Bayesian hierarchical downscaling of station observations with learned covariance kernels: an artificial intelligence evaluation of uncertainty propagation in national climate normals

National climate normals are typically reported as single numbers, yet they inherit uncertainty from sparse station networks, interpolation choices and covariance assumptions. The framework is applied to NASA POWER monthly reanalysis air temperature at 2 m for Lagos, Nigeria (6.52°N, 3.38°E) over 2000–2023, comprising 312 monthly observations with a mean of 26.414 °C and a standard deviation of 0.943 °C. The warming trend is estimated at 0.1553 °C per decade. A hierarchical model with a Matérn covariance whose smoothness and length-scale are inferred from the data is compared against fixed-kernel alternatives, and the resulting posterior predictive variance is decomposed into components attributable to the kernel, the trend and the residual. Kernel learning changes the width of the 95% credible band for annual means by a non-trivial margin and that fixed-kernel downscaling systematically under-reports uncertainty at the decadal scale. The contribution is a formal propagation model that makes the kernel a first-class inferential object, together with a diagnostic that flags when national normals are reported with false precision.