Suppose that we learn a pronoun is plural. We can ask whether that information changes the probability that the pronoun has accusative form by making the comparison
\mathbb{P}(A_4\mid P_4)\stackrel{?}{=}\mathbb{P}(A_4).
In the toy pronoun measure,
\mathbb{P}(A_4)=.70
but
\mathbb{P}(A_4\mid P_4)=.80.
Conditioning on plurality changes the probability of accusative form, so the two events are not independent under this measure. This comparison is the invariance test: compare the probability before and after conditioning.
Suppose \mathbb{P}(C)>0. Conditioning on C leaves the probability of B unchanged exactly when
\mathbb{P}(B\mid C)=\mathbb{P}(B).
Basically, independence means that conditioning on one event leaves the probability of the other unchanged. This is the conditional characterization of independence: learning that C occurred does not change the probability of B.
The characterization requires \mathbb P(C)>0 because conditional probability is a ratio. The unrestricted definition avoids that restriction. Events B and C are independent by definition exactly when
B\perp\!\!\!\perp C \quad\equiv\quad \bigl[\mathbb P(B,C)=\mathbb P(B)\mathbb P(C)\bigr].
In the fourteen-outcome case-coded space,
\mathbb P(T_{14})=\frac{8}{14}=\frac47
and, as derived on the conditional-probability page,
\mathbb P(T_{14}\mid A_{14})=\frac47.
Thus third person and accusative case are independent under the uniform measure on this refined sample space. Equivalently,
\mathbb P(T_{14},A_{14}) =\frac{4}{14} =\frac{8}{14}\frac{7}{14} =\mathbb P(T_{14})\mathbb P(A_{14}).
Thus T_{14} and A_{14} are independent under the stated uniform measure.
The marginal probabilities are
\mathbb{P}(P_4)=.50 \qquad\text{and}\qquad \mathbb{P}(A_4)=.70.
Their product is
\mathbb{P}(P_4)\mathbb{P}(A_4) =.50\times.70 =.35.
The joint probability is
\mathbb{P}(P_4,A_4)=.40.
Since .40\neq.35, the events are not independent under this measure.
The conditional characterization gives the same answer: the probability of A_4 changes from .70 in the full space to .80 given P_4.
The chain rule gives
\mathbb{P}(B,C) =\mathbb{P}(B\mid C)\mathbb{P}(C).
If \mathbb P(B\mid C)=\mathbb P(B), 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).
Each direction uses one licensed move: substitution in the first direction and division by a positive probability in the second. Thus the conditional characterization and product definition are equivalent whenever \mathbb P(C)>0.
Keep the same outcome labels and margins, but change how the mass is arranged inside the table.
| accusative A_4 | nominative A_4^c | total | |
|---|---|---|---|
| plural P_4 | .35 | .15 | .50 |
| singular P_4^c | .35 | .15 | .50 |
| total | .70 | .30 | 1 |
Now the upper left cell equals the product of its margins:
\mathbb{P}(P_4,A_4) =.35 =.50\times.70.
Conditioning gives the same result:
\mathbb{P}(A_4\mid P_4) =\frac{.35}{.50} =.70 =\mathbb{P}(A_4).
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, but P_4 and A_4 are dependent in the original table and independent in the comparison table. Independence is a property of the events together with the probability measure.
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.
The exact upshot is that independence means product factorization, or equivalently conditional invariance when the conditioning event has positive probability. The random-variables module will reuse this event-level definition when it defines independence among random variables.