Marginal Probability Function

A marginal probability function is a multivariate probability function that reports the marginal probability value for an event (without aposteriori knowledge of the events for the other random variables).

• AKA: Marginal Posterior Probability.
• Context:
  • input:
    • two or more Random Variables.
    • a Random Variable Event.
  • range: a marginal probability value.
  • It can be instantiated as a Marginal Probability Distribution Structure.
  • It can range from being a Marginal Probability Mass Function (for discrete random variable) to being a Marginal Probability Density Function.
  • It can be inferred by a Bayesian Inference Algorithm.
• Example(s):
  • [math]\displaystyle{ P(\text{dice}_{a=5}, \text{dice}_b) }[/math], associated with a two-dice experiment.
  • …
• Counter-Example(s):
  • a Conditional Probability Function.
  • a Joint Probability Function.
• See: Marginal Variable, Posterior Marginal Probability Function, Marginal Likelihood Function.

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References

2009

• (Wikipedia, 2009) ⇒ http://en.wikipedia.org/wiki/Marginal_distribution
  • In probability theory and statistics, the marginal distribution of a subset of a collection of random variables is the probability distribution of the variables contained in the subset. The term marginal variable is used to refer to those variables in the subset of variables being retained. These terms are dubbed "marginal" because they used to be found by summing values in a table along rows or columns, and writing the sum in the margins of the table. The distribution of the marginal variables (the marginal distribution) is obtained by "marginalising" over the distribution of the variables being discarded.
  • The context here is that the theoretical studies being undertaken, or the data analysis being done, involves a wider set of random variables but that attention is being limited to a reduced number of those variables. In many applications an analysis may start with a given collection of random variables, then first extend the set by defining new ones (such as the sum of the original random variables) and finally reduce the number by placing interest in the marginal distribution of a subset (such as the sum). Several different analyses may be done, each treating a different subset of variables as the marginal variables.
  • …
  • Given two random variables [math]\displaystyle{ X }[/math] and [math]\displaystyle{ Y }[/math] whose joint distribution is known, the marginal distribution of [math]\displaystyle{ X }[/math] is simply the probability distribution of [math]\displaystyle{ X }[/math] averaging over information about Y. This is typically calculated by summing or integrating the joint probability distribution over Y.
  • For discrete random variables, the marginal probability mass function can be written as Pr(X = x).

1987

• (Hogg & Ledolter, 1987) ⇒ Robert V. Hogg, and Johannes Ledolter. (1987). “Engineering Statistics." Macmillan Publishing. ISBN:0023557907
  • QUOTE: Probabilities such as … are called marginal probabilities because they are usually recorded in the margins of a joint probability table.
  • … In Example 2.4-1 we have illustrated the computation of the marginal probabilities.
    • [math]\displaystyle{ f }[/math]₁(x) = ∑ _y [math]\displaystyle{ f }[/math](x,y) = ∑ _y P(X=x, Y=y) = P(X=x),
    • and [math]\displaystyle{ f }[/math]₁(y) = ∑ _x [math]\displaystyle{ f }[/math](x,y) = ∑ _x P(X=x, Y=y) = P(Y=y).
  • We call [math]\displaystyle{ f }[/math]₁(x) and [math]\displaystyle{ f }[/math]₂(y) the marginal probability density function of [math]\displaystyle{ X }[/math] and [math]\displaystyle{ Y }[/math], respectively.

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