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lines 5-98 of file: xrst/model/random_value.xrst
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{xrst_begin random_value}
{xrst_spell
  ll
}

The Random Effects Value Density Function
#########################################

u_j
***
We use :math:`u_j` to denote
one component of the
:ref:`random effects<model_variables@Random Effects, u>` vector.

prior_id
********
We use *prior_id* for the prior that is attached to the
*j*-th random effect; see
:ref:`model_variables@Prior for a Variable` .

lambda_j
********
We use :math:`\lambda_j` to denote the
:ref:`smooth_table@mulstd_value_prior_id`
multiplier for the
:ref:`smooth_table@smooth_id`
corresponding to :math:`u_j`.

mu_j
****
We use :math:`\mu_j` to denote the
:ref:`prior_table@mean`
corresponding to *prior_id* .

epsilon_j
*********
We use :math:`\sigma_j` to denote the
:ref:`prior_table@std`
corresponding to *prior_id* .

eta_j
*****
We use :math:`\eta_j` to denote the
:ref:`prior_table@eta`
corresponding to *prior_id* .

d_j
***
We use :math:`d_j` to denote the
:ref:`prior_table@density_id` corresponding to *prior_id* .
In an abuse of notation, we include
:ref:`prior_table@eta` and
:ref:`eta<prior_table@nu>` in *d_j* ; see
:ref:`statistic@Notation@d` .

delta_j
*******
We use :math:`\delta_j` to denote the transformed standard deviation
corresponding to *prior_id*

.. math::

    \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.

V_j^u
*****
The value density for the *j*-th component of :math:`u` is

.. math::

    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 :math:`D` is the
:ref:`log-density function<statistic@Log-Density Function, D>` .

V^u
***
Let :math:`n` be the number of random effects.
The value density for all the random effects :math:`u` is defined by

.. math::

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

{xrst_end random_value}