MCMC

Normal Mixture

Let $f(x; \mu, \sigma)$ be the normal density function with mean $\mu$ and standard deviation (SD) $\sigma$ . Given parameters $\mu_1$ , $\sigma_1$ , $\mu_2$ , $\sigma_2$ , and $0 < \gamma < 1$ , the ``normal mixture'' density is defined as

$\displaystyle g(x; \mu_1, \sigma_1, \mu_2, \sigma_2, \gamma) = \gamma f(x; \mu_1, \sigma_1) + (1-\gamma) f(x; \mu_1, \sigma_1)$

The mixture of two normal distribution will be a bimodal distribution, and the two peaks may be approximated by the respective mean $\mu_1$ and $\mu_2$ .

> source("nm.r")
> x = seq(0,1,by=0.01)
> plot(x, nm(x), type="l", xlab="x", ylab="PDF", main="Normal mixture")

Programming note. The function ``nm()'' is defined in the source file nm.r and written in the form

nm = function(x, p=c(0.3, 0.2, 0.8, 0.1, 0.4)){
  p[5] * dnorm(x, p[1], p[2]) + (1-p[5]) * dnorm(x, p[3], p[4])
}

It is executed by

> y = mn(0.2)

The argument ``p'' is optional, and given ``c(0.3, 0.2, 0.8, 0.1, 0.4) when omitted. The function returns the value

  p[5] * dnorm(x, p[1], p[2]) + (1-p[5]) * dnorm(x, p[3], p[4])

in which dnorm() returns the normal density value.