Conditional Probability Function

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A conditional probability function is a multivariate probability function, [math]\displaystyle{ P(X|Y_1,...,Y_n) }[/math], that returns the probability for event [math]\displaystyle{ X }[/math] given aposteriori knowledge of some events, [math]\displaystyle{ Y_1,...,Y_n }[/math].



References

2013

  • http://en.wikipedia.org/wiki/Conditional_probability
    • In probability theory, a conditional probability is the probability that an event will occur, when another event is known to occur or to have occurred. If the events are A and B respectively, this is said to be "the probability of A given B". It is commonly denoted by P(A|B), or sometimes PTemplate:Sub(A). P(A|B) may or may not be equal to P(A), the probability of A. If they are equal, A and B are said to be independent. For example, if a coin is flipped twice, "the outcome of the second flip" is independent of "the outcome of the first flip".

      In the Bayesian interpretation of probability, the conditioning event is interpreted as evidence for the conditioned event. That is, P(A) is the probability of A before accounting for evidence E, and P(A|E) is the probability of A having accounted for evidence E.

2011

  • (Wikipedia, 2011) http://en.wikipedia.org/wiki/Conditional_probability
    • Given two jointly distributed random variables [math]\displaystyle{ X }[/math] and [math]\displaystyle{ Y }[/math], the conditional probability distribution of [math]\displaystyle{ Y }[/math] given [math]\displaystyle{ X }[/math] is the probability distribution of [math]\displaystyle{ Y }[/math] when [math]\displaystyle{ X }[/math] is known to be a particular value. For discrete random variables, the conditional probability mass function of [math]\displaystyle{ Y }[/math] given (the occurrence of) the value [math]\displaystyle{ x }[/math] of [math]\displaystyle{ X }[/math], can be written, using the definition of conditional probability, as: [math]\displaystyle{ p_Y(y\mid X = x)=P(Y = y \mid X = x) = \frac{P(X=x\ \cap Y=y)}{P(X=x)}. }[/math]
      As seen from the definition, and due to its occurrence, it is necessary that [math]\displaystyle{ P(X=x) \gt 0. }[/math]
      The relation with the probability distribution of [math]\displaystyle{ X }[/math] given [math]\displaystyle{ Y }[/math] is: [math]\displaystyle{ P(Y=y \mid X=x) P(X=x) = P(X=x\ \cap Y=y) = P(X=x \mid Y=y)P(Y=y). }[/math]
      Similarly for continuous random variables, the conditional probability density function of [math]\displaystyle{ Y }[/math] given (the occurrence of) the value [math]\displaystyle{ x }[/math] of [math]\displaystyle{ X }[/math], can be written as [math]\displaystyle{ f_Y(y \mid X=x) = \frac{f_{X, Y}(x, y)}{f_X(x)}, }[/math] where fX,Y(x, y) gives the joint density of [math]\displaystyle{ X }[/math] and [math]\displaystyle{ Y }[/math], while fX(x) gives the marginal density for X. Also in this case it is necessary that [math]\displaystyle{ f_X(x)\gt 0 }[/math].
      The relation with the probability distribution of [math]\displaystyle{ X }[/math] given [math]\displaystyle{ Y }[/math] is given by: [math]\displaystyle{ f_Y(y \mid X=x)f_X(x) = f_{X,Y}(x, y) = f_X(x \mid Y=y)f_Y(y). }[/math]
      The concept of the conditional distribution of a continuous random variable is not as intuitive as it might seem: Borel's paradox shows that conditional probability density functions need not be invariant under coordinate transformations.
      If for discrete random variables P(Y = [math]\displaystyle{ y }[/math] | [math]\displaystyle{ X }[/math] = x) = P(Y = y) for all [math]\displaystyle{ x }[/math] and [math]\displaystyle{ y }[/math], or for continuous random variables [math]\displaystyle{ f }[/math]Y(y | X=x) = [math]\displaystyle{ f }[/math]Y(y) for all x and y, then [math]\displaystyle{ Y }[/math] is said to be independent of [math]\displaystyle{ X }[/math] (and this implies that [math]\displaystyle{ X }[/math] is also independent of Y).
      Seen as a function of [math]\displaystyle{ y }[/math] for given [math]\displaystyle{ x }[/math], P(Y = [math]\displaystyle{ y }[/math] | [math]\displaystyle{ X }[/math] = x) is a probability and so the sum over all [math]\displaystyle{ y }[/math] (or integral if it is a conditional probability density) is 1. Seen as a function of [math]\displaystyle{ x }[/math] for given [math]\displaystyle{ y }[/math], it is a likelihood function, so that the sum over all [math]\displaystyle{ x }[/math] need not be 1.

2007

2004