at dependence between two facts is to decide whether they could be considered to be one fact or two facts. Consider the range of dependence with some examples of evidence from California v. Simpson:

3.1 Total Dependence

Two facts should be considered to be one fact if no values of significant characteristics differ; the two facts are clearly dependent:

• Trail of Blood. Several drops of Nicole's blood in a trail of blood differed only by being a few inches apart.

• Thatch of Hair. Several strands of Nicole's hair were found on the same glove at the murder scene.

To consider either the drops of blood in a trail or the hairs on a glove to be separate facts would be excessively incriminating. Two facts that are thought to be dependent should be grouped as a single compound fact and judged by a single probability.

3.2 Some Dependence

Two facts may or may not be considered to be two facts if one or more values differ; the two may or may not be dependent, depending upon whether these characteristics or their values are significant:

• Blood in Bronco. Simpson's blood was smeared on the passenger's side wall, on the instrument panel, on the center console, and on the driver's side wall. Even so, the values of the characteristic, location of Simpson's blood, differed only by a matter of inches (but over a wider area than was the trail of blood, above); they were dependent and considering them to be separate facts would excessively incriminate the defendant.

• Repeated Threats. Simpson's alleged, repeated threats against Nicole share many values except time, provocation, etc. Repeated threats are usually dependent facts.

• Clothing Fibers. The bloody glove at the murder scene contained fibers from Ron Goldman's jeans and from his shirt; they belonged to the same person but came from different facts of clothing.

Two facts that are thought to exhibit some dependence should be grouped as a single compound fact that is independent of the others and judged by a single probability.

3.3 No Dependence (i.e., Independence)

Two facts are considered to be two facts if all values differ; the two are clearly independent. However, it is not necessary that all or most values differ if they are significant:

• Threats and Gloves. Simpson's alleged threats to kill Nicole and his blood on either glove share no values (other than both belonged to the defendant).

• Right-handed Glove and Left-handed Glove. The two gloves were found many blocks apart. The single differing value (i.e., location) is significant enough to render the gloves to be independent facts. This example shows the importance of using reason, rather than simply counting similar values.

4. A Court Ruling on Dependence

The California Supreme Court (The People v. Collins, 1968) reversed a conviction for robbery because the prosecution used dependent characteristics associated with the defendant and unsubstantiated frequencies of occurrence. Both "facts" were intended to show that there was a very small probability that the defendant was not the robber. The following table lists the values of the characteristics and their frequencies of occurrence:

Table 1. Characteristics and frequencies of occurrence associated with a male defendant and his female accomplice.

The first and second characteristics involve facial hair of the defendant. The percentage of men with a mustache is stated as 25%, and the percentage of Negro men with a beard is stated as 10%. Not only are these characteristics dependent, but  they also conflict (i.e., a mustache is not a beard). The third and fourth characteristics involve the hair of his female companion. There is dependence between females with blond hair and females that wear ponytails (i.e., some females with blond hair wear ponytails).

Assuming that the characteristics are independent, the frequencies of occurrence should be multiplied. Also assuming the prosecution's frequencies of occurrence were roughly correct, the combined frequency of occurrence is 8.333 x . This means only 20 men in the United States had this appearance. In other words, there might have been only one such person in Los Angeles, where the robbery occurred. By this type of reasoning, the prosecution attempted to show that there was a very small probability that the defendant was not the robber.

Summary

Probabilities combined by FactLogic must relate to facts that are independent. If two or more facts might be dependent, combine them into a single compound fact that is independent of others.

Footnotes

¹ For example, drops of blood forming a trail of blood are not considered independent facts, but are grouped into a single fact and called a trail of blood.

²This definition is appropriate for facts that are judged by subjective probabilities. The strict definition is: Two events, A and B, are independent if and only if

P(AB) = P(A)P(B) and P(B|A) = P(B)

where

• P(A) is the positive probability of the event A,

• P(B) is the positive probability of the event B,

• P(AB) is the probability of the product (or intersection) of events A and B, and

• P(B|A) is the probability of the event B, given that the event A has occurred.