Documentation of alpha (#831)

* update documentation of alpha

* update vignette

* correctly use variance / sd in alpha doc

* update tests

* Revert "update tests"

This reverts commit 68c7b03.

* Revert prior change

* re-render docs

* refine wording (where GP applies depends on model)

* revert stray prior re-change

R/opts.R

@@ -454,12 +454,13 @@ backcalc_opts <- function(prior = c("reports", "none", "infections"),
 #' this is smaller.
 #'
 #' @param alpha_mean Numeric, defaults to 0. The mean of the magnitude parameter
-#' of the Gaussian process kernel. Should be approximately the expected variance
-#' of the logged Rt.
+#' of the Gaussian process kernel. Should be approximately the expected standard
+#' deviation of the Gaussian process (logged Rt in case of the renewal model,
+#' logged infections in case of the nonmechanistic model).
 #'
-#' @param alpha_sd Numeric, defaults to 0.01. The standard deviation of the
-#' magnitude parameter of the Gaussian process kernel. Should be approximately
-#' the expected standard deviation of the logged Rt.
+#' @param alpha_sd Numeric, defaults to 0.1. The standard deviation of the
+#' magnitude parameter of the Gaussian process kernel. Can be tuned to adjust
+#' how far alpha is allowed to deviate form its prior mean (`alpha_mean`).
 #'
 #' @param kernel Character string, the type of kernel required. Currently
 #' supporting the Matern kernel ("matern"), squared exponential kernel ("se"),

vignettes/gaussian_process_implementation_details.Rmd

@@ -44,21 +44,22 @@ with the following choices available for the kernel $k$
 ## Matérn 3/2 covariance kernel (the default)
 
 \begin{equation}
-k(\Delta t) = \alpha \left( 1 + \frac{\sqrt{3} \Delta t}{l} \right) \exp \left( - \frac{\sqrt{3} \Delta t}{l}\right)
+k(\Delta t) = \alpha^2 \left( 1 + \frac{\sqrt{3} \Delta t}{l} \right) \exp \left( - \frac{\sqrt{3} \Delta t}{l}\right)
 \end{equation}
 
 with $l>0$ and $\alpha > 0$ the length scale and magnitude, respectively, of the kernel.
+Note that here and later we use a slightly different definition of $\alpha$ compared to Riutort-Mayol et al. [@approxGP], where this is defined as our $\alpha^2$.
 
 ## Squared exponential kernel
 
 \begin{equation}
-k(\Delta t) = \alpha \exp \left( - \frac{1}{2} \frac{(\Delta t^2)}{l^2} \right)
+k(\Delta t) = \alpha^2 \exp \left( - \frac{1}{2} \frac{(\Delta t^2)}{l^2} \right)
 \end{equation}
 
 ## Ornstein-Uhlenbeck (Matérn 1/2) kernel
 
 \begin{equation}
-k(\Delta t) = \alpha \exp{\left( - \frac{\Delta t}{2 l^2}  \right)}
+k(\Delta t) = \alpha^2 \exp{\left( - \frac{\Delta t}{2 l^2}  \right)}
 \end{equation}
 
 ## Matérn 5/2 covariance kernel
@@ -142,7 +143,7 @@ t^* = \frac{t - \frac{1}{2}t_\mathrm{GP}}{\frac{1}{2}t_\mathrm{GP}}
 Relevant priors are
 
 \begin{align}
-\alpha &\sim \mathcal{Normal}(0, \sigma_{\alpha}) \\
+\alpha &\sim \mathcal{Normal}(\mu_\alpha, \sigma_{\alpha}) \\
 \rho   &\sim \mathcal{LogNormal} (\mu_\rho, \sigma_\rho)\\
 \end{align}
 
@@ -155,7 +156,8 @@ m_\rho &= 21 \\
 s_\rho &= 7 \\
 \rho_\mathrm{min} &= 0\\
 \rho_\mathrm{max} &= 60\\
-\sigma_\alpha &= 0.05\\
+\mu_\alpha &= 0\\
+\sigma_\alpha &= 0.01
 \end{align}
 
 # References