Algorithm 1. Model parameter estimation using integrated MLE and Jeffreys prior.

1 1em

Data: Observed dataset $\mathcal{D}=\{(y_1,x_1),\dots ,(y_N,x_N)\}$

Result: Posterior model parameters ${\hat{\mathrm{\Theta }}}_{\mathrm{p}\mathrm{o}\mathrm{s}\mathrm{t}}$

2 Initialize starting values for parameters $\mathrm{\Theta }$;

3 Initialize regularization parameters ${\lambda }_1,{\lambda }_2$;

4 Phase One: MLE Parameter Estimation;

5 while MLE stopping criteria not met do

6   Compute the gradient of the objective function (Eq.(15));

7   Update $\mathrm{\Theta }$ using advanced optimization method LBFGS;

8   Update model complexity and adaptability metrics;

9 Use MLE results as initial values for Jeffreys Prior calculation: ${\mathrm{\Theta }}_{\mathrm{i}\mathrm{n}\mathrm{i}\mathrm{t}}=\hat{\mathrm{\Theta }}$;

10 Phase Two: Bayesian Parameter Estimation Using Jeffreys Prior;

11 Calculate Fisher Information Matrix $I({\mathrm{\Theta }}_{\mathrm{i}\mathrm{n}\mathrm{i}\mathrm{t}})$, see Eq.(16);

12 Calculate Jeffreys Prior $\pi (\mathrm{\Theta })$, see Eq. (17);

13 while Bayesian stopping criteria not met do

14   Compute the gradient of the posterior distribution using Eqs. (18) and (21);

15   Update using Metropolis-Hastings algorithm for MCMC;

16   Update ${\lambda }_1,{\lambda }_2$ to control model complexity;

17 Calculate optimal posterior parameters ${\hat{\mathrm{\Theta }}}_{\mathrm{p}\mathrm{o}\mathrm{s}\mathrm{t}}$, see Eq. (22);

18 return ${\hat{\mathrm{\Theta }}}_{\mathrm{p}\mathrm{o}\mathrm{s}\mathrm{t}}$