\to 0$, because the measurements are performed at all times.

We write as $Y_n$ the set of measurement outcomes that are obtained up to time $t_n$, i.e. $Y_n := (y_{n_1}, y_{n_2}, \cdots, y_{[n]})$ where $[n]$ is the maximum $n_k$ satisfying $n_k \leq n$. If the measurement is continuous, then $Y_n = (y_0, y_1, \cdots, y_n)$.

We define $$P_{\rm c}[Y_n | X_n] := \prod_{k=1}^{M'} P[y_{n_k} | X_{n_k}],
\label{c}$$ where $M'$ is the maximum integer satisfying $n_{M'} \leq n$. Without feedback, Eq. (\[c\]) defines the conditional probability of obtaining outcomes $Y_n$ under the condition of $X_n$, while, with feedback, this interpretation of Eq. (\[c\]) is not necessarily correct as shown in the next section. To explicitly demonstrate this point and to distinguish $P_{\rm c}[Y_n | X_n]$ from the usual conditional probability, we put suffix “c”. Then, the joint distribution of $X_n$ and $Y_n$ is given by $$P[X_n, Y_n] = P_{\rm c}[Y_n | X_n] P[X_n].$$ The probability of obtaining outcomes $Y_n$ is given by $$\begin{aligned}
P[Y_n ] = \int dX_n P[X_n, Y_n] = \prod_{k=1}^{M'} P[y_{n_k} | Y_{n_k-1}],
\label{prob0}\end{aligned}$$ where the two equalities are just identities known in probability theory. We also note that $$\begin{split}
P[y_n | X_n, Y_{n-1}] &:= \frac{P[Y_n | X_n]}{P[Y_{n-1} | 