• Point process intensity
  • \(\lambda(\mathbf{s})\) events per acre
• Model
  • \(\log\lambda(\mathbf{s}) = \mu + Z(\mathbf{s})\)
  • \(Z(\mathbf{s})\) spatial Gaussian process
• Log-Gaussian Cox process (LGCP)

• Random continuous function
  • \(Z(\mathbf{s})\) a Gaussian random variable
  • Mean 0
  • Matèrn covariance function

• INLA: Integrated Nested Laplace Approximation (Rue, Martino, and Chopin 2009)
• Bayesian Hierarchical models
  • Many latent Gaussian variables
  • Few parameters
  • E.g. spatial prediction using Gaussian process model

• Laplace approximation in general
  • \(\int \exp[h(x)]\mathrm{d}x\)
  • Taylor expansion of \(h(x)\)

• Laplace approximation for likelihood
  • \(L(\boldsymbol{\theta}) = \exp[\ell(\boldsymbol{\theta})]\)
  • Taylor expansion of \(\ell(\boldsymbol{\theta})\)

• Example from Blangiardo and Cameletti (2015) section 4.9
  • \(\mathbf{y} = (y_{1}, \dots, y_{n})'\) independent Gaussian observations
  • \(y_{i} \sim \mathsf{N}(\theta, \sigma^{2})\)
  • \(\theta \sim \mathsf{N}(\mu_{0}, \sigma_{0}^{2})\)
  • \(\psi = 1/\sigma^{2}\), \(\psi \sim \mathrm{Gamma}(a, b)\)

• The posterior distribution of \(\psi\) \[p(\psi|\mathbf{y}) \propto \frac{p(\mathbf{y} | \theta, \psi) p(\theta) p(\psi)} {p(\theta | \psi, \mathbf{y})}\]

• Laplace approximation \[\tilde{p}(\psi|\mathbf{y}) \propto \frac{p(\mathbf{y} | \theta, \psi) p(\theta) p(\psi)} {\tilde{p}_{G}(\theta^{*} | \psi, \mathbf{y})}\]

• Repeat for \(\theta\)
• Will depend on \(\psi\)

• Priors: \(\mu_{0} = -3\), \(\sigma_{0}^{2} = 4\), \(a = 1.6\), \(b = 0.4\)
• 30 observed points

• INLA package for R
• Provides marginal posterior
  • Not joint posterior

\[ (\kappa^{2} - \Delta)^{\alpha / 2} (\tau Z(\mathbf{s})) = W(\mathbf{s}) \]

• SPDE approach (Lindgren, Rue, and Lindström 2011)
  • \((\kappa^{2} - \Delta)^{\alpha / 2} (\tau Z(\mathbf{s})) = W(\mathbf{s})\)
  • Choose nodes \(\mathbf{s}_{i}\) to model \(Z(\mathbf{s}_{i})\)
  • Build a triangular mesh
  • Autoregressive model on the nodes

\(\phantom{x + y + z = 1}\qquad x + y + z = 1\)

→

\[ f(\mathbf{s}) \approx \alpha f(1, 0, 0) + \beta f(0, 1, 0) + \gamma f(0, 0, 1) \]