Sum Rule

$$p(\mathbf{x})=\{\begin{array}{ll}\sum _{\mathbf{y}\in \mathcal{Y}}p(\mathbf{x},\mathbf{y})& \text{if}\mathbf{y}\text{is discrete}\\ {\int }_{\mathcal{Y}}p(\mathbf{x},\mathbf{y})\mathrm{d}\mathbf{y}& \text{if}\mathbf{y}\text{is continuous}\end{array}$$

$\mathcal{Y}$ 는 random variable $Y$ 의 target space 를 의미

we sum out (or integrate out) the set of states $\mathbf{y}$ of the random variable $Y$

2. 특징

Performing high-dimensional sums or integrals is generally computationally hard, in the sense that there is no known polynomial-time algorithm to calculate them exactly.

3. Related

product rule, marginal distribution

4. References