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@(@\newcommand{\B}[1]{ {\bf #1} } \newcommand{\R}[1]{ {\rm #1} } \newcommand{\W}[1]{ \; #1 \; }@)@ This is dismod_at-20221105 documentation: Here is a link to its current documentation .
The Random Effects Value Density Function

u_j
prior_id
lambda_j
mu_j
epsilon_j
eta_j
d_j
sigma_j
V_j^u
V^u

u_j
We use @(@ u_j @)@ to denote one component of the random effects vector.

prior_id
We use prior_id for the prior that is attached to the j-th random effect; see prior for a variable .

lambda_j
We use @(@ \lambda_j @)@ to denote the mulstd_value_prior_id multiplier for the smooth_id corresponding to @(@ u_j @)@.

mu_j
We use @(@ \mu_j @)@ to denote the mean corresponding to prior_id .

epsilon_j
We use @(@ \varepsilon_j @)@ to denote the std corresponding to prior_id .

eta_j
We use @(@ \eta_j @)@ to denote the eta corresponding to prior_id .

d_j
We use @(@ d_j @)@ to denote the density_id corresponding to prior_id . In an abuse of notation, we include eta and eta in d_j ; see d .

sigma_j
We use @(@ \sigma_j @)@ to denote the transformed standard deviation corresponding to prior_id @[@ \sigma_j = \left\{ \begin{array}{ll} \log ( \mu_j + \eta_j + \varepsilon_j ) - \log( \mu_j + \eta_j ) & \R{if \; log \; density} \\ \varepsilon_j & \R{otherwise} \end{array} \right. @]@

V_j^u
The value density for the j-th component of @(@ u @)@ is @[@ V_j^u ( u | \theta ) = \exp \left[ D \left( u_j \W{,} \mu_j \W{,} \lambda_j \sigma_j \W{,} d_j \right) \right] @]@ where @(@ D @)@ is the log-density function .

V^u
Let @(@ n @)@ be the number of random effects. The value density for all the random effects @(@ u @)@ is defined by @[@ V^u ( u | \theta ) = \prod_{j=0}^{n-1} V_j^u ( u | \theta ) @]@
Input File: omh/model/random_value.omh