Multiplication

Multiplication node has inputs s_k and the output is their product $\prod_k s_{k}$. The excepted exponential of the output cannot be evaluated without knowing the exact distribution of the inputs. Assuming independence between s_k, the mean and the variance of the output are

   

$$\left< s_{1} s_{2} \right> \left< s_{1} \right> \left< s_{2} \right>$$ = (4.30)

$$\mathrm{Var}\left\{s_{1} s_{2}\right\} \left< s_1^2 s_2^2 \right>-\left< s_1 s_2 \right>^2$$ =

$$\left< s_1^2 \right>\left< s_2^2 \right>-\left< s_1 \right>^2\left< s_2 \right>^2$$ = (4.31)

$$(\left< s_1 \right>^2+\mathrm{Var}\left\{s_1\right\})(\left< s_2 \right>^2+\mathrm{Var}\left\{s_2\right\})-\left< s_1 \right>^2\left< s_2 \right>^2$$ =

$$\left< s_{1} \right>^{2} \mathrm{Var}\left\{s_{2}\right\} + \math... ...\} \left( \left< s_{2} \right>^{2} + \mathrm{Var}\left\{s_{2}\right\} \right) .$$ = (4.32)

The propagated derivatives are obtained like with addition and they are

 

$$\frac{\partial C}{\partial \left< s_1 \right>} \frac{\partial C}{\partial \left< s_1 s_2 \right>} \frac{\partial... ...\frac{\partial \mathrm{Var}\left\{s_1 s_2\right\}}{\partial \left< s_1 \right>}$$ = (4.33)

$$\left< s_2 \right>\frac{\partial C}{\partial \left< s_1 s_2 \righ... ...rac{\partial C}{\partial \mathrm{Var}\left\{s_1 s_2\right\}} \left< s_1 \right>$$ =

$$\frac{\partial C}{\partial \mathrm{Var}\left\{s_1\right\}} \frac{\partial C}{\partial \left< s_1 s_2 \right>} \frac{\partial... ...al \mathrm{Var}\left\{s_1 s_2\right\}}{\partial \mathrm{Var}\left\{s_1\right\}}$$ = (4.34)

$$\left(\left< s_2 \right>^2 + \mathrm{Var}\left\{s_2\right\}\right)\frac{\partial C}{\partial \mathrm{Var}\left\{s_1 s_2\right\}}.$$ =

The equations for larger products are again obtained by induction, e.g. s₁s₂s₃=(s₁s₂)s₃.