Windham_populationinverse: Inverse Transform for the Population Parameters Under Windham...
In scorematchingad: Score Matching Estimation by Automatic Differentiation

Windham_populationinverse

R Documentation

Inverse Transform for the Population Parameters Under Windham Weights

Description

Returns the matrix which reverses the effect of weights on a population for certain models.

Usage

Windham_populationinverse(cW)

Windham_populationinverse_alternative(newtheta, previoustheta, cW, cWav)

Arguments

`cW`	A vector of tuning constants for the Windham robustification method performed by `Windham()`.
`newtheta`	The parameter vector most recently estimated
`previoustheta`	The parameter vector estimated in the previous step
`cWav`	The value of the non-zero elements of `cW`. That is `cW` have elements that are zero or equal to `cWav`.

Details

In the Windham robustification method (Windham()) the effect of weighting a population plays a central role. When the the model density is proportional to \exp(\eta(\theta) \cdot T(u)), where T(u) is a vector of sufficient statistics for a measurement u, and \eta is a linear function, Then weights proportional to \exp(\eta(c \circ \theta) \cdot t(u)), where c is a vector of tuning constants and \circ is the Hadamard (element-wise) product, have a very simple effect on the population parameter vector \theta: the weighted population follows a density of the same form, but with a parameter vector of (1 + c) \circ \theta. The inverse of this change to the parameter vector is then a matrix multiplication by a diagonal matrix with elements 1/(1+c_i), with c_i denoting the elements of c.

Value

A diagonal matrix with the same number of columns as cW.

Functions

Windham_populationinverse(): The matrix with diagonal elements 1/(1+c_i)
Windham_populationinverse_alternative(): The transform implemented as described by \insertCitescealy2024ro;textualscorematchingad. It is mathematically equivalent to multiplication by the result of Windham_populationinverse() in the situation in \insertCitescealy2024ro;textualscorematchingad.