(Y_{n-1})] := \prod_{k=0}^{k-1} P[x_{k+1} | X_k, \lambda_k (Y_k)].$$ On the other hand, along the trajectory $X_n$, the conditional probability of obtaining outcome $y_n$ at time $t_n$ is written as $P[y_n | X_n]$. We then define $$P_{\rm c}[Y_n | X_n] := \prod_{k=0}^{n-1} P[y_k | X_k],$$ which is to be compared with Eq. (\[c\]).

We then obtain the joint probability distribution of $X_n$ and $Y_n$ with feedback control as $$\begin{split}
P[X_n, Y_n] &= \prod _{k=0}^{n-1} P[y_{k+1} | X_{k+1}] P[x_{k+1} | X_{k}, \lambda_{k} (Y_{k-1})] \\
&= P_{\rm c}[Y_n | X_n] P [X_n | \Lambda_n (Y_{n-1})]. 
\end{split}
\label{joint_prob}$$ We can check that $$\int P[X_n, Y_n] dX_n dY_n = 1,$$ by integrating $X_n$ and $Y_n$ in Eq. (\[joint\_prob\]) in the order of $y_n \to x_n \to y_{n-1} \to x_{n-1} \to \cdots \to y_1 \to x_1 \to y_0 \to x_0$, where the causality of measurements and feedback play crucial roles.

The marginal distributions are given by $$P[X_n] = \int P[X_n, Y_n] dY_n, \ P[Y_n] = \int P[X_n, Y_n] dX_n,$$ and the conditional distributions by $$P[X_n | Y_n] = \frac{P[X_n, Y_n]}{P[Y_n]},  \ P[Y_n | X_n] = \frac{P[X_n, Y_n]}{P[X_n]}.$$ We stress that, in the presence of feedback control, $$P[Y_n | X_n] \neq P_{\rm c}[Y_n | X_n]$$ in general, 