lines 6-72 of file: lib/python/cppad_py/mixed.py

{xrst_begin_parent mixed}
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   laplace
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Laplace Approximation of Mixed Effects Models
#############################################

Notation
********

theta
=====
We use :math:`\theta` to denote a value for the fixed effects vector.

u
=
We use :math:`u` to denote a value for the random effects vector.

z
=
We use :math:`z` to denote
the data that does not depend on the random effects.

y
=
We use :math:`y` to denote
the data that depends on the random effects.

p(theta)
========
We use :math:`\B{p} ( \theta )` to denote the prior density for :math:`\theta`.

p(z|theta)
==========
We use :math:`\B{p} (z | \theta )` to denote the density of :math:`z`
given :math:`\theta`.

p(u|theta)
==========
We use :math:`\B{p} (u | \theta )` to denote the density of :math:`u`
given :math:`\theta`.

p(y|theta,u)
============
We use :math:`\B{p} (y | \theta , u)` to denote the density of :math:`y`
given :math:`\theta` and :math:`u`.

Fixed Effects Likelihood
========================
We refer to :math:`\B{p} (z | \theta ) \B{p} ( \theta )`
as the fixed effects likelihood.
The negative log of this, as function of :math:`\theta`, is computed by
:ref:`mixed_ctor@fix_likelihood` .

Random Effects Likelihood
=========================
We refer to :math:`\B{p} (y | \theta , u ) \B{p} ( u | \theta )`
as the random effects likelihood.
The negative log of this, as function of :math:`\theta, u`, is computed by
:ref:`mixed_ctor@ran_likelihood` .

Children
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