Practical Identifiability of a Minimal Adaptive Learning-Gain Model for Binary Feedback: A Simulation Study

Abstract

Social learning is jointly determined by several computationally distinct quantities, namely baseline belief (prior mean), sensitivity to feedback (learning gain), and the temporal uncertainty of belief (drift), which are rarely separated in empirical designs. Here we systematically characterise the practical identifiability and predictive adequacy of a minimal four-parameter adaptive learning-gain model of binary feedback: a latent belief v_t about the probability of a positive outcome is carried forward deterministically between observations while an internal confidence-precision state decays by a per-trial amount omega, and the belief is updated by a learning gain lambda from a deterministic prior mean mu with initial confidence precision sigma; sigma, omega and lambda are model-defined internal quantities, not validated psychological constructs. The simulated process is self-reinforcing: the same latent state both generates the binary outcome and represents the learner's belief, so the model does not represent inference about an independently evolving social environment. The study is therefore an analysis of a phenomenological adaptive filter under a specified binary stochastic process: its update equations are deliberately chosen rather than derived from the Bernoulli-logistic likelihood, and it does not reduce to a standard hierarchical Gaussian filter. Using fixed-seed simulation studies, we report four findings. First, joint maximum-likelihood recovery is poor. On the 24-condition parameter grid at T=80 (100 independent replications per condition), the belief mean mu shows comparatively smaller errors but is still not reliably recovered; sigma and omega show near-zero tolerance-based recovery that stays at most 0.002 across tolerance thresholds of 0.3, 0.6 and 1.0 transformed units, whereas lambda recovery is low but not near-zero and increases with tolerance (0.078, 0.182 and 0.263 at tolerances 0.3, 0.6 and 1.0). Boundary convergence is strongest for sigma (upper bound) and omega (lower bound), whereas lambda boundary solutions are not uniform across grid conditions. In a separate single-parameter-vector experiment that varies trial length (T in {40, 80, 120, 240, 480}), extending the sequence alone does not repair recovery of the positive parameters, including lambda. Tolerance recovery concerns point estimates only; calibrated interval coverage is not evaluated and is not substituted by a flat profile or a boundary estimate. Second, profile-likelihood and trade-off diagnostics show flat likelihood regions for sigma and omega, with an asymmetric profile for lambda that constrains its upper range but is uninformative over its low range; the flat sigma-omega geometry replicates in a second, low-gain regime (a within-study replication, not a claim across the full condition grid), whereas the exact steepness of the lambda profile is regime-dependent. Three optimisers converge to the same boundary solution, and the fitted estimates cannot reliably rank individual simulated subjects either. These patterns support weak practical identifiability under the tested design, not structural non-identifiability; no formal structural identifiability analysis is performed, so a structural claim is not made. Third, a fitting-parity audit found and corrected a return-order defect in the fixed-gain candidate: its fitted parameters were returned with sigma and omega interchanged, which artificially raised that candidate's fitted block loss. Under the corrected fit, overall block discrimination is 67/150 = 0.45 under the original wider bounds and 72/150 = 0.48 under common fitting bounds matched to the proposed model (56 common forecast observations per dataset, each scored exactly once, with an automated index check confirming no fit-forecast overlap and no look-ahead), modestly above the 0.33 chance level; a calibrated permutation test that preserves the asymmetric winner marginals is significant (p = 0.0011 and p < 0.001). This above-chance accuracy is almost entirely attributable to Rescorla-Wagner (0.88-0.90 correct), whereas the adaptive filter and its nested fixed-gain restriction remain heavily confused (row-wise accuracies 0.18-0.22 and 0.24-0.36, neither significant), and the result is robust to the fitting bounds (row-wise p-values are descriptive and uncorrected; none of this establishes model equivalence). Fourth, across 50 independent held-out datasets the model shows no predictive advantage over a training-set empirical-frequency baseline (the paired 95% confidence interval spans zero under both normal-approximation and bootstrap estimation; the proportion of datasets won is not used as evidence of superiority), and a paired ablation shows no consistent predictive gain from the added update components and no improvement in parameter stability; no predictive advantage over this empirical-frequency baseline has therefore been demonstrated.