Corrections

Given a predicted value $P(X_t|y_0, ..., y_{t-1})$ and the current observation $y_t$, we want to compute $P(X_t | y_0, ..., y_{t-1}, y_t)$ $$P(X_t | y_0, ..., y_{t-1}, y_t) \\ = \frac{P(y_t | X_t,y_0, ..., y_{t-1}) \cdot P(X_t | y_0, ..., y_{t-1})}{P(y_t|y_0, ..., y_{t-1})} \\ = \frac{P(y_t | X_t) \cdot P(X_t | y_0, ..., y_{t-1})}{P(y_t|y_0, ..., y_{t-1})} \\ = \frac{P(y_t | X_t) \cdot P(X_t | y_0, ..., y_{t-1})}{\int P(y_t|X_t)P(X_t | y_0, ..., y_{t-1})dX_t}$$ As we'll see, the dominator is just a normalized factor that ensures the probabilities sum to 1, so we'll never need to worry about it explicitly.