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fix two small typos
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HumphreyYang committed Jul 14, 2024
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8 changes: 4 additions & 4 deletions lectures/calvo_gradient.md
Original file line number Diff line number Diff line change
Expand Up @@ -953,25 +953,25 @@ $$
Then we have:
$$
\sum_{t=0}^\infty \beta^t \theta_t = \vec{\mathbf{1}}(\vec{\beta} \cdot (B \cdot \vec{\mu}))
\sum_{t=0}^\infty \beta^t \theta_t = \vec{\mathbf{1}}(\vec{\beta} \cdot (B \vec{\mu}))
$$
and
$$
\sum_{t=0}^\infty \beta^t \theta_t^2 = \vec{\beta} \cdot (\vec{\mu}^T B^T) \cdot (B \vec{\mu})
\sum_{t=0}^\infty \beta^t \theta_t^2 = \vec{\beta} \cdot (\vec{\mu}^T B^T) (B \vec{\mu})
$$
and
$$
\sum_{t=0}^\infty \beta^t \mu_t^2 = \vec{\mu}^T \vec{\beta} \vec{\mu}
\sum_{t=0}^\infty \beta^t \mu_t^2 = \vec{\beta} \cdot \vec{\mu}^T \vec{\mu}
$$
It follows that
$$
J = V - h_0 = \sum_{t=0}^\infty \beta^t (h_1 \theta_t + h_2 \theta_t^2 - \frac{c}{2} \mu_t^2) = h_1 \cdot \vec{\mathbf{1}} (\vec{\beta} \cdot (B \cdot \vec{\mu})) + h_2 \cdot \vec{\beta} \cdot (\vec{\mu}^T B^T) \cdot (B \vec{\mu}) - \frac{c}{2} \cdot \vec{\mu}^T \vec{\beta} \vec{\mu}
J = V - h_0 = \sum_{t=0}^\infty \beta^t (h_1 \theta_t + h_2 \theta_t^2 - \frac{c}{2} \mu_t^2) = h_1 \cdot \vec{\mathbf{1}} (\vec{\beta} \cdot (B \vec{\mu})) + h_2 \cdot \vec{\beta} \cdot (\vec{\mu}^T B^T)(B \vec{\mu}) - \frac{c}{2} \cdot \vec{\beta} \cdot \vec{\mu}^T \vec{\mu}
$$
So
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