{{Short description|Concept in probability theory}} [[File:Conditional Dependence.jpg|thumb|right|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.<ref>{{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}}</ref> It is the opposite of ''conditional independence''. For example, if <math>A</math> and <math>B</math> are two events that individually increase the probability of a third event <math>C,</math> and do not directly affect each other, then initially (when it has not been observed whether or not the event <math>C</math> occurs)<ref>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|url=https://web.archive.org/web/20131227164541/http://edlab-www.cs.umass.edu/cs589/2010-lectures/conditional%20independence%20in%20statistical%20theory.pdf|date=2013-12-27}}</ref><ref>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) "]</ref> <math display=block>\operatorname{P}(A \mid B) = \operatorname{P}(A) \quad \text{ and } \quad \operatorname{P}(B \mid A) = \operatorname{P}(B)</math> (<math>A \text{ and } B</math> are independent).
But suppose that now <math>C</math> is observed to occur. If event <math>B</math> occurs then the probability of occurrence of the event <math>A</math> will decrease because its positive relation to <math>C</math> is less necessary as an explanation for the occurrence of <math>C</math> (similarly, event <math>A</math> occurring will decrease the probability of occurrence of <math>B</math>). Hence, now the two events <math>A</math> and <math>B</math> are conditionally negatively dependent on each other because the probability of occurrence of each is negatively dependent on whether the other occurs. We have<ref>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|date=July 2020 |bot=InternetArchiveBot |fix-attempted=yes }}</ref> <math display=block>\operatorname{P}(A \mid C \text{ and } B) < \operatorname{P}(A \mid C).</math>
Conditional dependence of A and B given C is the logical negation of conditional independence <math>((A \perp\!\!\!\perp B) \mid C)</math>.<ref>{{Cite book |last=Bouckaert |first=Remco R. |title=Selecting Models from Data, Artificial Intelligence and Statistics IV |publisher=Springer-Verlag |year=1994 |isbn=978-0-387-94281-0 |editor-last=Cheeseman |editor-first=P. |series=Lecture Notes in Statistics |volume=89 |pages=101-111, especially 104 |language=EN |chapter=11. Conditional dependence in probabilistic networks |editor-last2=Oldford |editor-first2=R. W.}}</ref> In conditional independence two events (which may be dependent or not) become independent given the occurrence of a third event.<ref>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|url=https://web.archive.org/web/20131227164541/http://edlab-www.cs.umass.edu/cs589/2010-lectures/conditional%20independence%20in%20statistical%20theory.pdf |date=2013-12-27 }}</ref>
== Example ==
In essence probability is influenced by a person's information about the possible occurrence of an event. For example, let the event <math>A</math> be 'I have a new phone'; event <math>B</math> be 'I have a new watch'; and event <math>C</math> 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 <math>C</math> 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 <math>\Omega = \left\{ s_1, s_2, s_3, s_4 \right\},</math> given in the middle four columns of the following table, in which the occurrence of event <math>A</math> is signified by a <math>1</math> in row <math>A</math> and its non-occurrence is signified by a <math>0,</math> and likewise for <math>B</math> and <math>C.</math> That is, <math>A = \left\{ s_2, s_4 \right\}, B = \left\{ s_3, s_4 \right\},</math> and <math>C = \left\{ s_2, s_3, s_4 \right\}.</math> The probability of <math>s_i</math> is <math>1/4</math> for every <math>i.</math>
{| class="wikitable" |- ! Event !! <math>\operatorname{P}(s_1)=1/4</math> !! <math>\operatorname{P}(s_2)=1/4</math> !! <math>\operatorname{P}(s_3)=1/4</math> !! <math>\operatorname{P}(s_4)=1/4</math> !! Probability of event |- | <math>A</math> || 0 || 1 || 0 || 1 ! <math>\tfrac{1}{2}</math> |- | <math>B</math> || 0 || 0 || 1 || 1 ! <math>\tfrac{1}{2}</math> |- | <math>C</math> || 0 || 1 || 1 || 1 ! <math>\tfrac{3}{4}</math> |}
and so
{| class="wikitable" |- ! Event !! <math>s_1</math> !! <math>s_2</math> !! <math>s_3</math> !! <math>s_4</math> !! Probability of event |- | <math>A \cap B</math> || 0 || 0 || 0 || 1 ! <math>\tfrac{1}{4}</math> |- | <math>A \cap C</math> || 0 || 1 || 0 || 1 ! <math>\tfrac{1}{2}</math> |- | <math>B \cap C</math> || 0 || 0 || 1 || 1 ! <math>\tfrac{1}{2}</math> |- | <math>A \cap B \cap C</math> || 0 || 0 || 0 || 1 ! <math>\tfrac{1}{4}</math> |}
In this example, <math>C</math> occurs if and only if at least one of <math>A, B</math> occurs. Unconditionally (that is, without reference to <math>C</math>), <math>A</math> and <math>B</math> are independent of each other because <math>\operatorname{P}(A)</math>—the sum of the probabilities associated with a <math>1</math> in row <math>A</math>—is <math>\tfrac{1}{2},</math> while <math display=block>\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).</math> But conditional on <math>C</math> having occurred (the last three columns in the table), we have <math display=block>\operatorname{P}(A \mid C) = \operatorname{P}(A \text{ and } C) / \operatorname{P}(C) = \tfrac{1/2}{3/4} = \tfrac{2}{3}</math> while <math display=block>\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} < \operatorname{P}(A \mid C).</math> Since in the presence of <math>C</math> the probability of <math>A</math> is affected by the presence or absence of <math>B, A</math> and <math>B</math> are mutually dependent conditional on <math>C.</math>
== See also ==
* {{annotated link|Conditional independence}} * {{annotated link|de Finetti's theorem}} * {{annotated link|Conditional expectation}}
== References ==
{{reflist|group=note}} {{reflist}}
Category:Independence (probability theory)