Probabilistic framework for gravity field estimation that provides rigorous uncertainty quantification through posterior distributions, combining prior knowledge with observational data.
\[p(\mathbf{x}|\mathbf{y}) = \frac{p(\mathbf{y}|\mathbf{x})p(\mathbf{x})}{p(\mathbf{y})} \propto p(\mathbf{y}|\mathbf{x})p(\mathbf{x})\]
Posterior ∝ Likelihood × Prior
Accept proposal \(\mathbf{x}'\) with probability \(\alpha = \min\left(1, \frac{p(\mathbf{x}'|\mathbf{y})q(\mathbf{x}|\mathbf{x}')}{p(\mathbf{x}|\mathbf{y})q(\mathbf{x}'|\mathbf{x})}\right)\)
Formal errors from covariance matrix often underestimate true uncertainty. Calibration using external data or cross-validation provides more realistic estimates.
Uncertainty varies spatially due to non-uniform data coverage. Areas with dense observations have lower uncertainty than data-sparse regions.
Higher spherical harmonic degrees have larger uncertainties due to signal attenuation at satellite altitude and measurement noise amplification.
\[p(\mathbf{y}|\mathcal{M}) = \int p(\mathbf{y}|\mathbf{x},\mathcal{M})p(\mathbf{x}|\mathcal{M})d\mathbf{x}\]
Bayes Factor: \(BF_{12} = \frac{p(\mathbf{y}|\mathcal{M}_1)}{p(\mathbf{y}|\mathcal{M}_2)}\)
Bayesian model selection naturally penalizes complexity. Models with more parameters must provide substantially better fit to overcome the Occam factor.
log BF > 1: Substantial evidence
log BF > 2.5: Strong evidence
log BF > 5: Decisive evidence
Leave-one-out cross-validation (LOO-CV) provides an alternative for model comparison when computing evidence is intractable.