Addition and Multiplication

Addition and multiplication nodes can be used e.g. for constructing linear mappings and affine transformations between the variables. Denoting the inputs by $s_{i}$, the outputs are $\sum_i s_{i}$ for addition and $\prod_i s_{i}$ for multiplication nodes. The mean, variance and expected exponential of the addition node are

$$\left< s_{1}+s_{2} \right> = \left< s_{1} \right> + \left< s_{2} \right>$$ (3)

$$\mathrm{Var}\left\{s_{1}+s_{2}\right\} = \mathrm{Var}\left\{s_{1}\right\} + \mathrm{Var}\left\{s_{2}\right\}$$ (4)

$$\left< \exp(s_{1}+s_{2}) \right> = \left< \exp s_{1} \right> \left< \exp s_{2} \right>$$ (5)

assuming $s_{i}$ independent. For a multiplication node, the expected exponential cannot be evaluated without knowing the exact distribution of the inputs. Assuming independence between $s_i$, the mean and the variance of the output are

$$\left< s_{1} s_{2} \right> = \left< s_{1} \right> \left< s_{2} \right>$$ (6)

$$\mathrm{Var}\left\{s_{1} s_{2}\right\} = \left< s_{1} \right>^{2} \mathrm{Var}\left\{s_{2}\right\}$$ (7)

$$+ \mathrm{Var}\left\{s_{1}\right\} \left( \left< s_{2} \right>^{2} + \mathrm{Var}\left\{s_{2}\right\} \right) .$$

The equations for larger sums or products are obtained by induction, e.g.  $s_{1}s_{2}s_{3}=(s_{1}s_{2})s_{3}$.