bayespy.inference.vmp.nodes.gaussian.GaussianDistribution.compute_gradient

GaussianDistribution.compute_gradient(g, u, phi)[source]

Compute the standard gradient with respect to the natural parameters.

Gradient of the moments:

\mathrm{d}\overline{\mathbf{u}} &= \begin{bmatrix} \frac{1}{2} \phi_2^{-1} \mathrm{d}\phi_2 \phi_2^{-1} \phi_1 - \frac{1}{2} \phi_2^{-1} \mathrm{d}\phi_1 \\ - \frac{1}{4} \phi_2^{-1} \mathrm{d}\phi_2 \phi_2^{-1} \phi_1 \phi_1^{\mathrm{T}} \phi_2^{-1} - \frac{1}{4} \phi_2^{-1} \phi_1 \phi_1^{\mathrm{T}} \phi_2^{-1} \mathrm{d}\phi_2 \phi_2^{-1} + \frac{1}{2} \phi_2^{-1} \mathrm{d}\phi_2 \phi_2^{-1} + \frac{1}{4} \phi_2^{-1} \mathrm{d}\phi_1 \phi_1^{\mathrm{T}} \phi_2^{-1} + \frac{1}{4} \phi_2^{-1} \phi_1 \mathrm{d}\phi_1^{\mathrm{T}} \phi_2^{-1} \end{bmatrix} \\ &= \begin{bmatrix} 2 (\overline{u}_2 - \overline{u}_1 \overline{u}_1^{\mathrm{T}}) \mathrm{d}\phi_2 \overline{u}_1 + (\overline{u}_2 - \overline{u}_1 \overline{u}_1^{\mathrm{T}}) \mathrm{d}\phi_1 \\ u_2 d\phi_2 u_2 - 2 u_1 u_1^T d\phi_2 u_1 u_1^T + 2 (u_2 - u_1 u_1^T) d\phi_1 u_1^T \end{bmatrix}

Standard gradient given the gradient with respect to the moments, that is, given the Riemannian gradient \tilde{\nabla}:

\nabla = \begin{bmatrix} (\overline{u}_2 - \overline{u}_1 \overline{u}_1^{\mathrm{T}}) \tilde{\nabla}_1 + 2 (u_2 - u_1 u_1^T) \tilde{\nabla}_2 u_1 \\ (u_2 - u_1 u_1^T) \tilde{\nabla}_1 u_1^T + u_1 \tilde{\nabla}_1^T (u_2 - u_1 u_1^T) + 2 u_2 \tilde{\nabla}_2 u_2 - 2 u_1 u_1^T \tilde{\nabla}_2 u_1 u_1^T \end{bmatrix}