make_moment_function {aghq}

Moments of Positive Functions

Description

Given an object quad of class aghq returned by aghq::aghq(), aghq::compute_moment() will compute the moment of a positive function g(theta) of parameter theta. The present function, aghq::make_moment_function(), assists the user in constructing the appropriate input to aghq::compute_moment().

Usage

make_moment_function(...)

## S3 method for class 'aghqmoment'
make_moment_function(gg, ...)

## S3 method for class 'aghqtrans'
make_moment_function(gg, ...)

## S3 method for class ''function''
make_moment_function(gg, ...)

## S3 method for class 'character'
make_moment_function(gg, ...)

## S3 method for class 'list'
make_moment_function(gg, ...)

## Default S3 method:
make_moment_function(gg, ...)

Arguments

...

Used to pass arguments to methods.

gg

LOGARITHM of function ⁠R^p -> R^+⁠ of which the moment is to be computed along with its two derivatives. So for example providing gg = function(x) x will compute the moment of exp(x). Provided either as a function, a list, an aghqtrans object, or an aghqmoment object. See details.

Details

The approximation of moments of positive functions implemented in aghq::compute_moment() achieves the same asymptotic rate of convergence as the marginal likelihood. This involves computing a new mode and Hessian depending on the original posterior mode and Hessian, and g. These computations are handled by aghq::compute_moment(), re-using information from the original quadrature when feasible.

Computation of moments is defined only for scalar-valued functions, with a vector moment just defined as a vector of moments. Consequently, the user may input to aghq:compute_moment() a function g: R^p -> R^q+ for any q, and that function will return the corresponding vector of moments. This is handled within aghq::compute_moment(). The aghq::make_moment_function() interface accepts the logarithm of gg: R^p -> R^+, i.e. a multivariable, scalar-valued positive function. This is mostly to keep first and second derivatives as 1d and 2d arrays (i.e. the gradient and the Hessian); I deemed it too confusing for the user and the code-base to work with Jacobians and 2nd derivative tensors (if you're confused just reading this, there you go!). But, see aghq::compute_moment() for how to handle the very common case where the same transformation is desired of all parameter coordinates; for example when all parameters are on the log-scale and you want to compute E(exp(theta)) for vector theta.

If gg is a function or a character (like 'exp') it is first passed to match.fun, and then the output object is constructed using numDeriv::grad() and numDeriv::hessian(). If gg is a list then it is assumed to have elements fn, gr, and he of the correct form, and these elements are extracted and then passed back to make_moment_function(). If gg is an object of class aghqtrans returned by aghq::make_transformation(), then gg$fromtheta is passed back to make_moment_function(). If gg is an object of class aghqtrans then it is just returned.

Value

Object of class aghqmoment, which is a list with elements fn, gr, and he, exactly like the input to aghq::aghq() and related functions. Here gg$fn is log(gg(theta)), gg$gr is its gradient, and gg$he its Hessian. Object is suitable for checking with aghq::validate_moment() and for inputting into aghq::compute_moment().

See Also

Other moments: validate_moment()

Examples

# E(exp(x))
mom1 <- make_moment_function(force) # force = function(x) x
mom2 <- make_moment_function('force')
mom3 <- make_moment_function(list(fn=function(x) x,gr=function(x) 1,he = function(x) 0))