Conditional Dependence

In probability theory, conditional dependence is a relationship between two or more events that are dependent when a third event occurs.Introduction to Artificial Intelligence by Sebastian Thrun and Peter Norvig, 201
"Unit 3: Conditional Dependence"
/ref> For example, if $A$ and $B$ are two events that individually increase the probability of a third event $C,$ and do not directly affect each other, then initially (when it has not been observed whether or not the event $C$ occurs)
$\operatorname(A \mid B) = \operatorname(A) \quad \text \quad \operatorname(B \mid A) = \operatorname(B)$ ($A \text B$ are independent).

But suppose that now $C$ is observed to occur. If event $B$ occurs then the probability of occurrence of the event $A$ will decrease because its positive relation to $C$ is less necessary as an explanation for the occurrence of $C$ (similarly, event $A$ occurring will decrease the probability of occurrence of $B$). Hence, now the two events $A$ and $B$ are conditionally negatively dependent on each other because the probability of occurrence of each is negatively dependent on whether the other occurs. We have
$\operatorname(A \mid C \text B) < \operatorname(A \mid C).$

Conditional dependence of A and B given C is the logical negation of conditional independence $((A \perp\!\!\!\perp B) \mid C)$. In conditional independence two events (which may be dependent or not) become independent given the occurrence of a third event.Conditional Independence in Statistical theor
"Conditional Independence in Statistical Theory", A. P. Dawid

Example

In essence probability is influenced by a person's information about the possible occurrence of an event. For example, let the event $A$ be 'I have a new phone'; event $B$ be 'I have a new watch'; and event $C$ be 'I am happy'; and suppose that having either a new phone or a new watch increases the probability of my being happy. Let us assume that the event $C$ has occurred – meaning 'I am happy'. Now if another person sees my new watch, he/she will reason that my likelihood of being happy was increased by my new watch, so there is less need to attribute my happiness to a new phone.

To make the example more numerically specific, suppose that there are four possible states $\Omega = \left\,$ given in the middle four columns of the following table, in which the occurrence of event $A$ is signified by a $1$ in row $A$ and its non-occurrence is signified by a $0,$ and likewise for $B$ and $C.$ That is, $A = \left\, B = \left\,$ and $C = \left\.$ The probability of $s_i$ is $1/4$ for every $i.$

and so

In this example, $C$ occurs if and only if at least one of $A, B$ occurs. Unconditionally (that is, without reference to $C$), $A$ and $B$ are independent of each other because $\operatorname(A)$—the sum of the probabilities associated with a $1$ in row $A$—is $\tfrac,$ while
$\operatorname(A\mid B) = \operatorname(A \text B) / \operatorname(B) = \tfrac = \tfrac = \operatorname(A).$
But conditional on $C$ having occurred (the last three columns in the table), we have
$\operatorname(A \mid C) = \operatorname(A \text C) / \operatorname(C) = \tfrac = \tfrac$
while
$\operatorname(A \mid C \text B) = \operatorname(A \text C \text B) / \operatorname(C \text B) = \tfrac = \tfrac < \operatorname(A \mid C).$
Since in the presence of $C$ the probability of $A$ is affected by the presence or absence of $B, A$ and $B$ are mutually dependent conditional on $C.$

See also

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References

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Independence (probability theory)