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

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

x

This is the \(k \times k\) positive-definite
covariance matrix or precision matrix for `dyangberger`

or the
Cholesky factor \(\textbf{U}\) of the covariance matrix or
precision matrix for `dyangbergerc`

.

log

Logical. If `log=TRUE`

, then the logarithm of the
density is returned.

`dyangberger`

and `dyangbergerc`

give the density.

Application: Continuous Multivariate

Density: \(p(\theta) = \frac{1}{|\theta|^{\prod (d_j - d_{j-1})}}\), where \(d\) are increasing eigenvalues. See equation 13 in Yang and Berger (1994).

Inventor: Yang and Berger (1994)

Notation 1: \(\theta \sim \mathcal{YB}\)

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.

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`

```
# NOT RUN {
library(LaplacesDemon)
X <- matrix(c(1,0.8,0.8,1), 2, 2)
dyangberger(X, log=TRUE)
# }
```

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