Conditional dependence
Concept in probability theory
title: "Conditional dependence" type: doc version: 1 created: 2026-02-28 author: "Wikipedia contributors" status: active scope: public tags: ["independence-(probability-theory)"] description: "Concept in probability theory" topic_path: "science/mathematics" source: "https://en.wikipedia.org/wiki/Conditional_dependence" license: "CC BY-SA 4.0" wikipedia_page_id: 0 wikipedia_revision_id: 0
::summary Concept in probability theory ::
::figure[src="https://upload.wikimedia.org/wikipedia/commons/9/97/Conditional_Dependence.jpg" caption="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.{{cite book | last = Husmeier | first = Dirk | editor1-last = Husmeier | editor1-first = Dirk | editor2-last = Dybowski | editor2-first = Richard | editor3-last = Roberts | editor3-first = Stephen | contribution = Introduction to Learning Bayesian Networks from Data | doi = 10.1007/1-84628-119-9_2 | isbn = 1852337788 | pages = 17–57 | publisher = Springer-Verlag | series = Advanced Information and Knowledge Processing | title = Probabilistic Modeling in Bioinformatics and Medical Informatics}} It is the opposite of conditional independence. 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{P}(A \mid B) = \operatorname{P}(A) \quad \text{ and } \quad \operatorname{P}(B \mid A) = \operatorname{P}(B) (A \text{ 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 \operatorname{P}(A \mid C \text{ and } B)
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.
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{ s_1, s_2, s_3, s_4 \right}, 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{ s_2, s_4 \right}, B = \left{ s_3, s_4 \right}, and C = \left{ s_2, s_3, s_4 \right}. The probability of s_i is 1/4 for every i.
::data[format=table]
| Event | \operatorname{P}(s_1)=1/4 | \operatorname{P}(s_2)=1/4 | \operatorname{P}(s_3)=1/4 | \operatorname{P}(s_4)=1/4 | Probability of event | \tfrac{1}{2} | \tfrac{1}{2} | \tfrac{3}{4} |
|---|---|---|---|---|---|---|---|---|
| A | 0 | 1 | 0 | 1 | ||||
| B | 0 | 0 | 1 | 1 | ||||
| C | 0 | 1 | 1 | 1 | ||||
| :: |
and so
::data[format=table]
| Event | s_1 | s_2 | s_3 | s_4 | Probability of event | \tfrac{1}{4} | \tfrac{1}{2} | \tfrac{1}{2} | \tfrac{1}{4} |
|---|---|---|---|---|---|---|---|---|---|
| A \cap B | 0 | 0 | 0 | 1 | |||||
| A \cap C | 0 | 1 | 0 | 1 | |||||
| B \cap C | 0 | 0 | 1 | 1 | |||||
| A \cap B \cap C | 0 | 0 | 0 | 1 | |||||
| :: |
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{P}(A)—the sum of the probabilities associated with a 1 in row A—is \tfrac{1}{2}, while \operatorname{P}(A\mid B) = \operatorname{P}(A \text{ and } B) / \operatorname{P}(B) = \tfrac{1/4}{1/2} = \tfrac{1}{2} = \operatorname{P}(A). But conditional on C having occurred (the last three columns in the table), we have \operatorname{P}(A \mid C) = \operatorname{P}(A \text{ and } C) / \operatorname{P}(C) = \tfrac{1/2}{3/4} = \tfrac{2}{3} while \operatorname{P}(A \mid C \text{ and } B) = \operatorname{P}(A \text{ and } C \text{ and } B) / \operatorname{P}(C \text{ and } B) = \tfrac{1/4}{1/2} = \tfrac{1}{2} 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.
References
References
- Conditional Independence in Statistical theory [http://edlab-www.cs.umass.edu/cs589/2010-lectures/conditional%20independence%20in%20statistical%20theory.pdf "Conditional Independence in Statistical Theory", A. P. Dawid"] {{webarchive. link. (2013-12-27)
- Probabilistic independence on Britannica [http://www.britannica.com/EBchecked/topic/477530/probability-theory/32768/Applications-of-conditional-probability#toc32769 "Probability->Applications of conditional probability->independence (equation 7) "]
- Introduction to Artificial Intelligence by Sebastian Thrun and Peter Norvig, 2011 [https://www.ai-class.com/course/video/quizquestion/60 "Unit 3: Explaining Away"]{{Dead link. (July 2020)
- Bouckaert, Remco R.. (1994). "Selecting Models from Data, Artificial Intelligence and Statistics IV". [[Springer-Verlag]].
- Conditional Independence in Statistical theory [http://edlab-www.cs.umass.edu/cs589/2010-lectures/conditional%20independence%20in%20statistical%20theory.pdf "Conditional Independence in Statistical Theory", A. P. Dawid] {{webarchive. link. (2013-12-27)
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