random_value#

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The Random Effects Value Density Function#

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 \(\sigma_j\) 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 .

delta_j#

We use \(\delta_j\) to denote the transformed standard deviation corresponding to prior_id

\[\begin{split}\delta_j = \left\{ \begin{array}{ll} \log ( \mu_j + \eta_j + \sigma_j ) - \log( \mu_j + \eta_j ) & \R{if \; log \; density} \\ \sigma_j & \R{otherwise} \end{array} \right.\end{split}\]

V_j^u#

The value density for the j-th component of \(u\) is

\[V_j^u ( u | \theta ) = \exp \left[ D \left( u_j \W{,} \mu_j \W{,} \lambda_j \delta_j \W{,} d_j \right) \right]\]

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

\[V^u ( u | \theta ) = \prod_{j=0}^{n-1} V_j^u ( u | \theta )\]