#hierarchical-softmax

Hierarchical softmax over a 4-word vocabulary uses a balanced binary tree: a root, a left inner node with leaves $w_1, w_2$ and a right inner node with leaves $w_3, w_4$. Stepping to the left child of node $n$ multiplies by $\sigma(\mathbf{u}_n^\top\mathbf{v}_c)$, stepping to the right child by $\sigma(-\mathbf{u}_n^\top\mathbf{v}_c)$. For a center word with $\mathbf{u}_\textrm{root}^\top\mathbf{v}_c = 1.0$ and $\mathbf{u}_\textrm{right}^\top\mathbf{v}_c = 0.5$, what is $P(w_3 \mid w_c)$? Round to 3 decimals.