Conditional dependence

In probability theory, conditional dependence is a relationship between two or more events that are dependent when a third event occurs. For example, if and 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 occurs) ( are independent). But suppose that now is observed to occur. If event 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 ). Hence, now the two events and are conditionally negatively dependent on each other because the probability of occurrence of each is negatively dependent on whether the other occurs. We have Conditional dependence of A and B given C is the logical negation of conditional independence . In conditional independence two events (which may be dependent or not) become independent given the occurrence of a third event. In essence probability is influenced by a person's information about the possible occurrence of an event. For example, let the 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 is signified by a in row and its non-occurrence is signified by a and likewise for and That is, and The probability of is for every and so In this example, occurs if and only if at least one of occurs. Unconditionally