Robust Cognitive–Flexible Filtering
under Noisy Innovation Scores

What Is the Problem?

Cognitive Flexibility (CF) selects, at each step, the latent structure \(s\in\mathcal{S}\) minimising an innovation-based predictive score \(\Phi_t(s):=-\log\ell_{\theta,s}(y_{t+1}|\mathfrak{B}_t,u_t)\), leaving the underlying filter recursion intact. Under exact scores, three guarantees hold: descent in expectation, finite switching, and non-chattering.

In practice scores are estimated by a particle filter and carry additive perturbations \(\epsilon_t(s)\) with \(\mathbb{E}[\epsilon_t(s)\mid\mathcal{I}_t]=0\) and \(|\epsilon_t(s)|\leq\bar{\varepsilon}\) a.s. Unprotected score minimisation is vulnerable to spurious switching — and any margin \(\delta\) calibrated to suppress it must, in a nonstationary setting, be calibrated against a noise level that keeps changing. This letter asks — and answers — two questions: does the theory survive noisy scores, and can the margin itself be found online, without a stationary calibration window?

⚙ The Margin-Based Rule

We introduce the margin-based switching rule:

\[ s_{t+1} = \begin{cases} \hat{s}_t, & \hat{\Phi}_t(s_t) - \hat{\Phi}_t(\hat{s}_t) > \delta_t, \\ s_t, & \text{otherwise,} \end{cases} \quad \hat{s}_t\in\arg\min_{s}\hat{\Phi}_t(s), \]

with \(\delta_t\ge2\varepsilon_t+\underline{m}\) for a deterministic floor \(\underline{m}>0\). The static case \(\delta\equiv\text{const}>2\bar{\varepsilon}\) is recovered exactly as a special case, and restores all three noiseless-style guarantees with no change to the filter recursion.

📐 Main Theoretical Guarantees

• Theorem 1 (Descent): \(\mathbb{E}[\Phi_t(s_{t+1})\mid\mathcal{I}_t] \leq\mathbb{E}[\Phi_t(s_t)\mid\mathcal{I}_t]\) for all \(t\geq0\) — the expected score never increases after a switch, regardless of noise.
• Theorem 2 (Finite expected switching): \[ \mathbb{E}[N_T]\leq\frac{1}{\delta-2\bar{\varepsilon}} \sum_{t=0}^{T-1} \mathbb{E}\!\left[\Phi_t(s_t)-\min_s\Phi_t(s)\right] \]
• Theorem 3 (Non-chattering): if \(\limsup_{t\to\infty}[\Phi_t(s_t)-\min_s\Phi_t(s)] <\delta-2\bar{\varepsilon}\) a.s., switching stops a.s.
• Theorem 4 (adaptive margin, new): replacing \(\bar{\varepsilon},\delta\) by \(\mathcal{I}_t\)-adapted \(\varepsilon_t,\delta_t\), all three guarantees persist whenever the effective margin \(\delta_t-2\varepsilon_t\) stays above a floor. The operative quantity was never \(\delta\) itself.

⚡ Zero-Overhead Online Calibration

A sliding window across time cannot estimate \(\varepsilon_t\) safely: its variability mixes genuine dynamic variation of the state with Monte Carlo error, and so overestimates the noise exactly when the system is most nonstationary. We instead exploit variability within a single time step: partition the existing \(N_p\) particles into \(B\) independently resampled batches. All batches see the same belief and observation, so their spread is pure Monte Carlo noise, invariant to the dynamics.

\[ \hat{\varepsilon}_t=\frac{\kappa}{\sqrt{B}} \max_{s\in\mathcal{S}}\operatorname{std}_{b\leq B} \{\hat{\Phi}_t^{(b)}(s)\}, \qquad \varepsilon^{\mathrm{env}}_t=\max\{\hat{\varepsilon}_t,\, \lambda\,\varepsilon^{\mathrm{env}}_{t-1}\}, \qquad \delta_t=\max\{(2+\eta)\varepsilon^{\mathrm{env}}_{t-1}, \delta_{\min}\}. \]

The peak-hold envelope rises immediately when the noise grows and relaxes only geometrically when it falls — the correct asymmetry for a quantity that must upper-bound a perturbation. Because the batches partition an unchanged particle budget, the scheme adds zero extra Monte Carlo cost: only \(O(BK)\) scalar operations per step.

Numerical Experiments

We validate on the Li–Jilkov constant-velocity / coordinated-turn (CV/CT) manoeuvring target benchmark, a standard testbed for multiple-model tracking — not a synthetic scalar model. The target state is \(x_t=[p_x,\dot{p}_x,p_y,\dot{p}_y]^\top\in\mathbb{R}^4\), where \(p_x,p_y\) are position components in the horizontal plane and \(\dot{p}_x,\dot{p}_y\) the corresponding velocities, following \(x_{t+1}=F_tx_t+w_t\), \(w_t\sim\mathcal{N}(0,Q)\). The true structure switches as CV~(\(t{<}30\)) \(\to\) CT~(\(30{\leq}t{<}70\)) \(\to\) CV~(\(t{\geq}70\)).

Two candidate sets are used: the binary \(\mathcal{S}_2=\{s_\mathrm{CV},s_\mathrm{CT}\}\), and the four-model dictionary \(\mathcal{S}_4\) adding left/right coordinated turns and a near-constant-acceleration model, chosen so that several candidates stay plausible throughout the manoeuvre.

Crucially, score noise is not injected: scores are genuine particle-filter estimates, and nonstationarity is imposed physically by a particle-budget schedule — \(N_p=4000\) except \(N_p=60\) on \(t\in[40,75)\), as a radar resource manager reallocating effort across tracks would. This moves the noise scale directly: the 95th percentile of \(\hat{\varepsilon}_t\) rises from \(1.93\) to \(5.92\), a factor of \(3.07\). The schedule places the recovery transition at \(t=70\) inside the reduced-budget window, so a genuine structural change must be detected exactly when the scores are least reliable.

Table I: Switching and Detection Metrics

\(\mathbb{E}[N_T]\): switches; FA: false alarms outside transition windows; \(D_1\): detection delay at \(t=30\); ACC: selection accuracy. \(\mathbb{E}[N_T]\) alone is misleading — the largest fixed margin minimises it purely by failing to react.

Table II: Removing the Calibration Stage

Selection accuracy on \(\mathcal{S}_2\), \(M=50\) per point. The adaptive rule is significantly better at full budget and never worse in any regime tested.

Impact and Applications

• Adaptive state estimation under sensor noise and model uncertainty
• Particle filter-based systems where score approximation errors are unavoidable
• Manoeuvring target tracking with resource-managed (time-varying) particle budgets
• Neural-network-aided filtering (KalmanNet, Split-KalmanNet, Latent-KalmanNet) where learned scores carry structured, nonstationary errors
• Supervisory control and fault detection under drifting noise levels

Reproducibility

Julia source code is publicly available:

• radar_final_v6b_SPLrevision_2_1.jl — Li–Jilkov CV/CT benchmark, batch-split estimator, adaptive margin, three-regime sweep (Fig. 1, Tables I–II)

📦 Full repository