update_b: Update class means

In loelschlaeger/RprobitB: Bayesian Probit Choice Modeling

Update class means

Description

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

Usage

update_b(beta, Omega, z, m, xi, Dinv)

Arguments

beta	The matrix of the decision-maker specific coefficient vectors of dimension P_r x N. Set to NA if P_r = 0.
Omega	The matrix of class covariance matrices as columns of dimension P_r*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.
xi	The mean vector of length P_r of the normal prior for each b_c. Per default, xi = numeric(P_r).
Dinv	The precision matrix (i.e. the inverse of the covariance matrix) of dimension P_r x P_r of the normal prior for each b_c.

Details

The following holds independently for each class c. Let b_c be the mean of class number c. A priori, we assume that b_c is normally distributed with mean vector \xi and covariance matrix D. Let (\beta_n)_{z_n=c} be the collection of \beta_n that are currently allocated to class c, m_c the class size, and \bar{b}_c their arithmetic mean. Assuming independence across draws, (\beta_n)_{z_n=c} has a normal likelihood of

\prod_n \phi(\beta_n \mid b_c,\Omega_c),

where the product is over the values n for which z_n=c holds. Due to the conjugacy of the prior, the posterior \Pr(b_c \mid (\beta_n)_{z_n=c}) follows a normal distribution with mean

(D^{-1} + m_c\Omega_c^{-1})^{-1}(D^{-1}\xi + m_c\Omega_c^{-1}\bar{b}_c)

and covariance matrix

(D^{-1} + m_c \Omega_c^{-1})^{-1}.

Value

A matrix of updated means for each class in columns.

Examples

### N = 100 decider, P_r = 2 random coefficients, and C = 2 latent classes
N <- 100
(b_true <- cbind(c(0,0),c(1,1)))
Omega <- 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_true[,z], matrix(Omega[,z],2,2)))
### prior mean vector and precision matrix (inverse of covariance matrix)
xi <- c(0,0)
Dinv <- diag(2)
### updated class means (in columns)
update_b(beta = beta, Omega = Omega, z = z, m = m, xi = xi, Dinv = Dinv)

loelschlaeger/RprobitB documentation built on Oct. 15, 2024, 11:08 a.m.