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Figure 1: Time evolution of an unobserved particle with mass $m$ = 1 started at time $t_0$ = 0 from $x_0$ = 0 in the potential (16). Shown is the transition probability $p(x\vert\Delta t = t,x_0=1,v_{\rm true})$.

Figure 2: Time evolution of an observed particle with mass $m$ = 1 in the potential (16). The figure shows for each data point $i$ the probability $p(x\vert\Delta_i=5,x_{i-1},v_{\rm true})$, starting from $x_0$ = 0. (Hence, the probability at $i$ =1 corresponds to that shown in Fig. 1 at $t$ = 5.) The actual data points $x_i$ have been sampled from that probabilities and form the observed path shown on top as a thick line.

Figure 3: Numerical reconstruction of a potential from 50 coordinate measurements (see Fig. 2). Shown are the true potential $v_{\rm true}$ (thin line), the best parametric approximation used as reference potential $v_0$ (dashed line), and the reconstructed potential $v_{\rm ITDQ}$ (thick line). Parameters: $m$ = 1, $\Delta _i$ = 5, $v_{\rm true}$ of Eq. (16), $v_0$ of the form (18), Gaussian prior with ${\bf K}_0$ as in (17) with $\lambda$ = 0.1 and $\sigma _{0}$ = 3, $p_E$ (6) with $\mu$ = 10 and $\kappa$ = $E_0(v_{\rm true})$, periodic boundary conditions for $\psi _\alpha$, fixed boundary values $v(-10)$ = $v(10)$ = $10^5$ for $v_{\rm ITDQ}$, $v_0$ and $v_{\rm true}$, calculated on a lattice with 21 points. Errors: $\epsilon_D(v_{\rm ITDQ})$ = 99.1, $\epsilon_D(v_{\rm true})$ = 104.4, $\epsilon_g(v_{\rm ITDQ})$ = 1.891, $\epsilon_g(v_{\rm true})$ = 1.818.

Figure 4: Sum of empirical transition probabilities $p_{\rm emp}$ (bars), the corresponding true $p_{\rm true}$ (thick line) for $v_{\rm true}$, and the reconstructed $p_{\rm ITDQ}$ (thin line).

Figure 5: Same functions as in Fig. 4, but restricted to measurements of a particle which has been at position $x_{i-1}$ = 1 at the time of the previous measurement.