Conditional dependence

A Bayesian network illustrating conditional dependence

In probability theory, conditional dependence is a relationship between two or more events that are dependent when a third event occurs.[1][2] 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)[3][4] P⁡(A∣B)=P⁡(A) and P⁡(B∣A)=P⁡(B) (A and 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[5] P⁡(A∣C and B)<P⁡(A∣C).

Conditional dependence of A and B given C is the logical negation of conditional independence ((A⊥⊥B)∣C).[6] In conditional independence two events (which may be dependent or not) become independent given the occurrence of a third event.[7]

## 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 Ω={s1,s2,s3,s4}, 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={s2,s4},B={s3,s4}, and C={s2,s3,s4}. The probability of si is 1/4 for every i.

Event | P⁡(s1)=1/4 | P⁡(s2)=1/4 | P⁡(s3)=1/4 | P⁡(s4)=1/4 | Probability of event

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A | 0 | 1 | 0 | 1  | 12
B | 0 | 0 | 1 | 1  | 12
C | 0 | 1 | 1 | 1  | 34

and so

Event | s1 | s2 | s3 | s4 | Probability of event

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A∩B | 0 | 0 | 0 | 1  | 14
A∩C | 0 | 1 | 0 | 1  | 12
B∩C | 0 | 0 | 1 | 1  | 12
A∩B∩C | 0 | 0 | 0 | 1  | 14

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 P⁡(A)—the sum of the probabilities associated with a 1 in row A—is 12, while P⁡(A∣B)=P⁡(A and B)/P⁡(B)=1/41/2=12=P⁡(A). But conditional on C having occurred (the last three columns in the table), we have P⁡(A∣C)=P⁡(A and C)/P⁡(C)=1/23/4=23 while P⁡(A∣C and B)=P⁡(A and C and B)/P⁡(C and B)=1/41/2=12<P⁡(A∣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

* Conditional independence - Probability theory concept
* De Finetti's theorem - Conditional independence of exchangeable observations
* Conditional expectation - Expected value of a random variable given that certain conditions are known to occur

## References

1. ↑ Introduction to Artificial Intelligence by Sebastian Thrun and Peter Norvig, 2011 "Unit 3: Conditional Dependence"[yes|permanent dead link|dead link}}]
2. ↑ Introduction to learning Bayesian Networks from Data by Dirk Husmeier [1][yes|permanent dead link|dead link}}] "Introduction to Learning Bayesian Networks from Data -Dirk Husmeier"
3. ↑ Conditional Independence in Statistical theory "Conditional Independence in Statistical Theory", A. P. Dawid"
4. ↑ Probabilistic independence on Britannica "Probability->Applications of conditional probability->independence (equation 7) "
5. ↑ Introduction to Artificial Intelligence by Sebastian Thrun and Peter Norvig, 2011 "Unit 3: Explaining Away"[yes|permanent dead link|dead link}}]
6. ↑ Bouckaert, Remco R. (1994). "11. Conditional dependence in probabilistic networks". in Cheeseman, P. (in EN). Selecting Models from Data, Artificial Intelligence and Statistics IV. Lecture Notes in Statistics. 89. Springer-Verlag. pp. 101-111, especially 104. ISBN 978-0-387-94281-0.
7. ↑ Conditional Independence in Statistical theory "Conditional Independence in Statistical Theory", A. P. Dawid

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