Nested conditional Gaussian random variables for Kalman Filters

Question

I am trying to follow along the Kalman Filter notes from Stanford (EE 363). https://stanford.edu/class/ee363/lectures/kf.pdf

On Slide 15, when discussing the measurement update, the notes mention that

$x_t | Y_t$ is equal to $(x_t | Y_{t-1})|(y_t|Y_{t-1}) $

I am having trouble proving this, I tried using the tower property but did not go far.

For completeness, $x_t, Y_t$ are jointly Gaussian random variables. And $x_t|Y_{t−1}$ and $y_t|Y_{t−1}$ are also jointly Gaussian. And $y_t$ is the observation of Y at time t (if I am not mistaken).

Any help is much appreciated.

Answer 1

$(x_t | Y_{t-1})|(y_t|Y_{t-1})$ is not defined. There is no definition for "double" conditioning like this.

Perhaps your instructor means

$$ p(x_t \mid Y_{t}) = \frac{p(y_t \mid x_t) p(x_t \mid Y_{t-1})}{p(y_t \mid Y_{t-1})}, $$ which is Bayes' rule.