lines 8-90 of file: example/python/mixed/ran_likelihood_xam.py

{xrst_begin mixed_ran_likelihood_xam.py}
{xrst_spell
   laplace
}

ran_likelihood: Example and Test
################################

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

.. math::

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

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

.. math::

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

p(y|theta)
**********
For this example, Laplace approximation is equal to :math:`\B{p}(y|\theta)`
i.e, it is exact. Furthermore,

.. math::

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


Optimal Fixed Effects
*********************
For this example there is no fixed effects likelihood or constraints.
Hence the optimal fixed effects minimizes the following expression
w.r.t :math:`\theta`:

.. math::

   \frac{1}{2} \log \left[ \theta + \sigma^2 \right]
   +
   \frac{1}{2} ( y - \bar{y} )^2 / ( \theta + \sigma^2 )

Taking the derivative w.r.t. :math:`\theta` and setting
it equal to zero, the optimal fixed effects
:math:`\hat{\theta}` solves the equations

.. math::

   0 & = ( \hat{\theta} + \sigma^2 )^{-1}
   -
   ( y - \bar{y} )^2 ( \hat{\theta} + \sigma^2 )^{-2}
   \\
   1 & =
   ( y - \bar{y} )^2 ( \hat{\theta} + \sigma^2 )^{-1}
   \\
   \hat{\theta} & = (y - \bar{y})^2 - \sigma^2

{xrst_literal
   # BEGIN SOURCE
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}

{xrst_end mixed_ran_likelihood_xam.py}