Bayesians often use precision rather than variance. These are elementary
utility functions to facilitate conversions between precision,
standard deviation, and variance regarding scalars, vectors, and
matrices, and these functions are designed for those who are new to
Bayesian inference. The names of these functions consist of two
different scale parameters, separated by a `2', and capital letters
refer to matrices while lower case letters refer to scalars and
vectors. For example, the `Prec2Cov`

function converts a
precision matrix to a covariance matrix, while the `prec2sd`

function converts a scalar or vector of precision parameters to
standard deviation parameters.

The modern Bayesian use of precision developed because it was more straightforward in a normal distribution to estimate precision \(\tau\) with a gamma distribution as a conjugate prior, than to estimate \(\sigma^2\) with an inverse-gamma distribution as a conjugate prior. Today, conjugacy is usually considered to be merely a convenience, and in this example, a non-conjugate half-Cauchy prior distribution is recommended as a weakly informative prior distribution for scale parameters.

```
Cov2Prec(Cov)
Prec2Cov(Prec)
prec2sd(prec=1)
prec2var(prec=1)
sd2prec(sd=1)
sd2var(sd=1)
var2prec(var=1)
var2sd(var=1)
```

Cov

This is a covariance matrix, usually represented as \(\Sigma\).

Prec

This is a precision matrix, usually represented as \(\Omega\).

prec

This is a precision scalar or vector, usually represented as \(\tau\).

sd

This is a standard deviation scalar or vector, usually represented as \(\sigma\).

var

This is a variance scalar or vector, usually represented as \(\sigma^2\).

This returns a precision matrix, \(\Omega\), from a covariance matrix, \(\Sigma\), where \(\Omega = \Sigma^{-1}\).

This returns a covariance matrix, \(\Sigma\), from a precision matrix, \(\Omega\), where \(\Sigma = \Omega^{-1}\).

This returns a standard deviation, \(\sigma\), from a precision, \(\tau\), where \(\sigma = \sqrt{\tau^{-1}}\).

This returns a variance, \(\sigma^2\), from a precision, \(\tau\), where \(\sigma^2 = \tau^{-1}\).

This returns a precision, \(\tau\), from a standard deviation, \(\sigma\), where \(\tau = \sigma^{-2}\).

This returns a variance, \(\sigma^2\), from a standard deviation, \(\sigma\), where \(\sigma^2 = \sigma \sigma\).

This returns a precision, \(\tau\), from a variance, \(\sigma^2\), where \(\tau = \frac{1}{\sigma^2}\).

This returns a standard deviation, \(\sigma\), from a variance, \(\sigma^2\), where \(\sigma = \sqrt{\sigma^2}\).

Bayesians often use precision rather than variance, where precision is the inverse of the variance. For example, a linear regression may be represented equivalently as \(\textbf{y} \sim \mathcal{N}(\mu, \sigma^2)\), or \(\textbf{y} \sim \mathcal{N}(\mu, \tau^{-1})\), where \(\sigma^2\) is the variance, and \(\tau\) is the precision, which is the inverse of the variance.

```
# NOT RUN {
library(LaplacesDemon)
Cov2Prec(matrix(c(1,0.1,0.1,1),2,2))
Prec2Cov(matrix(c(1,0.1,0.1,1),2,2))
prec2sd(0.5)
prec2var(0.5)
sd2prec(1.4142)
sd2var(01.4142)
var2prec(2)
var2sd(2)
# }
```

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