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add missing partial
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yoyolicoris committed Apr 15, 2024
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### Gradients for the initial condition $`y_t|_{t \leq 0}`$

The initial conditions provide an entry point at $t=1$ for filtering, as we cannot evaluate $t=-\infty$.
Let us assume $`A_{t, :}|_{t \leq 0} = 0`$ so $`y_t|_{t \leq 0} = x_t|_{t \leq 0}`$, which also means $`\frac{\partial \mathcal{L}}{y_t}|_{t \leq 0} = \frac{\partial \mathcal{L}}{x_t}|_{t \leq 0}`$.
Let us assume $`A_{t, :}|_{t \leq 0} = 0`$ so $`y_t|_{t \leq 0} = x_t|_{t \leq 0}`$, which also means $`\frac{\partial \mathcal{L}}{\partial y_t}|_{t \leq 0} = \frac{\partial \mathcal{L}}{\partial x_t}|_{t \leq 0}`$.
Thus, the initial condition gradients are

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