Continuous prior for r...
model {
for (i in 1:6) {
for (j in 1:3) {
y[i,j] ~ dnegbin(p[i], r)
}
p[i] <- r/(mu[i] + r)
log(mu[i]) <- alpha + beta*log(x[i] + 10) + gamma*x[i]
}
r ~ dunif(1, max)
alpha ~ dnorm(0, 0.0001)
beta ~ dnorm(0, 0.0001)
gamma ~ dnorm(0, 0.0001)
}

Data:
   list(max = 1000,
y = structure(.Data = c(15,21,29,16,18,21,16,26,33,27,41,60,33,38,41,20,27,42),
.Dim = c(6, 3)),
x = c(0, 10, 33, 100, 333, 1000))

Inits:
   list(alpha = 0, beta = 0, gamma = 0, r = 10)
    Dbar = post.mean of -2logL; Dhat = -2LogL at post.mean of stochastic nodes
   Dbar   Dhat   pD   DIC   
y   130.739   129.179   1.560   132.298   
total   130.739   129.179   1.560   132.298   

   node   mean   sd   MC error   2.5%   median   97.5%   start   sample
   alpha   2.167   0.3164   0.0134   1.553   2.17   2.798   5001   100000
   beta   0.3209   0.08478   0.003641   0.1508   0.3202   0.4867   5001   100000
   gamma   -0.001011   3.678E-4   1.361E-5   -0.001739   -0.001011   -2.721E-4   5001   100000
   r   72.95   147.0   1.387   7.78   26.58   630.6   5001   100000

Continuous prior for log(r)...
model {
for (i in 1:6) {
for (j in 1:3) {
y[i,j] ~ dnegbin(p[i], r)
}
p[i] <- r/(mu[i] + r)
log(mu[i]) <- alpha + beta*log(x[i] + 10) + gamma*x[i]
}
logr.cont ~ dunif(0, 10)
log(r.cont) <- logr.cont
r <- round(r.cont)
alpha ~ dnorm(0, 0.0001)
beta ~ dnorm(0, 0.0001)
gamma ~ dnorm(0, 0.0001)
}

Data:
list(y = structure(.Data = c(15,21,29,16,18,21,16,26,33,27,41,60,33,38,41,20,27,42),
.Dim = c(6, 3)),
x = c(0, 10, 33, 100, 333, 1000))
      
Inits:
list(alpha = 0, beta = 0, gamma = 0, logr.cont = 1)
      
Dbar = post.mean of -2logL; Dhat = -2LogL at post.mean of stochastic nodes
   Dbar   Dhat   pD   DIC   
y   130.224   125.896   4.328   134.553   
total   130.224   125.896   4.328   134.553   

   node   mean   sd   MC error   2.5%   median   97.5%   start   sample
   alpha   2.198   0.3636   0.01521   1.48   2.206   2.91   5001   100000
   beta   0.3127   0.09866   0.004201   0.122   0.3107   0.5126   5001   100000
   gamma   -9.745E-4   4.334E-4   1.612E-5   -0.001841   -9.703E-4   -1.174E-4   5001   100000
   logr.cont   2.857   0.7325   0.007033   1.693   2.793   4.353   5001   100000
   r   43.9   489.4   2.045   5.0   16.0   78.0   5001   100000
   r.cont   43.89   489.4   2.045   5.437   16.32   77.69   5001   100000
   
   [example-8_6_4-salmonella0]