# This is the model comparing the Yes and No probability judgments in example 1
# The dependent variable is y
# x1 is the partition prime 
# x2 is whether the probabilities were precise or imprecise
model
	{
		for( i in 1 : N ) {
			for(j in 1:2){
			y[i,j] ~ dbeta(omega[i,j], tau[i,j])
# We reparameterize the beta distribution
			omega[i,j] <- mu[i,j]*phi[i,j]
			tau[i,j] <- (1-mu[i,j])*phi[i,j]
			mu[i,j] <- exp(lambda[i,j])/(1+exp(lambda[i,j]))
			P[i,j] <- exp(m[i,j])/(1+exp(m[i,j]))	
			}
# This is the location submodel
		lambda[i,1] <- (1-K1[i])*(beta1[1] + beta2[1]*x1[i] + beta3[1]*x2[i])
		lambda[i,2] <- (1-K2[i])*( beta1[1] + beta2[1]*x1[i] + beta3[1]*x2[i] + beta1[2] + beta2[2]*x1[i] + beta3[2]*x2[i])
# This is the composition submodel
		m[i,1] <- theta1[1] + theta2[1]*x1[i] + theta3[1]*x2[i] 
		m[i,2] <- theta1[1] + theta2[1]*x1[i] + theta3[1]*x2[i] + theta1[2] + theta2[2]*x1[i] + theta3[2]*x2[i]
# This is the dispersion submodel
		phi[i,1] <- exp(-kappa1[T1[i]])
		phi[i,2] <- exp(-kappa2[T2[i]])
		K1[i] ~ dbern(P[i,1])
		K2[i] ~ dbern(P[i,2])
		T1[i] <- K1[i] + 1
		T2[i] <- K2[i] + 1 
		}
		beta1[1] ~ dnorm(0.0, 1.0E-6)
		beta2[1] ~ dnorm(0.0, 1.0E-6)
		beta3[1] ~ dnorm(0.0, 1.0E-6)
		theta1[1] ~ dnorm(0.0, 1.0E-6)
		theta2[1] ~ dnorm(0.0, 1.0E-6)
		theta3[1] ~ dnorm(0.0, 1.0E-6)
		beta1[2] ~ dnorm(0.0, 1.0E-6)
		beta2[2] ~ dnorm(0.0, 1.0E-6)
		beta3[2] ~ dnorm(0.0, 1.0E-6)
		theta1[2] ~ dnorm(0.0, 1.0E-6)
		theta2[2] ~ dnorm(0.0, 1.0E-6)
		theta3[2] ~ dnorm(0.0, 1.0E-6)	
		kappa1[2] ~ dnorm(-8.0,10.0)
		kappa1[1] ~ dnorm(0.0, 1.0E-6)
		kappa2[2] ~ dnorm(-8.0,10.0)
		kappa2[1] ~ dnorm(0.0, 1.0E-6)
	}
# Data
list(N = 345,  y = structure(.Data = c(0.949, 0.051,0.5, 0.5,0.151, 0.849,0.35, 0.65,0.5, 0.5,0.151, 0.301,0.141, 0.856,0.151, 0.849,0.144, 0.856,0.091, 0.899,0.081, 0.799,0.301, 0.6,0.301, 0.699,0.144, 0.144,0.151, 0.849,0.141, 0.859,0.251, 0.749,0.45, 0.55,0.101, 0.101,0.251, 0.251,0.161, 0.4,0.151, 0.251,0.141, 0.859,0.051, 0.949,0.144, 0.144,0.151, 0.849,0.141, 0.839,0.101, 0.101,0.144, 0.144,0.201, 0.6,0.176, 0.874,0.201, 0.899,0.5, 0.5,0.5, 0.5,0.201, 0.45,0.5, 0.5,0.201, 0.201,0.151, 0.151,0.151, 0.749,0.151, 0.849,0.5, 0.5,0.101, 0.101,0.151, 0.749,0.151, 0.799,0.201, 0.65,0.151, 0.151,0.076, 0.924,0.201, 0.301,0.176, 0.774,0.151, 0.849,0.5, 0.5,0.301, 0.674,0.5, 0.799,0.5, 0.5,0.251, 0.251,0.251, 0.55,0.251, 0.4,0.5, 0.5,0.4, 0.5,0.181, 0.819,0.131, 0.819,0.151, 0.849,0.141, 0.859,0.201, 0.799,0.144, 0.856,0.4, 0.6,0.281, 0.719,0.051, 0.849,0.144, 0.856,0.5, 0.5,0.144, 0.856,0.021, 0.899,0.141, 0.839,0.144, 0.856,0.146, 0.839,0.141, 0.759,0.101, 0.101,0.141, 0.859,0.011, 0.141,0.131, 0.131,0.144, 0.856,0.141, 0.849,0.141, 0.859,0.151, 0.749,0.144, 0.856,0.144, 0.856,0.151, 0.849,0.151, 0.849,0.151, 0.849,0.141, 0.859,0.5, 0.849,0.5, 0.799,0.151, 0.849,0.151, 0.849,0.5, 0.151,0.201, 0.799,0.4, 0.6,0.251, 0.4,0.301, 0.55,0.176, 0.824,0.251, 0.301,0.35, 0.45,0.161, 0.849,0.226, 0.226,0.176, 0.176,0.211, 0.874,0.5, 0.5,0.5, 0.5,0.276, 0.45,0.126, 0.849,0.4, 0.849,0.101, 0.899,0.096, 0.575,0.051, 0.949,0.5, 0.5,0.101, 0.899,0.35, 0.65,0.4, 0.6,0.201, 0.4,0.101, 0.899,0.151, 0.849,0.144, 