Describing default priors of spde model in paper
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TheCorinna1994
Apr 10, 2026, 7:14:39 AM (13 days ago) Apr 10
to R-inla discussion group
Hi there,
I want to describe the difference of the default priors of the spde model in my paper.
I write:
The default priors are assigned to the SPDE hyperparameters $\tau$ (overall precision of the spatial field) and $\kappa$ (inverse range scaling), which lack a direct physical interpretation. However, they are coupled to the correlation range $\rho$ and marginal standard deviation $\sigma$ only through:
\begin{equation}
\kappa = \frac{\sqrt{8\nu}}{\rho} and
\end{equation}
which I have from another blog post within this discussion group?
Thank you for your valuable help.
I want to describe the difference of the default priors of the spde model in my paper.
I write:
The default priors are assigned to the SPDE hyperparameters $\tau$ (overall precision of the spatial field) and $\kappa$ (inverse range scaling), which lack a direct physical interpretation. However, they are coupled to the correlation range $\rho$ and marginal standard deviation $\sigma$ only through:
\begin{equation}
\kappa = \frac{\sqrt{8\nu}}{\rho} and
\tau = \frac{1}\sigma\,\kappa^{\nu}\sqrt{4\pi}},
\label{eq:kappa_tau}
\end{equation}
where the smoothness parameter $\nu$ is fixed by $\alpha=\nu+d/2\in [1,2]$. In our case $d=2$ and depending on the covariance function, we set $\alpha=2$ for the Matérn model or $\alpha=1.5$ for the Exponential one.
The SPDE hyperparameters are in INLA internally log-transformed with $ \log(\tau)=\theta_1 $ and $ \log(\kappa)=\theta_2$ and follow a
joint Gaussian prior, independent by default \cite{Lindgren2015}:
\begin{equation}
\theta_1 \sim \mathcal{N}(\mu_{\theta_1},\, 10),
\qquad
\theta_2 \sim \mathcal{N}(\mu_{\theta_2},\, 10),
\end{equation}
where the prior variance is very weakly informative, and the means $\mu_{\theta_1}$, $\mu_{\theta_2}$ are set heuristically by \texttt{inla.spde2.matern()} to match the mesh geometry: the prior median range is fixed at 20\% of the mesh diameter $d(\mathcal{M})$, giving
\begin{equation}
\mu_{\theta_2} = \log\!\left(\frac{\sqrt{8\nu}}{0.2\,d(\mathcal{M})}\right),
\end{equation}
with $\mu_{\theta_1}$ derived from $\mu_{\theta_2}$ under unit nominal
marginal variance.
Consequently, placing a flat prior on $\log \kappa$ does \emph{not} imply a flat prior on the physically meaningful range $\rho$, making it difficult to encode hydrogeological knowledge directly. Crucially, this means the default prior changes with
every mesh, conflating the effect of mesh resolution with the effect of
the prior — a confounding absent from the PC prior branch.
Is this correct since the priors are not really explained, besides that theta1 and theta2 have a multivariate gaussian distribution.? Do I need to cite this: \begin{equation}
\mu_{\theta_2} = \log\!\left(\frac{\sqrt{8\nu}}{0.2\,d(\mathcal{M})}\right),\label{eq:kappa_tau}
\end{equation}
where the smoothness parameter $\nu$ is fixed by $\alpha=\nu+d/2\in [1,2]$. In our case $d=2$ and depending on the covariance function, we set $\alpha=2$ for the Matérn model or $\alpha=1.5$ for the Exponential one.
