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lines 5-166 of file: xrst/model/fixed_prior.xrst
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{xrst_begin fixed_prior}

Prior for Fixed Effect Values
#############################

theta
*****

lambda
======
We use :math:`\lambda` to denote
the sub-vector of the fixed effects that are
:ref:`standard deviation multipliers<model_variables@Fixed Effects, theta@Smoothing Standard Deviation Multipliers, lambda>` .

beta
====
We use :math:`\beta` to denote
the sub-vector of the fixed effects that are
:ref:`model_variables@Fixed Effects, theta@Parent Rates`
or
:ref:`model_variables@Fixed Effects, theta@Group Covariate Multipliers` .

theta
=====
We use :math:`\theta` to denote the entire fixed effects vector; i.e.,
:math:`\theta = ( \lambda , \beta )`.

Value Constraints
*****************

theta_k
=======
We use :math:`\theta_k` to denote
one component of  :math:`\theta`.

L_k^v
=====
We use :math:`L_k^v` to denote the
:ref:`prior_table@lower` limit
corresponding to the
:ref:`smooth_grid_table@value_prior_id`
that corresponds to the fixed effect :math:`\theta_k`.

U_k^v
=====
We use :math:`U_k^v` to denote the
:ref:`prior_table@upper` limit
corresponding to the
:ref:`smooth_grid_table@value_prior_id`
that corresponds to the fixed effect :math:`\theta_k`.

Age Difference Limits
*********************
The fixed effects corresponding to the standard deviation multipliers
:ref:`smooth_table@mulstd_value_prior_id` ,
:ref:`smooth_table@mulstd_dage_prior_id` , and
:ref:`smooth_table@mulstd_dtime_prior_id`
are constant with respect to age and time.
Hence the constraints
below do not apply to the standard deviation multipliers.

a_i(k)
======
We use :math:`a_{i(k)}` to denote the age corresponding to the
:ref:`smooth_grid_table@age_id` for the fixed effect
:math:`\theta_k`.
If this is the maximum age for the corresponding
:ref:`smooth_table@smooth_id` ,
then there is no age difference for this fixed effect.
Otherwise, we use :math:`a_{i(k)+1}` to denote the next larger age
in the smoothing grid and
:math:`\theta_{\ell(k)}` denote the corresponding component of
:math:`\theta` corresponding to the next larger age.

Delta^a
=======
If :math:`a_{i(k)}` is not the maximum age, we use the notation

.. math::

   \Delta^a_k \theta = \theta_{\ell(k)} - \theta_k

L_k^a
=====
We use :math:`L_k^a` to denote the
:ref:`prior_table@lower` limit
corresponding to the
:ref:`smooth_grid_table@dage_prior_id`
that corresponds to the fixed effect :math:`\theta_k`.

U_k^a
=====
We use :math:`U_k^a` to denote the
:ref:`prior_table@upper` limit
corresponding to the
:ref:`smooth_grid_table@dage_prior_id`
that corresponds to the fixed effect :math:`\theta_k`.

Time Difference Limits
**********************
The time difference :math:`\Delta^t_k \theta`,
the index :math:`j(k)`, and limits
:math:`L_k^t` and :math:`U_k^t` are defined in a fashion
similar as for the age differences.

Capital Theta
*************
The constraint set :math:`\Theta` is defined as the set of
:math:`\theta` such that the following conditions hold:

#. For all :math:`k`,

   .. math::

      L_k^v \leq \theta_k \leq U_k^v

#. For :math:`k`,
   that are not standard deviation multipliers,
   and such that :math:`a_{i(k)}` is not the maximum age
   for the corresponding smoothing,

   .. math::

      L_k^a \leq \Delta^a_k \theta \leq U_k^a

#. For :math:`k`,
   that are not standard deviation multipliers,
   and such that :math:`t_{j(k)}` is not the maximum time
   for the corresponding smoothing,

   .. math::

      L_k^t \leq \Delta^t_k \theta \leq U_k^t

Normalization Constant, C_theta
*******************************
The normalization constant for the fixed effects density is

.. math::

   C_\theta
   =
   \int_{\Theta}
      V^\theta ( \theta ) A^\theta ( \theta ) T^\theta ( \theta )
         \; \B{d} \theta

See :ref:`fixed_value@V^theta` ,
:ref:`fixed_diff@A^theta` , and
:ref:`fixed_diff@T^theta`  for the definitions of
:math:`V^\theta`, :math:`A^\theta` and :math:`T^\theta`.

p(theta)
********
We define the prior for the fixed effects by

.. math::

   C_\theta \; \B{p} ( \theta )
   =
   V^\theta ( \theta ) A^\theta ( \theta ) T^\theta ( \theta )

{xrst_end fixed_prior}