# dist.YangBerger: Yang-Berger Distribution In LaplacesDemon: Complete Environment for Bayesian Inference

## Description

This is the density function for the Yang-Berger prior distribution for a covariance matrix or precision matrix.

## Usage

 ```1 2``` ```dyangberger(x, log=FALSE) dyangbergerc(x, log=FALSE) ```

## Arguments

 `x` This is the k x k positive-definite covariance matrix or precision matrix for `dyangberger` or the Cholesky factor U of the covariance matrix or precision matrix for `dyangbergerc`. `log` Logical. If `log=TRUE`, then the logarithm of the density is returned.

## Details

• Application: Continuous Multivariate

• Density: p(theta) = 1 / |theta|^(prod (d[j] - d[j-1])), where d are increasing eigenvalues. See equation 13 in Yang and Berger (1994).

• Inventor: Yang and Berger (1994)

• Notation 1: p(theta) ~ YB

• Mean:

• Variance:

• Mode:

Yang and Berger (1994) derived a least informative prior (LIP) for a covariance matrix or precision matrix. The Yang-Berger (YB) distribution does not have any parameters. It is a reference prior for objective Bayesian inference. The Cholesky parameterization is also provided here.

The YB prior distribution results in a proper posterior. It involves an eigendecomposition of the covariance matrix or precision matrix. It is difficult to interpret a model that uses the YB prior, due to a lack of intuition regarding the relationship between eigenvalues and correlations.

Compared to Jeffreys prior for a covariance matrix, this reference prior encourages equal eigenvalues, and therefore results in a covariance matrix or precision matrix with a better shrinkage of its eigenstructure.

## Value

`dyangberger` and `dyangbergerc` give the density.

## References

Yang, R. and Berger, J.O. (1994). "Estimation of a Covariance Matrix using the Reference Prior". Annals of Statistics, 2, p. 1195-1211.

`dinvwishart` and `dwishart`
 ```1 2 3``` ```library(LaplacesDemon) X <- matrix(c(1,0.8,0.8,1), 2, 2) dyangberger(X, log=TRUE) ```