normnorm: Normal sampling distribution

In yijin71/statg012: statg012: Statistical Inference at UCL

Description

Define the posterior distribution function for π ( μ |x ) if we have a normal sampling distribution f( x | μ, σ^2 ) where μ is unknown and σ^2 is known, and the normal prior distribution π ( μ; m, s ) for unknown μ. Define the posterior distirbution π ( μ,τ |x ) if we have a normal sampling distribution where both μ and σ^2 are unknown, and the normal-gamma prior distribution π( μ,τ; m,s,α,β ) for unknown mean and precison τ = 1/σ^2.

Usage

normnorm(x, m, s, alpha = NULL, beta = NULL, mu = NULL, sigma = NULL)

Arguments

x	a vector of observations from a normal distribution with mean mu and variance sigma^2.
m, s	two parameters of the prior normal distribution. If mu is NULL, they are mean and standard deviation of prior normal distribution for mu. If both mu and sigma are NULL, they are mean and standard deviation of normal distribution π (μ | τ).
alpha, beta	two parameters of a gamma distribution. Only used when mu and sigma are both NULL, then the prior distribution on μ and precision τ has a Normal-Gamma distribution. The distribution of τ is gamma(τ; α, β)
mu	the mean of x vector from a normal distribution. If it is NULL, the mean of x is unknown.
sigma	the standard deviation of x vector from a normal distribution. If it is NULL, the standard deviation of x is unknown.

Details

Suppose we have a random sample x from normal distribution N(μ,σ^2) whose μ is assumed to be unknow, the likelihood of this is f (x | μ) and the conjugate prior for this unknown μ is a normal distributin π (μ; m, s ). Then the posterior distribution π (μ | x ), proportional to f (x | μ)π (μ; m, s ), is also a normal distribution N(μ_n, σ_n^2).

Suppose we have a random sample x from normal distribution N(μ,σ^2) whose μ and σ^2 both are assumed to be unknow, the likelihood of this is f (x | μ, σ^2). The pricision τ equals 1/σ^2, then the conjugate prior for unknown μ and τ is a normal-gamma distribution where NG(μ, τ; m, s, α,β)= N(μ | τ ; m, (sτ)^-1) G(τ; α,β). The posterior distribution is normal-gamma with four updated parameters m_n, s_n, α_n, β_n. In addtion, the marginal distribution of μ is student t distribution.

Value

An object of class "g12post" is returned.

prior	the prior distribution.
likelihood	the likelihood function.
posterior	the posterior distribution.
mu	the mean of the x vector from a normal distribution.
pri.mean	the mean of the prior distribution.
pri.std	the standard deviation of the prior distribution.
pri.precision	the precision of the prior distribution.
pos.mean	the mean of the posterior distribution.
pos.std	the standard deviation of the posterior distribution.
pos.precision	the precision of the posterior distribution.
model	the prior and likelihood type to produce the posterior.

If both both mu and sigma are NULL, the addtional returns will be

tau	the precision of the x vector from a normal distribution.
mu.prior	the prior distribution for mu.
tau.prior	the prior distribution for tau.
mu.posterior	the posterior distribution for mu.
tau.posterior	the posterior distribution for tau.
pos.m	the parameter m in posteior normal-gamma distribution.
pos.s	the parameter s in posteior normal-gamma distribution.
pos.alpha	the parameter α in posteior normal-gamma distribution.
pos.beta	the parameter β in posteior normal-gamma distribution.

Notice pri.mean, pri.std, pos.mean and pos.std will not return in this situation.

Source

The slides 6 of STATG012 on Moodle

References

Hoff. 2010.Conjugate Priors for Normal Data, PowerPoint presentation, STA290: Bayesian and Modern Data Analysis, Duke University. Available from: Weblink.

Murphy, KP. 2007. Conjugate Bayesian analysis of the Gaussian distribution. Available from: Weblink.

See Also

summary.g12post for summararies of prior and posterior distribution.

plot.g12post for plots of prior and posterior distribution.

Examples

## this example is from slides6 Example 5.7
## generate a sample of 9 from a normal distribution with sd=2
x <- rnorm(9, sd = 2)
## the observed sample mean is 20
xx <- x- mean(x) + 20
## find the posterior density
exmp1 <- normnorm(xx, m = 25, s = sqrt(10), sigma = 2)
## summay and plot the example
summary(exmp1)
plot(exmp1, leg_pos = "right", cex = 0.8)

## this example assumes both mu and sigma are unknown
y <- rnorm(9)
exmp2 <- normnorm(y, m = 1, s = 2, a = 1, b = 1)
summary(exmp2)
## show the first plot : Prior and Posterior Distribution of mu
plot(exmp2, which = 1, main = "Prior and Posterior Distribution of mu")
## show the second plot : Prior and Posterior Distribution of tau
plot(exmp2, which = 2, col = 1:2)
## show the third plot : Prior Contour
plot(exmp2, which = 3, main = "Prior Contour",
           xlim = c(-1,3),ylim = c(0,5) )
## show the fourth plot : Posterior Contour
plot(exmp2, which = 4, main = "Posterior Contour",
           xlim = c(-3,3), ylim = c(0,5))
## show the fifth plot : Prior and Posterior Contour
plot(exmp2, which = 5, main = "Prior and Posterior Contour",
           xlim = c(-3,3), ylim = c(0,5))

yijin71/statg012 documentation built on May 23, 2019, 4:04 p.m.