Marginal Distribution

Definition

The marginal distribution of a subset of random variables is obtained by integrating (or summing) the joint distribution over the remaining variables.

  • Continuous case:
$$ f_X(x) = \int_{-\infty}^{\infty} f_{X,Y}(x, y)\, dy. $$
  • Discrete case:
$$ p_X(x) = \sum_{y} p_{X,Y}(x, y). $$

Role in Inference

Computing marginals from a joint distribution is a fundamental operation in probabilistic graphical models. When the state space is large, exact marginalization becomes intractable, motivating approximate methods such as Belief Propagation (BP) and Variational Inference (VI).