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lines 6-109 of file: example/user/hes_fixed_obj.cpp
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{xrst_begin hes_fixed_obj.cpp}

Hessian of Fixed Effects Objective: Example and Test
####################################################

Model
*****

.. math::

    \B{p}( y_i | \theta , u ) \sim \B{N} ( u_i + \theta_0 , \theta_1^2 )

.. math::

    \B{p}( u_i | \theta ) \sim \B{N} ( 0 , 1 )

.. math::

    \B{p}( \theta ) \sim \B{N} ( 4 , 1 )

It follows that the Laplace approximation is exact and

.. math::

    \B{p}( y_i | \theta ) \sim \B{N} \left( \theta_0 , 1 + \theta_1^2 \right)

The corresponding fixed effects objective is

.. math::

    L( \theta ) =  C + \frac{1}{2} \left[
        ( \theta_0 - 4 )^2 + ( \theta_1 - 4 )^2 +
        N \log \left( 1 + \theta_1^2 \right) +
        \left( 1 + \theta_1^2 \right)^{-1} \sum_{i=0}^{N-1} ( y_i - \theta_0 )^2
    \right]

The partial of the objective w.r.t :math:`\theta_0` is

.. math::

    \partial_{\theta(0)} L ( \theta )
    =
    ( \theta_0 - 4 )
    -
    \left( 1 + \theta_1^2 \right)^{-1} \sum_{i=0}^{N-1} ( y_i - \theta_0 )

The partial of the objective w.r.t :math:`\theta_1` is

.. math::

    \partial_{\theta(1)} L ( \theta )
    =
    ( \theta_1 - 4 )
    +
    N \left( 1 + \theta_1^2 \right)^{-1} \theta_1
    -
    \left( 1 + \theta_1^2 \right)^{-2} \theta_1
        \sum_{i=0}^{N-1} ( y_i - \theta_0 )^2

The second partial w.r.t :math:`\theta_0`, :math:`\theta_0` is

.. math::

    \partial_{\theta(0)} \partial_{\theta(0)} L ( \theta )
    =
    1 + N \left( 1 + \theta_1^2 \right)^{-1}

The second partial w.r.t :math:`\theta_1`, :math:`\theta_1` is

.. math::

    \partial_{\theta(1)} \partial_{\theta(1)} L ( \theta )
    =
    1
    +
    N \left( 1 + \theta_1^2 \right)^{-1}
    -
    2 N \left( 1 + \theta_1^2 \right)^{-2} \theta_1^2
    -
    \left( 1 + \theta_1^2 \right)^{-2} \sum_{i=0}^{N-1} ( y_i - \theta_0 )^2
    +
    4 \left( 1 + \theta_1^2 \right)^{-3} \theta_1^2
        \sum_{i=0}^{N-1} ( y_i - \theta_0 )^2

The second partial w.r.t :math:`\theta_0`, :math:`\theta_1` is

.. math::

    \partial_{\theta(0)} \partial_{\theta(1)} L ( \theta )
    =
    2 \left( 1 + \theta_1^2 \right)^{-2} \theta_1
        \sum_{i=0}^{N-1} ( y_i - \theta_0 )

trace_init
**********
If you change *trace_init* to ``true`` ,
a trace of the initialization will be printed on standard output.

{xrst_literal
    // BEGIN C++
    // END C++
}

{xrst_end hes_fixed_obj.cpp}