Conditional dependence

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In probability theory, conditional dependence is a relationship between two or more events that are dependent when a third event occurs.^[1] It is the opposite of conditional independence. For example, if ${\displaystyle A}$ and ${\displaystyle B}$ are two events that individually increase the probability of a third event ${\displaystyle C,}$ and do not directly affect each other, then initially (when it has not been observed whether or not the event ${\displaystyle C}$ occurs)^[2]^[3] ${\displaystyle \operatorname {P} (A\mid B)=\operatorname {P} (A)\quad {\text{ and \quad \operatorname {P} (B\mid A)=\operatorname {P} (B)}$ ( ${\displaystyle A{\text{ and B}$ are independent).

But suppose that now ${\displaystyle C}$ is observed to occur. If event ${\displaystyle B}$ occurs then the probability of occurrence of the event ${\displaystyle A}$ will decrease because its positive relation to ${\displaystyle C}$ is less necessary as an explanation for the occurrence of ${\displaystyle C}$ (similarly, event ${\displaystyle A}$ occurring will decrease the probability of occurrence of ${\displaystyle B}$ ). Hence, now the two events ${\displaystyle A}$ and ${\displaystyle B}$ are conditionally negatively dependent on each other because the probability of occurrence of each is negatively dependent on whether the other occurs. We have^[4] ${\displaystyle \operatorname {P} (A\mid C{\text{ and B)<\operatorname {P} (A\mid C).}$

Conditional dependence of A and B given C is the logical negation of conditional independence ${\displaystyle ((A\perp \!\!\!\perp B)\mid C)}$ .^[5] In conditional independence two events (which may be dependent or not) become independent given the occurrence of a third event.^[6]

Example#edit

In essence probability is influenced by a person's information about the possible occurrence of an event. For example, let the event ${\displaystyle A}$ be 'I have a new phone'; event ${\displaystyle B}$ be 'I have a new watch'; and event ${\displaystyle C}$ 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 ${\displaystyle C}$ 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 ${\displaystyle \Omega =\left\{s_{1},s_{2},s_{3},s_{4}\right\},}$ given in the middle four columns of the following table, in which the occurrence of event ${\displaystyle A}$ is signified by a ${\displaystyle 1}$ in row ${\displaystyle A}$ and its non-occurrence is signified by a ${\displaystyle 0,}$ and likewise for ${\displaystyle B}$ and ${\displaystyle C.}$ That is, ${\displaystyle A=\left\{s_{2},s_{4}\right\},B=\left\{s_{3},s_{4}\right\},}$ and ${\displaystyle C=\left\{s_{2},s_{3},s_{4}\right\}.}$ The probability of ${\displaystyle s_{i$ is ${\displaystyle 1/4}$ for every ${\displaystyle i.}$

Event	${\displaystyle \operatorname {P} (s_{2})=1/4}$	${\displaystyle \operatorname {P} (s_{3})=1/4}$	${\displaystyle \operatorname {P} (s_{4})=1/4}$	Probability of event
${\displaystyle A}$	1	0	1	${\displaystyle {\tfrac {1}{2$
${\displaystyle B}$	0	1	1	${\displaystyle {\tfrac {1}{2$
${\displaystyle C}$	1	1	1	${\displaystyle {\tfrac {3}{4$

and so

Event	${\displaystyle s_{2$	${\displaystyle s_{3$	${\displaystyle s_{4$	Probability of event
${\displaystyle A\cap B}$	0	0	1	${\displaystyle {\tfrac {1}{4$
${\displaystyle A\cap C}$	1	0	1	${\displaystyle {\tfrac {1}{2$
${\displaystyle B\cap C}$	0	1	1	${\displaystyle {\tfrac {1}{2$
${\displaystyle A\cap B\cap C}$	0	0	1	${\displaystyle {\tfrac {1}{4$

In this example, ${\displaystyle C}$ occurs if and only if at least one of ${\displaystyle A,B}$ occurs. Unconditionally (that is, without reference to ${\displaystyle C}$ ), ${\displaystyle A}$ and ${\displaystyle B}$ are independent of each other because ${\displaystyle \operatorname {P} (A)}$ —the sum of the probabilities associated with a ${\displaystyle 1}$ in row ${\displaystyle A}$ —is ${\displaystyle {\tfrac {1}{2,}$ while ${\displaystyle \operatorname {P} (A\mid B)=\operatorname {P} (A{\text{ and B)/\operatorname {P} (B)={\tfrac {1/4}{1/2={\tfrac {1}{2=\operatorname {P} (A).}$ But conditional on ${\displaystyle C}$ having occurred (the last three columns in the table), we have ${\displaystyle \operatorname {P} (A\mid C)=\operatorname {P} (A{\text{ and C)/\operatorname {P} (C)={\tfrac {1/2}{3/4={\tfrac {2}{3$ while ${\displaystyle \operatorname {P} (A\mid C{\text{ and B)=\operatorname {P} (A{\text{ and C{\text{ and B)/\operatorname {P} (C{\text{ and B)={\tfrac {1/4}{1/2={\tfrac {1}{2&lt$ Since in the presence of ${\displaystyle C}$ the probability of ${\displaystyle A}$ is affected by the presence or absence of ${\displaystyle B,A}$ and ${\displaystyle B}$ are mutually dependent conditional on ${\displaystyle C.}$

References#edit

^ Husmeier, Dirk. "Introduction to Learning Bayesian Networks from Data". In Husmeier, Dirk; Dybowski, Richard; Roberts, Stephen (eds.). Probabilistic Modeling in Bioinformatics and Medical Informatics. Advanced Information and Knowledge Processing. Springer-Verlag. pp. 17–57. doi:10.1007/1-84628-119-9_2. ISBN 1852337788.
^ Conditional Independence in Statistical theory "Conditional Independence in Statistical Theory", A. P. Dawid" Archived 2013-12-27 at the Wayback Machine
^ Probabilistic independence on Britannica "Probability->Applications of conditional probability->independence (equation 7) "
^ Introduction to Artificial Intelligence by Sebastian Thrun and Peter Norvig, 2011 "Unit 3: Explaining Away"
^ Bouckaert, Remco R. (1994). "11. Conditional dependence in probabilistic networks". In Cheeseman, P.; Oldford, R. W. (eds.). Selecting Models from Data, Artificial Intelligence and Statistics IV. Lecture Notes in Statistics. Vol. 89. Springer-Verlag. pp. 101–111, especially 104. ISBN 978-0-387-94281-0.
^ Conditional Independence in Statistical theory "Conditional Independence in Statistical Theory", A. P. Dawid Archived 2013-12-27 at the Wayback Machine

Categories Independence (probability theory)

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Conditional dependence

Example#edit

See also#edit

References#edit