Conditional dependence
In probability theory, conditional dependence is a relationship between two or more events that are dependent when a third event occurs.
are two events that individually increase the probability of a third event
and do not directly affect each other, then initially (when it has not been observed whether or not the event
is observed to occur.
occurs then the probability of occurrence of the event
will decrease because its positive relation to
is less necessary as an explanation for the occurrence of
(similarly, event
occurring will decrease the probability of occurrence of
are conditionally negatively dependent on each other because the probability of occurrence of each is negatively dependent on whether the other occurs.
Conditional dependence of A and B given C is the logical negation of conditional independence
[6] In conditional independence two events (which may be dependent or not) become independent given the occurrence of a third event.
[7] In essence probability is influenced by a person's information about the possible occurrence of an event.
be 'I have a new phone'; event
be 'I have a new watch'; and event
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
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
given in the middle four columns of the following table, in which the occurrence of event
in row
and its non-occurrence is signified by a
Unconditionally (that is, without reference to
—the sum of the probabilities associated with a
in row
But conditional on
having occurred (the last three columns in the table), we have
is affected by the presence or absence of
are mutually dependent conditional on
