Table 1.

Sum-product update rules some standard nodes with Gaussian messages, see Korl (2005, ch. 4) and Loeliger et al. (2007) for more elaborate tables.

#No	Node	Factor update rule
1		$\begin{array}{l}f(x,y,z)=\delta (x+y-z)\\ m_z=m_x+m_y\\ {\vartheta }_z={\vartheta }_x+{\vartheta }_y\end{array}$
	Addition
2		$\begin{array}{l}f(x,y,z)=\delta (x+y-z)\\ m_y=m_z-m_x\\ {\vartheta }_y={\vartheta }_z+{\vartheta }_x\end{array}$
	Subtraction
3		$\begin{array}{l}f(x,y,a)=\delta (y-ax)\\ m_y=a{m}_x\\ {\vartheta }_y=a^2{\vartheta }_x\end{array}$
	Multiplication (forward)
4		$\begin{array}{l}f(x,y,a)=\delta (y-ax),a\ne 0\\ m_x=a^{-1}{m}_y\\ {\vartheta }_x=a^{-2}{\vartheta }_y\end{array}$
	Multiplication (backward)
5		$\begin{array}{l}f(x,y,z)=\delta (x-y)\delta (x-z)\\ m_z={\vartheta }_z({\vartheta }_x^{-1}{m}_x+{\vartheta }_y^{-1}{m}_y)\\ {\vartheta }_z={\vartheta }_x{\vartheta }_y{\left({\vartheta }_x+{\vartheta }_y\right)}^{-1}\end{array}$
	Equality
6		$\begin{array}{l}f(x)=\mathcal{N}(m,\vartheta )\\ m_x=m\\ {\vartheta }_x=\vartheta \end{array}$
	Gaussian