C++ Laplace Approximation of Mixed Effects Models: cppad

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@(@ \newcommand{\R}[1]{ {\rm #1} } \newcommand{\B}[1]{ {\bf #1} } \newcommand{\W}[1]{ \; #1 \; } @)@C++ Laplace Approximation of Mixed Effects Models: cppad_mixed-20220519
Old Documentation
Current Documentation
License
Source Code Repository
Notation
     Fixed Effects, theta
     Random Effects, u
     Data, y, z
     Fixed Prior Density, p(theta)
     Fixed Data Density, p(z|theta)
     Random Prior Density, p(u|theta)
     Random Data Density, p(y|theta,u)
     Fixed Constraint Function, c(theta)
     Optimal Random Effects, u^(theta)
     Random Constraint Matrix, A
     Random Constraint Function, A*u^(theta)
Problem
     Maximum Likelihood
     No Random Effects
     Fixed Constraints, c
     Random Constraints
Negative Log-Density Vector
Contents

	One section per page	All sections in one page
MathJax	cppad_mixed.htm	_printable.htm
MathML	cppad_mixed.xml	_printable.xml

License
GNU Affero General Public License version 3.0 or later

Source Code Repository
https://github.com/bradbell/cppad_mixed

Notation
The densities below are known functions of @(@ y @)@, @(@ z @)@, @(@ \theta @)@, and @(@ u @)@:

Fixed Effects, theta
For each cppad_mixed object, there is a vector of fixed effects theta . The number of the fixed effects, and their order is constant (for each cppad_mixed object). We sometimes use @(@ \theta @)@ to denote the vector of fixed effects.

Random Effects, u
For each cppad_mixed object, there is a vector of random effects u . The number of the random effects, and their order is constant (for each cppad_mixed object).

Data, y, z
We use @(@ y @)@, (@(@ z @)@) to denote the set of data points that are dependent on (independent of) the random effects.

Fixed Prior Density, p(theta)
The fixed effects prior density is denoted by @(@ \B{p}( \theta ) @)@.

Fixed Data Density, p(z|theta)
The data density, that does not depend on the random effects, is denoted by @(@ \B{p}( z | \theta ) @)@.

Random Prior Density, p(u|theta)
The random effects prior density, given the fixed effects, is denoted by @(@ \B{p}( u | \theta) @)@.

Random Data Density, p(y|theta,u)
The density for the data, given both the fixed and random effects, is denoted by @(@ \B{p}( y | \theta, u) @)@.

Fixed Constraint Function, c(theta)
The fixed effects constraint function, denoted by @(@ c( \theta ) @)@, is a smooth function of the fixed effects,

Optimal Random Effects, u^(theta)
Given a value for the fixed effects @(@ \theta @)@, the corresponding optimal random effects are defined by @[@ \hat{u} ( \theta ) = \R{argmax} \; \B{p} ( y | \theta , u) \; \B{p} ( u | \theta ) @]@ Note that this definition agrees with the other definition for u^(theta) .

Random Constraint Matrix, A
The random constraint matrix is denoted by @(@ A @)@. It has row dimension equal to the number of constraints and column dimension equal to the number of random effects.

Random Constraint Function, A*u^(theta)
We refer to @(@ A \; \hat{u} ( \theta ) @)@ as the random constraint function.

Problem

Maximum Likelihood
We are given the problem of estimating the fixed effects by maximizing the likelihood with respect to @(@ \theta @)@; i.e., @[@ \B{p}( y , z , \theta ) = \B{p} ( \theta ) \B{p} ( z | \theta ) \int_{-\infty}^{+\infty} \B{p} ( y | \theta, u ) \B{p} ( u | \theta ) \; \B{d} u @]@

No Random Effects
In the case where there are not random effects, the vector @(@ u @)@ is empty and the optimal fixed effects maximize @[@ \B{p} ( \theta ) \B{p} ( z | \theta ) @]@

Fixed Constraints, c
The fixed effects have constraints of the form @[@ c_L \leq c(\theta) \leq c_U @]@ where and @(@ c_L @)@, @(@ c_U @)@ are vectors in the range of @(@ c(x) @)@. In the case where there are no constraints on the fixed effects, the range space is the empty vector.

Random Constraints
The random constraints are defined by the equation @[@ 0 = A \; \hat{u} ( \theta ) @]@

Negative Log-Density Vector
If vec is a density vector corresponding to @(@ \B{p}(x) @)@, the corresponding negative log-density is given by

@(@ - \log [ \B{p} (x) ] = @)@

vec[0] + fabs(vec[1]) + ... fabs(vec[s-1])

where s = vec.size() .

Contents

_contents	Table of Contents
install_unix	Installing cppad_mixed in Unix
theory	Laplace Approximation for Mixed Effects Models
base_class	cppad_mixed: Public Declarations
namespace	The CppAD::mixed Namespace Public Declarations
user_examples	User API Examples
release_notes	Changes and Additions to cppad_mixed
wish_list	CppAD Mixed Wish List
math_notation	Mathematical Notation
_reference	Alphabetic Listing of Cross Reference Tags
_index	Keyword Index
_search	Search This Web Site
_external	External Internet References

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