Marginal Distributions

Marginal distributions focus on a single variable from a multivariate distribution by averaging out the effects of the others. Mathematically, if we have a joint distribution P(X,Y), the marginal distribution of X is found by summing or integrating over all possible values of Y: P(X) = ∑ᵧ P(X,Y=y) or P(X) = ∫ P(X,Y=y) dy.

In machine learning, marginal distributions help us understand individual variables' behavior while acknowledging they exist within a multivariate context. Feature importance analysis, dimensionality reduction, and univariate analysis all leverage the concept of marginalization to focus on specific aspects of complex data.