Skip to content

index.md

Latest commit

 

History

History
62 lines (44 loc) · 4.21 KB

index.md

File metadata and controls

62 lines (44 loc) · 4.21 KB
Raw

Estimation, Comparison, and Diagnostics

Technical Reference | Models | Data likelihood

BestFit is Bayesian-first: MCMC is the primary route for parameter and predictive uncertainty. MLE provides likelihood point estimates and local asymptotics; MAP provides a posterior mode; GMM is the only moments-based estimator and underlies the specialized Bulletin 17C path. These methods target different objectives and must not be interchanged merely because each returns a parameter vector.

Chapter Map

Chapter Target Primary output
Maximum likelihood IModel.DataLogLikelihood MLE, curvature/sandwich covariance, AIC/BIC
Maximum a posteriori IModel.LogLikelihood posterior mode and local Laplace curvature
Generalized method of moments moment quadratic plus optional penalty GMM estimate and asymptotic covariance
Bayesian MCMC full posterior kernel joint posterior draws and credible uncertainty
Model comparison likelihood or pointwise predictive criteria AIC, BIC, DIC, WAIC, verified PSIS-LOOIC
Predictive checks prior/posterior predictive simulation discrepancy p-values and replicate summaries
Diagnostics optimization and chain behavior status, trace interpretation, R-hat, ESS
Influence diagnostics pointwise likelihood/moments/prior curvature observation and prior sensitivity measures

Objective Crosswalk

For data log likelihood $\ell(\boldsymbol\theta)$, prior log kernel $r(\boldsymbol\theta)$, sample moment vector $\mathbf g_n$, weight $\mathbf W$, and optional penalty $P$:

$$ \begin{aligned} \text{MLE:}&& \max_{\boldsymbol\theta};&\ell(\boldsymbol\theta),\\ \text{MAP:}&& \max_{\boldsymbol\theta};&{\ell(\boldsymbol\theta)+r(\boldsymbol\theta)},\\ \text{MCMC:}&& \boldsymbol\theta^{(s)}\sim&;p(\boldsymbol\theta\mid\mathbf y) \propto\exp{\ell+r},\\ \text{GMM:}&& \min_{\boldsymbol\theta};&\mathbf g_n^{\mathsf T}\mathbf W\mathbf g_n \quad\text{or}\quad \tfrac12\mathbf g_n^{\mathsf T}\mathbf W\mathbf g_n+P. \end{aligned} \tag{EST.1} $$

MLE and MAP require IModel. GMM requires IGMMModel or moment delegates and does not imply a likelihood. Bulletin17CDistribution is an IGMMModel, not an IModel; its uncertainty ensembles are frequentist/asymptotic or bootstrap products, not posterior chains.

Required Reporting

Every fitted result should state:

  • model and exact parameterization, units, support, fixed parameters, and bounds;
  • objective decomposition and observation-information units;
  • estimator/sampler, starts, seeds, tolerances, iterations, and dependency versions;
  • termination and convergence evidence, including open diagnostic limitations;
  • uncertainty construction and whether it is asymptotic, bootstrap, or posterior;
  • predictive checks and sensitivity to scientifically material choices;
  • verification source and tolerance for every numerical claim.

Evidence Boundaries

MLE and MAP nuisance profiling, PSIS tail smoothing, covariance failure signaling, GMM Hansen J, overidentified one-step fitting and covariance, ARWMH realized-state adaptation, NUTS acceptance/gradient routing, rank-normalized R-hat and conservative bulk/tail ESS, and combined-influence terminology are implemented and verified. The legacy GMM PSIS-shaped compatibility overloads are explicitly obsolete. Joint-prior sampling remains a documented limitation. The chapters describe current implemented behavior and identify any output whose scientific interpretation remains limited.

Validation Boundary

The marked C# examples compile against the .NET 10/RMC.BestFit 2.0 API and establish API conformance. Independent numerical comparisons in RMC.BestFit.Verification support the specific estimation, recovery, coverage, and published-result claims identified in each chapter. A compiled example alone does not establish numerical accuracy.

Start with MLE | Bayesian MCMC | Diagnostics