Inverse Gamma Distribution
This is the density function and random generation from the inverse gamma distribution.
dinvgamma(x, shape=1, scale=1, log=FALSE) rinvgamma(n, shape=1, scale=1)
n |
This is the number of draws from the distribution. |
x |
This is the scalar location to evaluate density. |
shape |
This is the scalar shape parameter alpha, which defaults to one. |
scale |
This is the scalar scale parameter beta, which defaults to one. |
log |
Logical. If |
Application: Continuous Univariate
Density: p(theta) = (beta^alpha / Gamma(alpha)) * theta^(-(alpha + 1)) * exp(-beta / theta), theta > 0
Inventor: Unknown (to me, anyway)
Notation 1: theta ~ G^-1(alpha, beta)
Notation 2: p(theta) = G^-1(theta | alpha, beta)
Parameter 1: shape alpha > 0
Parameter 2: scale beta > 0
Mean: E(theta) = beta / (alpha - 1), for alpha > 1
Variance: var(theta) = beta^2 / ((alpha - 1)^2 * (alpha - 2)), alpha > 2
Mode: mode(theta) = beta / (alpha + 1)
The inverse-gamma is the conjugate prior distribution for the normal or Gaussian variance, and has been traditionally specified as a vague prior in that application. The density is always finite; its integral is finite if alpha > 0. Prior information decreases as alpha, beta -> 0.
These functions are similar to those in the MCMCpack package.
dinvgamma gives the density and
rinvgamma generates random deviates. The parameterization
is consistent with the Gamma Distribution in the stats package.
library(LaplacesDemon)
x <- dinvgamma(4.3, 1.1)
x <- rinvgamma(10, 3.3)
#Plot Probability Functions
x <- seq(from=0.1, to=20, by=0.1)
plot(x, dinvgamma(x,1,1), ylim=c(0,1), type="l", main="Probability Function",
ylab="density", col="red")
lines(x, dinvgamma(x,1,0.6), type="l", col="green")
lines(x, dinvgamma(x,0.6,1), type="l", col="blue")
legend(2, 0.9, expression(paste(alpha==1, ", ", beta==1),
paste(alpha==1, ", ", beta==0.6), paste(alpha==0.6, ", ", beta==1)),
lty=c(1,1,1), col=c("red","green","blue"))Please choose more modern alternatives, such as Google Chrome or Mozilla Firefox.