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On this webage concerning the likelihood function of the Kalman filtering, the following lines of equations can be found:

\begin{align} P[y_2\mid y_1]&=\int P[y_2,x_2\mid y_1]dx_2\\ &= \int P[y_2 \mid x_2]P[x_2 \mid y_1]dx_2 \end{align}

For some reason, I cannot replicate the second line - i.e.,transition from $P[y_2,x_2\mid y_1]$ to the product $P[y_2 \mid x_2]P[x_2 \mid y_1]$. I tried playing around with Bayes Theorem to no avail. How was this derived?

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In general, you have: $$ P[y_2, x_2\;|\; y_1] = P[y_2\;|\; x_2, y_1] P[x_2\;|\; y_1], $$ which follows from the definition of conditional probability.

But for Kalman filters, the observation $y_2$ is independent of the observation $y_1$ given the state $x_2$, i.e.: $$ P[y_2\;|\; x_2, y_1] = P[y_2\;|\; x_2]. $$ This results in your identity: $$ P[y_2, x_2\;|\; y_1] = P[y_2 \mid x_2]P[x_2 \mid y_1]. $$

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