0.856,0.201, 0.799,0.5, 0.5,0.201, 0.699,0.301, 0.699,0.141, 0.141,0.151, 0.849,0.141, 0.859,0.251, 0.749,0.5, 0.5,0.101, 0.101,0.151, 0.301,0.38, 0.47,0.301, 0.6,0.121, 0.879,0.101, 0.899,0.144, 0.144,0.101, 0.899,0.121, 0.799,0.5, 0.5,0.301, 0.6,0.101, 0.6,0.4, 0.4,0.151, 0.849,0.5, 0.301,0.5, 0.5,0.151, 0.55,0.5, 0.5,0.226, 0.55,0.151, 0.849,0.141, 0.141,0.151, 0.849,0.201, 0.799,0.201, 0.45,0.076, 0.076,0.475, 0.475,0.071, 0.929,0.5, 0.5,0.126, 0.126,0.251, 0.749,0.141, 0.65,0.251, 0.749,0.201, 0.6,0.5, 0.5,0.5, 0.5,0.5, 0.35,0.5, 0.5,0.5, 0.5,0.4, 0.251,0.126, 0.949,0.4, 0.6,0.5, 0.5,0.151, 0.849,0.301, 0.699,0.201, 0.799,0.161, 0.839,0.301, 0.699,0.101, 0.899,0.151, 0.749,0.4, 0.6,0.301, 0.201,0.101, 0.799,0.101, 0.699,0.5, 0.5,0.151, 0.849,0.101, 0.699,0.151, 0.151,0.141, 0.859,0.051, 0.849,0.201, 0.799,0.031, 0.6,0.101, 0.899,0.151, 0.849,0.051, 0.949,0.141, 0.859,0.301, 0.5,0.176, 0.824,0.176, 0.859,0.326, 0.4,0.126, 0.874,0.146, 0.854,0.201, 0.4,0.051, 0.849,0.151, 0.799,0.35, 0.4,0.151, 0.6,0.5, 0.5,0.301, 0.301,0.251, 0.35,0.151, 0.749,0.5, 0.5,0.251, 0.749,0.201, 0.799,0.5, 0.5,0.151, 0.849,0.201, 0.799,0.251, 0.6,0.201, 0.5,0.061, 0.939,0.4, 0.5,0.5, 0.301,0.5, 0.5,0.251, 0.799,0.326, 0.251,0.151, 0.849,0.101, 0.899,0.5, 0.5,0.131, 0.131,0.141, 0.141,0.151, 0.151,0.251, 0.051,0.141, 0.141,0.5, 0.5,0.5, 0.5,0.141, 0.141,0.141, 0.859,0.699, 0.301,0.5, 0.5,0.021, 0.979,0.151, 0.45,0.5, 0.5,0.5, 0.5,0.5, 0.849,0.076, 0.475,0.201, 0.6,0.6, 0.4,0.5, 0.151,0.5, 0.5,0.126, 0.874,0.126, 0.874,0.301, 0.699,0.151, 0.849,0.016, 0.016,0.126, 0.65,0.201, 0.799,0.201, 0.699,0.151, 0.849,0.141, 0.859,0.051, 0.051,0.031, 0.969,0.141, 0.859,0.201, 0.5,0.141, 0.859,0.141, 0.859,0.699, 0.301,0.141, 0.859,0.141, 0.859,0.144, 0.856,0.151, 0.849,0.301, 0.699,0.141, 0.859,0.151, 0.151,0.201, 0.6,0.151, 0.849,0.216, 0.799,0.141, 0.859,0.276, 0.326,0.161, 0.839,0.151, 0.749,0.151, 0.849,0.4, 0.35,0.151, 0.45,0.146, 0.854,0.276, 0.5,0.151, 0.849,0.151, 0.849,0.101, 0.899,0.5, 0.5,0.201, 0.101,0.5, 0.5,0.301, 0.301,0.5, 0.5,0.141, 0.141,0.5, 0.5,0.5, 0.5,0.141, 0.141,0.121, 0.879,0.699, 0.301,0.5, 0.5,0.051, 0.949,0.5, 0.5,0.151, 0.849,0.051, 0.65,0.5, 0.5,0.5, 0.5,0.301, 0.301,0.201, 0.799,0.35, 0.35,0.5, 0.5,0.849, 0.251,0.151, 0.35,0.5, 0.5,0.301, 0.5,0.5, 0.5,0.141, 0.859,0.201, 0.201,0.151, 0.799,0.051, 0.899,0.101, 0.899,0.141, 0.859,0.101, 0.899,0.141, 0.859,0.201, 0.799,0.6, 0.4,0.151, 0.161,0.101, 0.699,0.151, 0.849,0.011, 0.061,0.141, 0.859,0.066, 0.435,0.151, 0.65,0.151, 0.899,0.151, 0.749,0.57, 0.859,0.251, 0.65,0.301, 0.301,0.5, 0.5,0.101, 0.899,0.126, 0.874,0.141, 0.5,0.151, 0.849,0.151, 0.849,0.146, 0.854,0.151, 0.5), .Dim = c(345, 2)), x1 = c(-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1), x2 = c(-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1))

# Inits for one chain
list(kappa1 = c(-2.0, -8.0), kappa2 = c(-2.0, -8.0), beta1 = c(2.0, 1.0), beta2 = c(1.0, 0.5), beta3 = c(0.5, 1.0), theta1 = c(-1.0, 0.5), theta2 = c(0.1, 0.5), theta3 = c(-0.5, 0.4), K1 = c(0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1), K2 = c(0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1))