bayespy.inference.vmp.nodes.categorical.CategoricalDistribution.compute_gradient

CategoricalDistribution.compute_gradient(g, u, phi)

Compute the Euclidean gradient.

In order to compute the Euclidean gradient, we first need to derive the gradient of the moments with respect to the variational parameters:

\mathrm{d}\overline{u}_i = N \cdot \frac {e^{\phi_i} \mathrm{d}\phi_i \sum_j e^{\phi_j}} {(\sum_k e^{\phi_k})^2} - N \cdot \frac {e^{\phi_i} \sum_j e^\phi_j \mathrm{d}\phi_j} {(\sum_k e^{\phi_k})^2} = \overline{u}_i \mathrm{d}\phi_i - \overline{u}_i \sum_j \frac{\overline{u}_j}{N} \mathrm{d}\phi_j

Now we can make use of the chain rule. Given the Riemannian gradient \tilde{\nabla} of the variational lower bound \mathcal{L} with respect to the variational parameters \phi, put the above result to the derivative term and re-organize the terms to get the Euclidean gradient \nabla:

\mathrm{d}\mathcal{L} = \tilde{\nabla}^T \mathrm{d}\overline{\mathbf{u}} = \sum_i \tilde{\nabla}_i \mathrm{d}\overline{u}_i = \sum_i \tilde{\nabla}_i ( \overline{u}_i \mathrm{d}\phi_i - \overline{u}_i \sum_j \frac {\overline{u}_j} {N} \mathrm{d}\phi_j ) = \sum_i \left(\tilde{\nabla}_i \overline{u}_i \mathrm{d}\phi_i - \frac{\overline{u}_i}{N} \mathrm{d}\phi_i \sum_j \tilde{\nabla}_j \overline{u}_j \right) \equiv \nabla^T \mathrm{d}\phi

Thus, the Euclidean gradient is:

\nabla_i = \tilde{\nabla}_i \overline{u}_i - \frac{\overline{u}_i}{N} \sum_j \tilde{\nabla}_j \overline{u}_j

See also

compute_moments_and_cgf

Computes the moments \overline{\mathbf{u}} given the variational parameters \phi.