The SPDE hyperparameters are in INLA internally log-transformed with $ \log(\tau)=\theta_1 $ and $ \log(\kappa)=\theta_2$ and follow a
joint Gaussian prior, independent by default \cite{Lindgren2015}:
\begin{equation}
\theta_1 \sim \mathcal{N}(\mu_{\theta_1},\, 10),
\qquad
\theta_2 \sim \mathcal{N}(\mu_{\theta_2},\, 10),
\end{equation}
where the prior variance is very weakly informative, and the means $\mu_{\theta_1}$, $\mu_{\theta_2}$ are set heuristically by \texttt{inla.spde2.matern()} to match the mesh geometry: the prior median range is fixed at 20\% of the mesh diameter $d(\mathcal{M})$, giving
\begin{equation}
\mu_{\theta_2} = \log\!\left(\frac{\sqrt{8\nu}}{0.2\,d(\mathcal{M})}\right),
\end{equation}
with $\mu_{\theta_1}$ derived from $\mu_{\theta_2}$ under unit nominal
marginal variance.
Consequently, placing a flat prior on $\log \kappa$ does \emph{not} imply a flat prior on the physically meaningful range $\rho$, making it difficult to encode hydrogeological knowledge directly. Crucially, this means the default prior changes with
every mesh, conflating the effect of mesh resolution with the effect of
the prior — a confounding absent from the PC prior branch.
Is this correct since the priors are not really explained, besides that theta1 and theta2 have a multivariate gaussian distribution.? Do I need to cite this: \begin{equation}
\end{equation}
which I have from another blog post within this discussion group?
Thank you for your valuable help.
Best, Corinna
TheCorinna1994
Apr 10, 2026, 7:16:27 AM (13 days ago) Apr 10
to R-inla discussion group
P.S the variance of 10 in
\begin{equation}
\theta_1 \sim \mathcal{N}(\mu_{\theta_1},\, 10),
\qquad
\theta_2 \sim \mathcal{N}(\mu_{\theta_2},\, 10),
\end{equation}
I have from theta.prior.prec=0.1 in the function call
inla.spde2.matern(mesh, alpha = 2, param = NULL, constr = FALSE, extraconstr.int = NULL, extraconstr = NULL, fractional.method = c("parsimonious", "null"), B.tau = matrix(c(0,1,0),1,3), B.kappa = matrix(c(0,0,1),1,3), prior.variance.nominal = 1, prior.range.nominal = NULL, prior.tau = NULL, prior.kappa = NULL, theta.prior.mean = NULL, theta.prior.prec = 0.1, n.iid.group = 1) .
Do I need to justify that any further?
Thanks!
\theta_1 \sim \mathcal{N}(\mu_{\theta_1},\, 10),
\qquad
\theta_2 \sim \mathcal{N}(\mu_{\theta_2},\, 10),
\end{equation}
Do I need to justify that any further?
Thanks!
Helpdesk (Haavard Rue)
Apr 10, 2026, 7:30:24 AM (13 days ago) Apr 10
to TheCorinna1994, R-inla discussion group
could you share a pdf of the text?
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Helpdesk (Haavard Rue)
Apr 10, 2026, 8:03:55 AM (13 days ago) Apr 10
to TheCorinna1994, R-inla discussion group
A spesific reason you're not using the PC priors?
Message has been deleted
Finn Lindgren
Apr 10, 2026, 9:43:46 AM (12 days ago) Apr 10
to TheCorinna1994, R-inla discussion group
Hi,
I think we tried to explain in our 2015 JSS paper (Lindgren and Rue, Bayesian spatial modelling with R-INLA) that one shouldn’t use the default priors, and we showed how to construct problem specific vague priors. After the pcprior paper for matern models came out, there isn’t much reason to even use those vague priors, except for nonstationary models.
The default priors only still exist of backwards compatibility with code written before 2015…
So if you want to make your life easier, just refer to that paper to say that the default priors are a relic of the past.
Finn
On 10 Apr 2026, at 13:17, TheCorinna1994 <corinn...@hotmail.com> wrote:
Maybe I give context:We want to test how different combinations of modelling assumptions influence our spatial prediction. We set up a few different meshes, different combinations of formulas and we are also testing default priors vs pc priors. So yes I also explain PC Priors too, but in order to understand the difference between default and pc I wanted to introduce default priors as well.
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