Dugongs: nonlinear growth curve

Carlin and Gelfand (1991) present a nonconjugate Bayesian analysis of the following data set from Ratkowsky (1983):
[dugongs1]The data are length and age measurements for 27 captured dugongs (sea cows). Carlin and Gelfand (1991) model this data using a nonlinear growth curve with no inflection point and an asymptote as Xi tends to infinity:

   Yi ~ Normal(μi, τ),   i = 1,...,27
   
   μi = α - βγXi   α, β > 0; 0 < γ < 1

Standard noninformative priors are adopted for α, β and τ, and a uniform prior on (0,1) is assumed for γ. However, this specification leads to a non conjugate full conditional distribution for γ which is also non log-concave. The graph and corresponding BUGS code is given below

   model
   {
      for( i in 1 : N ) {
         Y[i] ~ dnorm(mu[i], tau)
         mu[i] <- alpha - beta * pow(gamma,x[i])   
      }
      alpha ~ dunif(0, 100)
      beta ~ dunif(0, 100)
      gamma ~ dunif(0.5, 1.0)
      tau ~ dgamma(0.001, 0.001)
      sigma <- 1 / sqrt(tau)
      U3 <- logit(gamma)   
   }

Data
   list(x = c( 1.0, 1.5, 1.5, 1.5, 2.5, 4.0, 5.0, 5.0, 7.0,
   8.0, 8.5, 9.0, 9.5, 9.5, 10.0, 12.0, 12.0, 13.0,
   13.0, 14.5, 15.5, 15.5, 16.5, 17.0, 22.5, 29.0, 31.5),
   Y = c(1.80, 1.85, 1.87, 1.77, 2.02, 2.27, 2.15, 2.26, 2.47,
   2.19, 2.26, 2.40, 2.39, 2.41, 2.50, 2.32, 2.32, 2.43,
   2.47, 2.56, 2.65, 2.47, 2.64, 2.56, 2.70, 2.72, 2.57), N = 27)
Inits for chain 1
list(alpha = 1, beta = 1, tau = 1, gamma = 0.9)
   
Inits for chain 2
list(alpha = 0.1, beta = 0.1, tau = 0.1, gamma = 0.6)


Results

[dugongs3]



   model
   {
      for( i in 1 : N ) {
         Y[i] ~ dnorm(mu[i], tau)
         mu[i] <- alpha - beta * pow(gamma,x[i])
         diffMuAlpha[i] <- diff(mu[i], alpha)
         diffMuBeta[i] <- diff(mu[i], beta)
         diffMuGamma[i] <- diff(mu[i], gamma)
         pow[i] <- pow(gamma,x[i])
      }
      alpha ~ dunif(0, 100)
      beta ~ dunif(0, 100)
      gamma ~ dunif(0.5, 1.0)
      tau ~ dgamma(0.001, 0.001)
      sigma <- 1 / sqrt(tau)
      U3 <- logit(gamma)   
   }

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