Normal Distribution: Precision Parameterization
These functions provide the density, distribution function, quantile function, and random generation for the univariate normal distribution with mean mu and precision tau.
dnormp(x, mean=0, prec=1, log=FALSE) pnormp(q, mean=0, prec=1, lower.tail=TRUE, log.p=FALSE) qnormp(p, mean=0, prec=1, lower.tail=TRUE, log.p=FALSE) rnormp(n, mean=0, prec=1)
x, q |
These are each a vector of quantiles. |
p |
This is a vector of probabilities. |
n |
This is the number of observations, which must be a positive integer that has length 1. |
mean |
This is the mean parameter mu. |
prec |
This is the precision parameter tau, which must be positive. |
log, log.p |
Logical. If |
lower.tail |
Logical. If |
Application: Continuous Univariate
Density: p(theta) = sqrt(tau/(2*pi)) * exp(-(tau/2)*(x-mu)^2)
Inventor: Carl Friedrich Gauss or Abraham De Moivre
Notation 1: theta ~ N(mu, tau^(-1))
Notation 2: p(theta) = N(theta | mu, tau^(-1))
Parameter 1: mean parameter mu
Parameter 2: precision parameter tau > 0
Mean: E(theta) = mu
Variance: var(theta) = tau^(-1)
Mode: mode(theta) = mu
The normal distribution, also called the Gaussian distribution and the
Second Law of Laplace, is usually parameterized with mean and variance,
or in Bayesian inference, with mean and precision, where precision is
the inverse of the variance. In contrast, Base R parameterizes
the normal distribution with the mean and standard deviation. These
functions provide the precision parameterization for convenience and
familiarity.
Some authors attribute credit for the normal distribution to Abraham de Moivre in 1738. In 1809, Carl Friedrich Gauss published his monograph “Theoria motus corporum coelestium in sectionibus conicis solem ambientium”, in which he introduced the method of least squares, method of maximum likelihood, and normal distribution, among many other innovations.
Gauss, himself, characterized this distribution according to mean and precision, though his definition of precision differed from the modern one. The modern Bayesian use of precision tau developed because it was more straightforward to estimate tau with a gamma distribution as a conjugate prior, than to estimate sigma^2 with an inverse-gamma distribution as a conjugate prior.
Although the normal distribution is very common, it often does not fit data as well as more robust alternatives with fatter tails, such as the Laplace or Student t distribution.
A flat distribution is obtained in the limit as tau -> 0.
For models where the dependent variable, y, is specified to be
normally distributed given the model, the Jarque-Bera test (see
plot.demonoid.ppc or plot.laplace.ppc) may
be used to test the residuals.
These functions are similar to those in base R.
dnormp gives the density,
pnormp gives the distribution function,
qnormp gives the quantile function, and
rnormp generates random deviates.
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dlaplace,
dnorm,
dnormv,
prec2var,
dst,
dt,
plot.demonoid.ppc, and
plot.laplace.ppc.
library(LaplacesDemon)
x <- dnormp(1,0,1)
x <- pnormp(1,0,1)
x <- qnormp(0.5,0,1)
x <- rnormp(100,0,1)
#Plot Probability Functions
x <- seq(from=-5, to=5, by=0.1)
plot(x, dnormp(x,0,0.5), ylim=c(0,1), type="l", main="Probability Function",
ylab="density", col="red")
lines(x, dnormp(x,0,1), type="l", col="green")
lines(x, dnormp(x,0,5), type="l", col="blue")
legend(2, 0.9, expression(paste(mu==0, ", ", tau==0.5),
paste(mu==0, ", ", tau==1), paste(mu==0, ", ", tau==5)),
lty=c(1,1,1), col=c("red","green","blue"))Please choose more modern alternatives, such as Google Chrome or Mozilla Firefox.