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Let $x_{0:t}^{(i)}$ be the states from time $0$ to $t$ from sample $i$. Similarly for the observations $y_{1:t}$.

The normalized weights are updated according to

enter image description here

Where does the term $p(y_t|x_t^{(i)})$ come from? $\pi$ is the proposal/importance distribution, and $p$ the true distribution.

From the unnormalized weight formula:

$$\omega^{(i)}_t=\frac{p(x_{0:t}^{(i)}|y_{1:t})}{\pi(x_{0:t}^{(i)}|y_{1:t})}= \frac{p(x_{0:t-1}^{(i)}|y_{1:t})p(x_{t}^{(i)}|x_{0:t-1}^{(i)},y_{1:t})}{\pi(x_{0}^{(i)})\prod_k^{t-1} \pi(x_{k}^{(i)}|x_{0:k-1}^{(i)},y_{1:k})\pi(x_{t}^{(i)}|x_{0:t-1}^{(i)},y_{1:t})}=\omega^{(i)}_{t-1}\frac{p(x_{t}^{(i)}|x_{t-1}^{(i)})}{\pi(x_{t}^{(i)}|x_{0:t-1}^{(i)},y_{1:t})}$$

But I fail to get that term

An old man in the sea.
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  • I'll leave this question here, but I've solved this. There's a mistake since the weight at time $t-1$ should have the term $p(x_{0:t-1}|y_{1:t-1})$ not $p(x_{0:t-1}|y_{1:t})$. Also, using a recursive formula for $p(x_{0:t}|y_{1:t})$ we get equation (10) in http://cecas.clemson.edu/~ahoover/ece854/lecture-notes/lecture-import-samp.pdf – An old man in the sea. Aug 10 '18 at 12:55

2 Answers2

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Something like this: \begin{align*} \omega^{(i)}_t &=\frac{p(x_{0:t}^{(i)}|y_{1:t})}{\pi(x_{0:t}^{(i)}|y_{1:t})} \\ &\propto \frac{p(x_{0:t}^{(i)}, y_{1:t})}{\pi(x_{0:t}^{(i)}|y_{1:t})} \\ &= \frac{p(y_t \mid x_{1:t}^{(i)},y_{0:t-1}) p(x_{t}^{(i)}\mid x_{0:t-1}^{(i)}, y_{1:t-1})p(x_{0:t-1}^{(i)}, y_{1:t-1})}{\pi(x_{0:t}^{(i)}|y_{1:t})} \\ &= \frac{p(y_t \mid x_{t}^{(i)}) p(x_{t}^{(i)}\mid x_{t-1}^{(i)})p(x_{0:t-1}^{(i)}, y_{1:t-1})}{\pi(x_{0:t}^{(i)}|y_{1:t})} \\ &= \frac{p(y_t \mid x_{t}^{(i)}) p(x_{t}^{(i)}\mid x_{t-1}^{(i)})p(x_{0:t-1}^{(i)}, y_{1:t-1})}{\pi(x_{t}^{(i)} \mid x_{0:t-1}^{(i)},y_{1:t})\pi(x_{0:t-1}^{(i)}|y_{1:t-1})} \\ &= \frac{p(y_t \mid x_{t}^{(i)}) p(x_{t}^{(i)}\mid x_{t-1}^{(i)})}{\pi(x_{t}^{(i)} \mid x_{0:t-1}^{(i)},y_{1:t})} \omega^{(i)}_{t-1} \end{align*}

Taylor
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$$w(x_{0:t}^{(i)})=\frac{p(x_{0:t-1}^{(i)}|y_{1:t-1})p(y_t|x_{t}^{(i)})p(x_{t}^{(i)}|x_{t-1}^{(i)})}{\pi(x_{0:t-1}^{(i)}|y_{1:t-1})\pi(x_{t}^{(i)}|x_{0:t-1}^{(i)},y_{1:t-1})p(y_t|y_{t-1})}\propto w(x_{0:t}^{(i)}) \frac{p(y_t|x_{t}^{(i)})p(x_{t}^{(i)}|x_{t-1}^{(i)})}{\pi(x_{t}^{(i)}|x_{0:t-1}^{(i)},y_{1:t-1})}$$

An old man in the sea.
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