transforms.Rmd

# Changes of Variables

When working with an inference or optimization algorithm, it is
convenient to have functions defined on all of $\mathbb{R}^N.$ Yet
most models are more naturally formulated in terms of constrained
parameters like probabilities (values in $[0, 1]$), correlations
(values in $[-1, 1]$), variances (in $[0, \infty)$).  Even more
complex constraints are imposed on covariance matrices (positive
definite matrices) or simplexes (non-negative sequence of values
summing to one).

In all of these cases, it is possible to transform the constrained
variable to the unconstrained scale.  When transforming back from the
constrained scale to the unconstrained scale, a Jacobian adjustment is
necessary to account for the change of variables.  And to make the
resulting function differentiable, this Jacobian must be
differentiable.

This chapter collects autodiff results for the inverses of several
popular constraining transforms.