#### Bayes factors using pseudo priors

Bayes factors using the Carlin and Chib method. For full description see Page 47 of Classic BUGS examples Vol 2.

model{
# standardise data
for(i in 1:N){
Ys[i] <- (Y[i] - mean(Y[])) / sd(Y[])
xs[i] <- (x[i] - mean(x[])) / sd(x[])
zs[i] <- (z[i] - mean(z[])) / sd(z[])
}

# model node
j ~ dcat(p[])
p[1] <- 0.9995 p[2] <- 1 - p[1] # use for joint modelling
# p[1] <- 1 p[2] <- 0 # include for estimating Model 1
# p[1] <- 0 p[2] <-1 # include for estimating Model 2
pM2 <- step(j - 1.5)

# model structure
for(i in 1 : N){
mu[1, i] <- alpha + beta * xs[i]
mu[2, i] <- gamma + delta*zs[i]
Ys[i] ~ dnorm(mu[j, i], tau[j])
}

# Model 1
alpha ~ dnorm(mu.alpha[j], tau.alpha[j])
beta ~ dnorm(mu.beta[j], tau.beta[j])
tau[1] ~ dgamma(r1[j], l1[j])
# estimation priors
mu.alpha[1]<- 0 tau.alpha[1] <- 1.0E-6
mu.beta[1] <- 0 tau.beta[1] <- 1.0E-4
r1[1] <- 0.0001 l1[1] <- 0.0001
# pseudo-priors
mu.alpha[2]<- 0 tau.alpha[2] <- 256
mu.beta[2] <- 1 tau.beta[2] <- 256
r1[2] <- 30 l1[2] <- 4.5

# Model 2
gamma ~ dnorm(mu.gamma[j], tau.gamma[j])
delta ~ dnorm(mu.delta[j], tau.delta[j])
tau[2] ~ dgamma(r2[j], l2[j])
# pseudo-priors
mu.gamma[1] <- 0 tau.gamma[1] <- 400
mu.delta[1] <- 1 tau.delta[1] <- 400
r2[1] <- 46 l2[1] <- 4.5
# estimation priors
mu.gamma[2] <- 0 tau.gamma[2] <- 1.0E-6
mu.delta[2] <- 0 tau.delta[2] <- 1.0E-4
r2[2] <- 0.0001 l2[2] <- 0.0001
}

##### Data
``` list(N=42,Y = c(3040, 2470, 3610, 3480, 3810, 2330, 1800, 3110, 3160, 2310,4360, 1880, 3670, 1740, 2250, 2650, 4970, 2620, 2900, 1670,2540, 3840, 3800, 4600, 1900, 2530, 2920, 4990, 1670, 3310,3450, 3600, 2850, 1590, 3770, 3850, 2480, 3570, 2620, 1890,3030,3030),x = c(29.2, 24.7, 32.3, 31.3, 31.5, 24.5, 19.9, 27.3, 27.1, 24.0,33.8, 21.5, 32.2, 22.5, 27.5, 25.6, 34.5, 26.2, 26.7, 21.1,24.1, 30.7, 32.7, 32.6, 22.1, 25.3, 30.8, 38.9, 22.1, 29.2,30.1, 31.4, 26.7, 22.1, 30.3, 32.0, 23.2, 30.3, 29.9, 20.8,33.2, 28.2),z = c(25.4, 22.2, 32.2, 31.0, 30.9, 23.9, 19.2, 27.2, 26.3, 23.9, 33.2, 21.0, 29.0, 22.0, 23.8, 25.3, 34.2, 25.7, 26.4, 20.0,23.9, 30.7, 32.6, 32.5, 20.8, 23.1, 29.8, 38.1, 21.3, 28.5,29.2, 31.4, 25.9, 21.4, 29.8, 30.6, 22.6, 30.3, 23.8, 18.4,29.4, 28.2)) ```
##### Inits for chain 1
``` list(j = 2, tau = c(1,1), alpha = 0, beta = 0, gamma = 0, delta = 0) ```

##### Inits for chain 2
``` list(j = 1, tau = c(0.1,0.1), alpha = 1.0, beta = 1.0, gamma = 1.0, delta = 1.0) ```

Results

corresponding to a Bayes factor of 0.6402 / (1 - 0.6402) x 0.9995 / 0.0005 = 3557