update_Omega function

Update class covariances

This function updates the class covariances (independent from the other classes).

update_Omega(beta, b, z, m, nu, Theta)

Arguments

• beta: The matrix of the decision-maker specific coefficient vectors of dimension P_r x N. Set to NA if P_r = 0.
• b: The matrix of class means as columns of dimension P_r x C. Set to NA if P_r = 0.
• z: The vector of the allocation variables of length N. Set to NA if P_r = 0.
• m: The vector of class sizes of length C.
• nu: The degrees of freedom (a natural number greater than P_r) of the Inverse Wishart prior for each Omega_c. Per default, nu = P_r + 2.
• Theta: The scale matrix of dimension P_r x P_r of the Inverse Wishart prior for each Omega_c. Per default, Theta = diag(P_r).

Returns

A matrix of updated covariance matrices for each class in columns.

Details

The following holds independently for each class $c$. Let $\Omega_c$ be the covariance matrix of class number c. A priori, we assume that $\Omega_c$ is inverse Wishart distributed with $\nu$ degrees of freedom and scale matrix $\Theta$. Let $(\beta_n)_{z_n=c}$ be the collection of $\beta_n$ that are currently allocated to class $c$, $m_c$ the size of class $c$, and $b_c$ the class mean vector. Due to the conjugacy of the prior, the posterior $\Pr(\Omega_c \mid (\beta_n)_{z_n=c})$ follows an inverted Wishart distribution with $\nu + m_c$ degrees of freedom and scale matrix $\Theta^{-1} + \sum_n (\beta_n - b_c)(\beta_n - b_c)'$, where the product is over the values $n$ for which $z_n=c$ holds.

Examples

### N = 100 decider, P_r = 2 random coefficients, and C = 2 latent classes
N <- 100
b <- cbind(c(0,0),c(1,1))
(Omega_true <- matrix(c(1,0.3,0.3,0.5,1,-0.3,-0.3,0.8), ncol=2))
z <- c(rep(1,N/2),rep(2,N/2))
m <- as.numeric(table(z))
beta <- sapply(z, function(z) rmvnorm(b[,z], matrix(Omega_true[,z],2,2)))
### degrees of freedom and scale matrix for the Wishart prior
nu <- 1
Theta <- diag(2)
### updated class covariance matrices (in columns)
update_Omega(beta = beta, b = b, z = z, m = m, nu = nu, Theta = Theta)