Chapter 4 Inference for Lattice Models

4.1 Parameter Estimation for Lattice Models

4.1.1 Pseudolikelihood

Pseudolikelihood is to be writen as \[\begin{equation} p(\boldsymbol{\eta})=\prod_{i=1}^{n} \mathbf{P}\left(z\left(s_{i}\right) \mid\left\{z\left(s_{j}\right): j \neq i\right\} ; \boldsymbol{\eta}\right) \tag{4.1} \end{equation}\] Also denote the negative log-pseudolikelihood function as \[\begin{equation} P(\boldsymbol{\eta})=-\log (p(\boldsymbol{\eta}))=-\sum_{i=1}^{n} \log \mathbf{P}\left(z\left(s_{i}\right) \mid\left\{z\left(s_{j}\right): j \neq i\right\} ; \boldsymbol{\eta}\right) \tag{4.2} \end{equation}\]

The goal is to maximizing p seudolikelihood (??). Equally, we aims to find \(\hat \eta_p\) which minimize formula (4.2).

4.1.2 Gaussian Maximum Likelihood Estimation

Under Gaussian assumptions, The SG and CG model can be estimated by maximum likelihood method.

  • If \(\mathbf{Z} \sim \mathrm{N}(\boldsymbol{\mu}, \sigma)\), the negative log-likelihood is \[\begin{equation} L(\boldsymbol{\eta})=\frac{n}{2} \log (2 \pi)+\frac{1}{2} \log (|\Sigma|)+\frac{1}{2}(\mathbf{z}-\boldsymbol{\mu})^{\top} \Sigma^{-1}(\mathbf{z}-\boldsymbol{\mu}) \tag{4.3} \end{equation}\]
  • For the covariance matrix, \[\begin{equation} \begin{array}{ll} \text { SG : } & \Sigma=(I-B)^{-1} \Lambda\left(I-B^{\prime}\right)^{-1} \\ \text { CG : } & \Sigma=(I-C)^{-1} M \end{array} \tag{4.4} \end{equation}\]

4.2 Conditional Gaussian Model

Recall Conditional Gaussian Model introduced in section The negative log-pseudolikelihood function here is \[\begin{equation} P(\boldsymbol{\eta})=\frac{1}{2} \sum_{i=1}^{n} \frac{\left(z\left(s_{i}\right)-\theta\left(\left\{z\left(s_{j}\right): j \neq i\right\}\right)\right)^{2}}{\tau_{i}^{2}}+\log \left(2 \pi \tau_{i}^{2}\right) \tag{4.5} \end{equation}\] where \(\theta\left(\left\{z\left(s_{j}\right): j \neq i\right\}\right)=\mu_{i}+\sum_{j=1}^{n} c_{i j}\left(z\left(s_{j}\right)-\mu_{j}\right) .\)

It can be seen from forumula (4.5) that

  • The maximum pseudolikelihood estimator is a weighted least squares estimator under heterogeneous variances \(\tau_i\)
  • If \(\tau_{i}^{2} \equiv \sigma^{2}\) for all \(i=1, \cdots, n\), then maximum pseudolikelihood estimator is the ordinary least squares estimator.

Now that we have the goal function, MLE illustrated in section 4.1.2 can be applied for parameter estimation

4.3 Auto-logistic Model

The conditional distribution is \[\begin{equation} \operatorname{logit}\left(Z\left(s_{i}\right)=1 \mid\left\{z\left(s_{j}\right): j \neq i\right\}\right)=\alpha_{i}+\sum_{j=1}^{n} \theta_{i j} z\left(s_{j}\right) . \tag{4.6} \end{equation}\]

Consider again the isotropic Ising model \(\gamma_{1}=\gamma_{2}=\gamma\), the model is simplified to \[\begin{equation} \operatorname{logit}\left(Z\left(s_{i}\right)=1 \mid\left\{z\left(s_{j}\right): j \neq i\right\}\right)=\alpha+\gamma n_{i} . \tag{4.7} \end{equation}\]

Estimation: parameters can be estimated by a logistic regression with pseudo-data \(\left\{\left(z_{i}, n_{i}\right), i=1, \cdots, n\right\}\).

4.4 Auto-Poisson Model

The negative log-likelihood

\[\begin{equation} \begin{aligned} \mathbf{P}\left(\left\{\alpha_{i}\right\},\left\{\theta_{i j}\right\}\right) &=\sum_{i=1}^{n} \lambda_{i}\left(\left\{z\left(s_{j}\right): j \neq i\right\}\right) \\ &-z\left(s_{i}\right) \log \left(\lambda_{i}\left(\left\{z\left(s_{j}\right): j \neq i\right\}\right)\right)+\log \left(z\left(s_{i}\right) !\right), \end{aligned} \tag{4.8} \end{equation}\] which is a Poisson regression model.