Conditional independence on arvau

Wikipedia / Conditional independence

In probability theory, conditional independence describes situations in which an observation is irrelevant or redundant when evaluating the certainty of a hypothesis. It is the opposite of conditional dependence. Conditional independence is usually formulated in terms of conditional probability are observations, conditional independence can be stated as an equality:

are said to be conditionally independent.

The concept of conditional independence is essential to graph-based theories of statistical inference, as it establishes a mathematical relation between a collection of conditional statements and a graphoid.

Conditional independence of events

be events are said to be conditionally independent. This property is symmetric (more on this below.

is the joint probability are independent events, .

Proof of the equivalent definition

Examples

Coloured boxes

are represented by the areas shaded , and is shaded .

because:

Proximity and delays

Let events A and B be defined as the probability that person A and person B will be home in time for dinner where both people are randomly sampled from the entire world. Events A and B can be assumed to be independent i.e. knowledge that A is late has minimal to no change on the probability that B will be late. However, if a third event is introduced, person A and person B live in the same neighborhood, the two events are now considered not conditionally independent. Traffic conditions and weather-related events that might delay person A, might delay person B as well. Given the third event and knowledge that person A was late, the probability that person B will be late does meaningfully change.

Dice rolling

Conditional independence depends on the nature of the third event. If you roll two dice, one may assume that the two dice behave independently of each other. Looking at the results of one die will not tell you about the result of the second die. (That is, the two dice are independent.) If, however, the 1st die's result is a 3, and someone tells you about a third event - that the sum of the two results is even - then this extra unit of information restricts the options for the 2nd result to an odd number. In other words, two events can be independent, but not conditionally independent.

Height and vocabulary

Height and vocabulary are dependent since very small people tend to be children, known for their more basic vocabularies. But knowing that two people are 19 years old (i.e., conditional on age) there is no reason to think that one person's vocabulary is larger if we are told that they are taller.

Conditional independence of random variables

Two discrete random variables if and only if they are independent in their conditional probability distribution. Formally:

is the conditional cumulative distribution function.

are conditionally independent given a σ-algebra if

denotes the conditional expectation of the indicator function. That is,

if they are independent given σ(W. This is commonly written:

, given".

is independent."

assumes a countable set of values, this is equivalent to the conditional independence of X and Y. Conditional independence of more than two events, or of more than two random variables, is defined analogously.

neither implies nor is implied by.

is 0 with probability 0.5 and 1 otherwise. When W= are dependent, because Pr(X=0) < Pr(X=0|Y=0). This is because Pr(X=0) =0.5, but if Y=0 then it's very likely that W=0 and thus that X=0 as well, so Pr(X=0|Y=0)>0.5.

, each taking the values 0 and 1 with probability, Pr(X=0)=2/3, but Pr(X=0|Y=0)= is false. This is also an example of Explaining Away. See Kevin Murphy's tutorial take the values "brainy" and "sporty".

Conditional independence of random vectors

Two random vectors. Formally:

and the conditional cumulative distributions are defined as follows.

Uses in Bayesian inference

Let p be the proportion of voters who will vote "yes" in an upcoming referendum. In taking an opinion poll, one chooses n voters randomly from the population. For i=1,...,n, let Xi=1 or 0 corresponding, respectively, to whether or not the ith chosen voter will or will not vote "yes".

In a frequentist approach to statistical inference one would not attribute any probability distribution to p (unless the probabilities could be somehow interpreted as relative frequencies of occurrence of some event or as proportions of some population) and one would say that X1, ..., Xn are independent random variables.

By contrast, in a Bayesian approach to statistical inference, one would assign a probability distribution to p regardless of the non-existence of any such "frequency" interpretation, and one would construe the probabilities as degrees of belief that p is in any interval to which a probability is assigned. In that model, the random variables X1,...,Xn are not independent, but they are conditionally independent given the value of p. In particular, if a large number of the Xs are observed to be equal to 1, that would imply a high conditional probability, given that observation, that p is near 1, and thus a high conditional probability, given that observation, that the next X to be observed will be equal to 1.

Rules of conditional independence

A set of rules governing statements of conditional independence have been derived from the basic definition.

These rules were termed "Graphoid Axioms" by Pearl and Paz, is interpreted to mean: "All paths from X to A are intercepted by the set B".

Symmetry

Proof:

From the definition of conditional independence,

Decomposition

Proof From the definition of conditional independence, we seek to show that:

. The left side of this equality is:

. Further decomposing,

. Special cases of this property include

Proof:. Then:
Proof:. Then:

Weak union

Proof:

, we aim to show

. We begin with the left side of the equation

. From the given condition

Special Cases:

Some textbooks present the property as

Both versions can be shown to follow from the weak union property given initially via the same method as in the decomposition section above.

Contraction

Proof

, respectively.

Intersection

For strictly positive probability distributions, the following also holds: