Data | Knot lines | Fit | Summary | Estimated field

check by shard | Profile derivative checks | Joint density derivative checks

Introduction | Data | Model specification | Fit | AD development check | Joint log density | Likelihood

Data | Model and AD setup | Laplace log-likelihood at starting values | Outer optimization | Fitted values and density overlay | Derivative check along one parameter

Model | Simulated data | Building AD shards | Observations and hyperparameters | Random effects | Observation likelihood check | Combining shards | Inner optimization | Profile likelihood | Outer optimization | Profile derivative checks | Joint density derivative checks | Nicer interface

Introduction | Data | Building AD shards | Combining shards | Joint log density derivatives | Inner optimization | Joint density derivative checks | Profile likelihood | Outer optimization | Spatial effects

Model | Simulated data | Building AD shards | Observations and hyperparameters | Random effects | Observation likelihood check | Combining shards | Joint log density derivatives | Inner optimization | likelihood | Outer optimization | Derivative checks | Joint density derivative checks | nicer interface | mean estimation

\begin{equation}\label{eq:multinom}P(\mathbf{y} \mid \boldsymbol{\lambda}, \sum_{k=1}^K Y_k = N) | \begin{equation}\label{eq:dm}P(\mathbf{y} \mid \alpha, \boldsymbol{\mu}, N) | Simplificaiton | \begin | \begin{equation}\label{eq:dm_ratio}P(\mathbf{y} \mid \alpha, \boldsymbol{\mu}, N)

FEM ingredients | Hierarchical toy model | Joint density, gradient, and Hessian | Finite-difference checks

Continuous model: Matérn via Whittle SPDE | $$\operatorname{Cov}\bigl(u(\mathbf{s}),u(\mathbf{s}')\bigr) | Parameterization: range and sd | Finite-element weights and Grams | Tensor-product B-splines | Numerical comparison: Matérn vs FEM | References