Performance Evaluation of Circular Inference

To measure performance, in Figs. 2I, 3 B and D and 4 G and H, we computed the circular average distance (72 ) of the estimate μ_T from the true HD ϕ_T at the end of a simulation of length T = 20 from P = 5′000 simulated trajectories by $m_1=\frac{1}{P}\sum _{k=1}^{P}exp\left(i\left({\mu }_T^{(k)}-{\varphi }_T^{(k)}\right)\right)$ . The absolute value of the imaginary-valued circular average, 0 ≤ |m₁|≤1, is unitless and denotes an empirical accuracy (or “inference accuracy”) and thus measures how well the estimate μ_T matches the true HD ϕ_T. Here, a value of 1 denotes an exact match. The reported inference accuracy is related to the circular variance via Var_circ = 1 − |m₁|. In SI Appendix, Fig. S5, we provide histograms with samples μ_T − ϕ_T with different numerical values of |m₁| to provide some intuition for the spread of estimates for a given value of the performance measure.
We estimated performance through such averages for a range of HD observation information rates in Figs. 2I, 3B and 4G. This information rate is a simulation time-step size-independent quantity, which measures the Fisher information that HD observations provide about true HD per unit time. For individual HD observations of duration dt, Eq. 6, this Fisher information approaches I_zt(ϕ_t)→(κ_zdt)²/2 with dt → 0 (31 , Theorem 2]. Per unit time, we observe 1/dt independent observations, leading to a total Fisher information (or information rate) of γ_z = κ_z²dt/2. As in simulations, γ_z needs to remain constant with changing Δt to avoid increasing the amount of provided information, the HD observation reliability κ_z needs to change with the size of simulation time-step size Δt. To keep our plots independent of this time-step size, we thus plot performance as a function of the HD observation information rate rather than κ_z. For the inset of Fig. 3B, and for Figs. 3 D and F, we additionally performed a grid search over the fixed-point κ^* (Fig. 3B, inset) or both the fixed-point κ^* and of the decay speed β (Figs. 3 D and F). For each setting of κ^* and β, we assessed the performance by computing an average over this performance for a range of HD observation information rates, weighted by how likely each observation reliability is assumed to be a priori. The latter was specified by a log-normal prior, p(γ_z)=Lognormal(γ_z|μ_γz, σ_γz²), favoring intermediate reliability levels. We chose μ_γz = 0.5 and σ_γz² = 1 for the prior parameters, but our results did not strongly depend on this parameter choice. The performance loss shown in Fig. 3D also relied on such a weighted average across information rates γ_z for a particle filter benchmark (PF, SI for details). The loss itself was then defined as $1-\frac{\text{Performance}}{\text{Performance PF}}$ .