Suppose that we learn a pronoun is plural. We can ask whether conditioning on plurality changes the probability that the pronoun has accusative form:
\mathbb{P}(A\mid P)\stackrel{?}{=}\mathbb{P}(A).
In the toy pronoun measure,
\mathbb{P}(A)=.70
but
\mathbb{P}(A\mid P)=.80.
Conditioning on plurality changes the probability of accusative form, so the two events are not independent under this measure.
Suppose \mathbb{P}(C)>0. We define events B and C to be independent when conditioning on C leaves the probability of B unchanged:
\mathbb{P}(B\mid C)=\mathbb{P}(B).
This conditional statement gives the direct interpretation of independence. Learning that C occurred supplies no information about whether B occurred.
The marginal probabilities are
\mathbb{P}(P)=.50 \qquad\text{and}\qquad \mathbb{P}(A)=.70.
Their product is
\mathbb{P}(P)\mathbb{P}(A) =.50\times.70 =.35.
The joint probability is
\mathbb{P}(P,A)=.40.
Since .40\neq.35, the events are not independent under this measure.
The conditional interpretation gives the same answer: the probability of A changes from .70 in the full space to .80 among plural forms.
The chain rule gives
\mathbb{P}(B,C) =\mathbb{P}(B\mid C)\mathbb{P}(C).
Under independence, substitute \mathbb{P}(B) for \mathbb{P}(B\mid C):
\mathbb{P}(B,C) =\mathbb{P}(B)\mathbb{P}(C).
Conversely, if this factorization holds and \mathbb{P}(C)>0, divide both sides by \mathbb{P}(C):
\mathbb{P}(B\mid C)=\mathbb{P}(B).
Thus the conditional-invariance and product-factorization criteria are equivalent whenever the conditional probability is defined. The product criterion also covers zero-probability events, so the standard unrestricted definition is
B\perp\!\!\!\perp C \quad\Longleftrightarrow\quad \mathbb{P}(B,C)=\mathbb{P}(B)\mathbb{P}(C).
Keep the same outcome labels and margins, but change how the mass is arranged inside the table.
| accusative A | nominative A^c | total | |
|---|---|---|---|
| plural P | .35 | .15 | .50 |
| singular P^c | .35 | .15 | .50 |
| total | .70 | .30 | 1 |
Now the upper left cell equals the product of its margins:
\mathbb{P}(P,A) =.35 =.50\times.70.
Conditioning gives the same result:
\mathbb{P}(A\mid P) =\frac{.35}{.50} =.70 =\mathbb{P}(A).
The accusative probability is also .70 in the singular row. Under this measure, knowing whether the form is singular or plural does not change the probability that it is accusative.
joint_independent <- matrix(
c(.35, .35, .15, .15),
nrow = 2,
dimnames = list(
number = c("plural", "singular"),
case = c("accusative", "nominative")
)
)
p_joint <- joint_independent["plural", "accusative"]
p_plural <- sum(joint_independent["plural", ])
p_acc <- sum(joint_independent[, "accusative"])
c(observed = p_joint, product = p_plural * p_acc)Both returned values are .35.
The two tables use the same outcome labels and the same events. In the original table, P and A are dependent. In the comparison table, they are independent. Whether the events are independent depends on the probability measure, not on the labels plural and accusative.
For the same reason, a construction choice and an animacy feature may be independent in one corpus genre but associated in another. We can use the same sample space and event definitions for both populations while assigning different probabilities to their intersections.
Mutually exclusive events cannot occur together. If two mutually exclusive events B and C both have positive probability, then
\mathbb{P}(B,C)=0
while
\mathbb{P}(B)\mathbb{P}(C)>0.
They are not independent. Observing C tells us that B did not occur, which is strong information.
Mutual exclusivity concerns whether the intersection is empty. Independence concerns whether the intersection receives the product of the marginal probabilities.
Independence is an exact property of a probability measure, but sample proportions are estimates. They will rarely be exactly equal in a finite sample, even when the represented events are independent. They may also be equal by chance when the events are dependent.
Later chapters introduce statistical models for uncertainty about such claims. At this stage, we can determine only whether a fully specified probability measure satisfies the independence equation.