lines 8-67 of file: example/python/mixed/optimize_random_xam.py

{xrst_begin mixed_optimize_random_xam.py}

optimize_random: Example and Test
#################################

p(y|theta, u)
*************
In this example math:`y` given :math:`( \theta , u )`
is distributed normally with mean :math:`u`
and variance one; i.e.,

.. math::

   - \log [ \B{p} ( y | \theta , u ) ]
   =
   \log \left[ \sqrt{ 2 \pi } \right]
   +
   \frac{1}{2} ( y - u )^2


p(u|theta)
**********
In this example, the prior for :math:`u` given :math:`\theta`
is a normal with mean :math:`\theta` and  variance one; i.e.

.. math::

   - \log [ \B{p} ( u | \theta ) ]
   =
   \log \left[ \sqrt{ 2 \pi } \right]
   +
   \frac{1}{2} (u - \theta )^2

Optimal Random Effects
**********************
Given a value for the fixed effects :math:`\theta`,
the optimal random effects minimizes the following expression
w.r.t :math:`u`:

.. math::

   \frac{1}{2} ( u - y )^2 + \frac{1}{2} (u - \theta )^2

Taking the derivative w.r.t. :math:`u` and setting
it equal to zero, the optimal random effects,
as a function of the fixed effects,
:math:`\hat{u} ( \theta )` solves the equations

.. math::

   0 & = \hat{u} ( \theta ) - y + \hat{u} ( \theta ) - \theta
   \\
   \hat{u} ( \theta ) & = \frac{y + \theta}{2}

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{xrst_end mixed_optimize_random_xam.py}