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lines 6-43 of file: example/user/abs_density.cpp
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{xrst_begin abs_density.cpp}

Absolute Value In Log-Density: Example and Test
###############################################

Model
*****

.. math::

   \B{p}( z_i | \theta ) \sim \B{L} ( \theta_i , \sigma )

where :math:`\B{L} ( \mu , \sigma )` is the Laplace distribution
with mean :math:`\mu` and standard deviation :math:`\sigma`.
The corresponding fixed likelihood
:ref:`g(theta)<theory@Fixed Likelihood, g(theta)>`
is

.. math::

   g( \theta ) = \sum_{i} \left[
      \log ( \sigma \sqrt{2} )
      +
      \sqrt{2} \; \left| \frac{ z_i - \exp( \theta_i )}{\sigma} \right|
   \right]

The optimal solution, with no constraints and no prior on :math:`\theta` is

.. math::

   \hat{\theta}_i = \log( z_i )

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

{xrst_end abs_density.cpp}