reparametrized

Reparameterized is the past participle of reparameterize, describing something that has been expressed or described with a different set of parameters or a different parameterization. The term appears in mathematics, statistics, and machine learning to indicate a change in how a model, function, or distribution is described, while preserving or adjusting its essential meaning or output.

In mathematics, reparameterization commonly refers to changing the parameter that traces a curve or a curve-like function. If a curve is given by a parameter t, a reparameterization uses a new parameter s = f(t) and defines the curve as the composition gamma(f(t)) or gamma~(s) = gamma(t(s)). This can alter the speed of traversal or simplify analysis, while the geometric image of the curve remains the same (up to orientation and regularity considerations).

In probability and statistics, reparameterization often means expressing a random variable as a function of another variable and parameters. A notable example is the reparameterization trick used in variational inference and deep learning, where a random variable is written as a deterministic function of a base random variable with fixed distribution and the model parameters. This form enables gradient-based optimization by moving stochasticity into a known, differentiable transformation.

Reparameterization is also employed to improve numerical stability and optimization in statistical modeling, such as transforming constrained parameters to unconstrained spaces or choosing alternative parameterizations that reduce correlation among parameters. The concept emphasizes changing the descriptive framework rather than the underlying data or outcomes, and “reparameterized” characterizes models, equations, or distributions that have undergone such a change.