Cognitive Flexibility as a Latent Structural Operator

When the model itself is wrong — adapt the representation, not just the parameters.

Detect structural mismatch, Select the most predictively consistent representation, and Preserve Bayesian well-posedness — always.

Cognitive Flexibility for Bayesian State Estimation

Belief → Innovation Score → CF Rule → Structure Selection → Bayesian Update

CF acts solely at the representation level — the Bayesian filtering recursion remains structurally unchanged.

Cognitive Flexibility (CF) is a belief-level mechanism for online latent-structure selection in Bayesian filtering under structural mismatch. Rather than adapting parameters within a fixed model class, CF selects at each step the latent structure that minimises an innovation-based predictive score, leaving the underlying Bayesian recursion intact.

What Is Structural Mismatch?

In Bayesian filtering, the posterior belief \(\mathfrak{B}_t\) evolves under a parameterised model class. When the true dynamics lie outside this class, parameter adaptation alone cannot restore predictive consistency — the belief remains well-posed but becomes systematically misaligned with the true data-generating process. This is structural mismatch: an intrinsic failure mode that cannot be eliminated by tuning \(\theta\).

⚙ The CF Pipeline

Fig. 1. The CF pipeline as a latent structural operator. Dashed regions correspond to the three analytical layers: well-posedness (Layer 1), mechanism (Layer 2), and consequences (Layer 3).

At each time step, the innovation scores \(\{\Phi(\mathfrak{B}_t,s)\}_{s\in\mathcal{S}}\) are evaluated against the current belief \(\mathfrak{B}_t\) and passed to the CF rule, which selects \(s_{t+1}\) and parameterises the Bayesian update \(\mathfrak{B}_{t+1}= \mathcal{F}_{\theta,s_{t+1}}(\mathfrak{B}_t,u_t,y_{t+1})\). The belief update remains fully Bayesian; CF acts only through the structural update.

📐 Three-Layer Theoretical Analysis

Layer 1 — Well-posedness: The coupled belief–structure recursion is well posed and forward invariant on \(\mathcal{P}(\mathcal{Z})\times\mathcal{S}\). Structural mismatch is irreducible: no parameter adaptation can eliminate it.
Layer 2 — Mechanism: CF satisfies a one-step structural descent property, switches finitely under persistent score separation, and defines a hybrid dynamical system on \(\mathcal{P}(\mathcal{Z})\times\mathcal{S}\).
Layer 3 — Consequences: CF reduces to standard Bayesian filtering once the optimal structure is identified, and is non-intrusive when the model is correctly specified.

🤖 Core Insight

CF enlarges the set of admissible belief trajectories from \(\mathcal{R}_s\) (fixed structure) to \(\mathcal{R}_{\mathrm{CF}}=\bigcup_{s\in\mathcal{S}}\mathcal{R}_s\), strictly expanding representational capacity beyond any fixed model class — and beyond what IMM filtering achieves through probabilistic mixing.

Impact and Applications

CF establishes a principled belief-level framework for representation adaptation in Bayesian state estimation. Its implications span:

Adaptive filtering under latent-dynamics and observation-model shifts
Learning-based control where the latent structure may become restrictive
Deep state-space models under distribution shift
Model predictive control under structural mismatch

Reproducibility

All Julia code for numerical experiments is available at thanana.github.io.

exp1_cf_benchmark_corrected.jl — Experiment 4.1 (dynamics mismatch)
exp2_cf_benchmark_corrected.jl — Experiment 4.2 (observation shift)
exp3_cf_benchmark_corrected.jl — Experiment 4.3 (negative control)
automatica_exp44.jl — Experiment 4.4 (2D latent state)

Publication

Cognitive Flexibility as a Latent Structural Operator for Bayesian State Estimation

When the model is wrong, change the representation — not just the parameters.