Parameter Estimation Methods

1 Maximum Likelihood Estimation

For the energy metabolism model, parameters are estimated by minimizing the weighted residual sum of squares between model predictions and CPET-derived measurements (peak VOβ‚‚, anaerobic threshold, lactate kinetics). The objective function is:

\[ \mathcal{L}(\mathbf{\theta}) = -\frac{1}{2} \sum_{i=1}^{N} lr([ \frac{(y_i - \hat{y}_i (\mathbf{\theta}))^2}{\sigma_i^2} + \log(2 \pi \sigma_i^2) ]) \tag{1}\]

where \(y_i\) are observed values, \(\hat{y}_i (\mathbf{\theta})\) are model predictions at parameter vector \(\mathbf{\theta}\), and \(\sigma_i\) are measurement standard deviations. Optimization uses the Nelder–Mead simplex algorithm for initial exploration followed by gradient-based refinement (L-BFGS-B with box constraints on biologically plausible parameter ranges).

2 Bayesian Inference

Bayesian parameter estimation uses informative priors derived from published literature. For each parameter \(\theta_j\), the prior distribution encodes existing knowledge:

  • Well-characterized parameters (enzyme \(K_m\) values, degradation rates): Log-normal priors with mean and variance from published enzymology
  • Moderately characterized parameters (impairment factors \(\alpha_\text{CI}\), \(\beta\)): Uniform priors on biologically plausible ranges
  • Poorly characterized parameters (symptom weighting coefficients): Weakly informative half-normal priors

Posterior sampling uses the No-U-Turn Sampler (NUTS), a variant of Hamiltonian Monte Carlo that automatically tunes the step size and trajectory length. Convergence is assessed by the \(\hat{R}\) statistic (\(\hat{R} < 1.01\) required) and effective sample size (\(n_\text{eff} > 400\) per chain). Four independent chains of 2000 post-warmup samples each are used.