These functions make it easier for the user to represent marginal parameter transformations
for which inferences are to be made. Suppose quadrature is done on the posterior for parameter theta,
but interest lies in parameter lambda = g(theta) for smooth, monotone, univariate
g. This interface lets the user provide g, g^-1, and (optionally)
the Jacobian dtheta/dlambda, and aghq will do quadrature on the theta scale
but report summaries on the lambda scale. See a note in the Details below about
multidimensional parameters.
make_transformation(...)# S3 method for aghqtrans
make_transformation(transobj, ...)
# S3 method for list
make_transformation(translist, ...)
# S3 method for default
make_transformation(totheta, fromtheta, jacobian = NULL, ...)
Object of class aghqtrans, which is simply a list with elements totheta,
fromtheta, and jacobian. Object is suitable for checking with aghq::validate_transformation
and for inputting into any function in aghq which takes a transformation argument.
Used to pass arguments to methods.
An object of class aghqtrans. Just returns this object. This is for internal
compatibility.
A list with elements totheta, fromtheta, and, optionally, jacobian.
Inverse function g^-1(theta). Specifically, takes vector
g_1(theta_1)...g_p(theta_p) and returns vector theta_1...theta_p.
Function g: R^p -> R^p, where p = dim(theta).
Must take vector theta_1...theta_p and return vector g_1(theta_1)...g_p(theta_p), i.e.
only independent/marginal transformations are allowed (but these are the only ones
of interest, see below). For j=1...p, the parameter of
inferential interest is lambda_j = g_j(theta_j) and the parameter whose posterior is being
normalized via aghq is theta_j. Passed to match.fun.
(optional) Function taking theta and returning the absolute value of the determinant of
the Jacobian dtheta/dg(theta). If not provided, a numerically differentiated Jacobian is used as
follows: numDeriv::jacobian(totheta,fromtheta(theta)). Passed to match.fun.
Often, the scale on which quadrature is done is not the scale on which the user
wishes to make inferences. For example, when a parameter lambda>0 is
of interest, the posterior for theta = log(lambda) may be better approximated
by a log-quadratic than that for lambda, so running aghq on the
likelihood and prior for theta may lead to faster and more stable optimization
as well as more accurate estimates. But, interest is still in the original parameter
lambda = exp(theta).
These considerations are by no means unique to the use of quadrature-based approximate
Bayesian inferences. However, when using (say) MCMC, inferences for summaries
of transformations of the parameter are just as easy as for the un-transformed parameter.
When using quadrature, a little bit more work is needed.
The aghq package provides an interface for computing
posterior summaries of smooth, monotonic parameter transformations. If quadrature
is done on parameter theta and g(theta) is a univariate, smooth, monotone function,
then inferences are made for lambda = g(theta). In the case that theta is
p-dimensional, p > 1, the supplied function g is understood to
take in theta_1...theta_p and return g_1(theta_1)...g_p(theta_p). The
Jacobian is diagonal.
To reiterate, all of this discussion applies only to marginal parameter transformations.
For the full joint parameter, the only summary statistics you can even calculate at all
(at present?) are moments, and you can already calculate the moment of any function h(theta)
using aghq::compute_moment, so no additional interface is needed here.
Other transformations:
default_transformation(),
validate_transformation()
make_transformation('log','exp')
make_transformation(log,exp)
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