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{xrst_begin math_notation}

Mathematical Notation
#####################

A
*
The notation :math:`A` is used for the
:ref:`random constraint matrix<problem@Notation@Random Constraint Matrix, A>` .

B
*
The notation :ref:`B(beta, theta, u)<theory@Approximate Random Constraint Function, B(beta, theta, u)>`
( :math:`B( \beta , \theta , u )` )
is used for the second order random constraint approximation.

c
*
The notation
:ref:`c(theta)<problem@Notation@Fixed Constraint Function, c(theta)>`
( :math:`c( \theta )` )
is used for the fixed effects constraints as a function.

c_L
***
The notation
:ref:`c_L<problem@Notation@Fixed Constraint Function, c(theta)>`
( :math:`c_L` )
is used for the lower limit of the constraints on the fixed effects.

c_U
***
The notation
:ref:`c_U<problem@Notation@Fixed Constraint Function, c(theta)>`
( :math:`c_U` )
is used for the upper limit of the constraints on the fixed effects.

f
*
The notation :ref:`f(theta, u)<theory@Random Likelihood, f(theta, u)>`
( :math:`f( \theta , u )` )
is used for the random likelihood function.

g
*
The notation :ref:`g(theta)<theory@Fixed Likelihood, g(theta)>`
( :math:`g( \theta )` )
is used for the fixed likelihood function.

Lower h
*******
The notation
:ref:`h(theta, u)<theory@Objective@Laplace Approximation, h(theta, u)>`
( :math:`h( \theta , u )` )
is used for the Laplace approximation function.

Capital H
*********
The notation :ref:`H(beta, theta, u)<theory@Approximate Laplace Objective, H(beta, theta, u)>`
( :math:`H( \beta , \theta , u )` )
is used for the second order Laplace objective.

p
*
The notations
:math:`\B{p} ( \cdot )` and
:math:`\B{p} ( \cdot | \cdot )` are use for the
probability density (conditional probability density) functions;
see
:ref:`p(theta)<problem@Notation@Fixed Prior Density, p(theta)>` ,
:ref:`p(z | theta)<problem@Notation@Fixed Prior Density, p(theta)>` ,
:ref:`p(u | theta)<problem@Notation@Fixed Prior Density, p(theta)>` ,
:ref:`p(y | theta, u)<problem@Notation@Fixed Prior Density, p(theta)>` .

r
*
The notation
:ref:`r(theta)<theory@Objective@Laplace Objective, r(theta)>`
( :math:`r( \theta )` )
is used for the Laplace objective function.

L
*
The notation
:ref:`L(theta)<theory@Objective@Fixed Effects Objective, L(theta)>`
( :math:`L( \theta )` )
is used for the fixed effects objective function.

Lower u
*******
The notation :ref:`u<problem@Notation@Random Effects, u>`
( :math:`\theta` )
is used for the vector of fixed effects.

u^(theta)
*********
The notation
:ref:`u^(theta)<problem@Notation@Optimal Random Effects, u^(theta)>`
( :math:`\hat{u}( \theta )` )
is used for the optimal random effects.

Capital U
*********
The notation :ref:`U(beta, theta, u)<theory@Approximate Optimal Random Effects@First Order, U(beta, theta, u)>`
( :math:`U( \beta , \theta , u )` )
is used for the first order optimal random effects.

W
*
The notation :ref:`W(beta, theta, u)<theory@Approximate Optimal Random Effects@Second Order, W(beta, theta, u)>`
( :math:`W( \beta , \theta , u )` )
is used for the second order approximation of optimal random effects.

theta
*****
The notation :ref:`theta<problem@Notation@Fixed Effects, theta>`
( :math:`\theta` )
is used for the vector of fixed effects.

y
*
The notation :ref:`y<problem@Notation@Data, y, z>`
is used for data that depends on the random effects.

z
*
The notation :ref:`z<problem@Notation@Data, y, z>`
is used for data that does **not** depend on the random effects.

{xrst_end math_notation}