model {

for(i in 1:N){
mu_M1[i,1] <- 1  #baseline is the 1st category
mu_M1[i,2] <- exp(alpha1*x[i])
mu_M1[i,3] <- exp(alpha2*x[i])
sum_M[i] <-mu_M1[i,1]+mu_M1[i,2]+mu_M1[i,3]

for (k in 1:3)
{mu_M[i,k] <- mu_M1[i,k]/sum_M[i]}

M[i]~dcat(mu_M[i,])

mu_y[i] <-c*x[i]+beta1*equals(M[i],2)+beta2*equals(M[i],3)
y[i] ~ dnorm(mu_y[i],prec2)

ie[i]<-(alpha1*mu_M[i,2]*(1-mu_M[i,2])-alpha2*mu_M[i,2]*mu_M[i,3])*beta1+(alpha2*mu_M[i,3]*(1-mu_M[i,3])-alpha1*mu_M[i,2]*mu_M[i,3])*beta2
}

alpha1 ~ dnorm(0.0,0.01)
alpha2 ~ dnorm(0.0,0.01)
beta1 ~ dnorm(0.0,0.000001)  
beta2 ~ dnorm(0.0,0.000001)  
c ~ dnorm(0.0,0.000001) 
DE<-c
var2 ~ dgamma(1,0.1)
prec2 <-1/var2
}