Modelling data as mixture of two normal distributions

[jeromyanglim]

• How can data be modelled as a mixture of two normal distributions?

My first attempt had issues. In particular, mu1 and mu2 had the posterior distributions eventhough conceptually I wanted to somehow think of mu1 as the lower group mean and mu2 as the higher group mean.

[jeromyanglim]

I have since learnt that this problem is called the 'label switching problem'. The issue is discussed with links in this question on CrossValidated.

This tutorial from a course on Bayesian and Modern Data analysis by Merlise Clyde

They suggest the following bugs model code:

for( i in 1 : N ) {
    y[i] ~ dnorm(mu[i], tau)
    mu[i] <- lambda[T[i]]
    T[i] ~ dcat(pi[]) 
}
pi[1:2] ~ ddirch(alpha[])
theta ~ dnorm(0.0, 1.0E-6)%_%I(0.0, )
lambda[1] ~ dnorm(0.0, 1.0E-6)
lambda[2] <- lambda[1] + theta
tau ~ dgamma(0.001,0.001)
sigma <- 1 / sqrt(tau)

Notes on the above code:

• lambda is a vector of length 2 that stores the mean for the first and second groups
• The first group is defined as the one with the lower mean by defining lamba two as greater than lambda by the quantity theta; and theta is constrained to be greater than zero through the use of the I notation
• T[i] stores a vector with length equal to the number of observations N, and made up of values of 1 and 2, thereby permitting the extraction of group means for each case lambda[T[i]].