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More to the point (to which I’m slowly but surely coming), if you google “inla” the first hit is the Irish national liberation army, again not quite what you’d expect (if you’re a statistician with only limited interests outside your field, that is…).
But, speaking of INLA (I mean our INLA $-$ not the Irish separatists), Marta, Michela, Håvard and I have just finished our paper, reviewing its use for spatial and spatio-temporal Bayesian data analysis.
In comparison to the original series of INLA papers, we slightly changed the notation. Originally, Håvard and his co-authors had defined a linear predictor on (a suitable transformation of) the mean of the observed data as
$$ \eta_i = \alpha + \sum_{j=1}^{n_f}f^{(j)}(u_{ij}) \sum_{k=1}^{n_\beta} \beta_k z_{ki} + \epsilon_i, $$
and the parameters were defined as
$$\mathbf{x} = \left( \alpha,\{\beta_k\}, \{f^{(j)}(\cdot)\}, \{\epsilon_i\} \right) $$
and given a Gaussian Markov Random prior, as functions of a set of hyper-parameters $\boldsymbol\theta$.
I’ve always thought that this was a bit confusing, as we’re normally used to thinking of $\boldsymbol\theta$ as “level-one” parameters and of $\mathbf{x}$ as observable covariates. In the paper, we’ve modified this to clarify (in my opinion) how this works; so for us $\boldsymbol\theta$ represents the set of parameters, while we use $\boldsymbol\phi$ for the hyper-parameters (ie the variances of the structured effects).
Also, in the original formulation, $\epsilon_i \sim \mbox{Normal} (0, \sigma_{\epsilon})$, with $\sigma_\epsilon \rightarrow 0$ is just a technical device used in the code to allow INLA to monitor directly the linear predictor $\eta_i$. But, as far as the model specification is concerned, there is no real need to include it (that’s why we didn’t).
In the paper there are also several worked examples (links to the data and the R code are available here).
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