simple x-y 2 dimension linear bayes fit for non informed prior .

test1.bugs

model {
  for (i in 1:N) {
    y[i] ~ dnorm(mu[i],tau)
    mu[i] <- alpha + beta*pow(gamma,x[i])
  }
  alpha ~ dnorm(0, 1.0E-6)
  beta ~ dnorm(0, 1.0E-6)
  gamma ~ dunif(0.0, 1.0)
  tau ~ dgamma(0.01,0.01)
}

test1.R

library(R2WinBUGS)
library(coda)

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 <- function() { list(alpha=1,beta=1,tau=1,gamma=0.9) }
parameters <- c("alpha","beta","tau","gamma")

result.sim <- bugs(data, inits, parameters, 
                   model.file="R/test1.bugs",
                   n.chains = 4, n.iter = 1000,
                   debug=FALSE,
                   working.directory=getwd())

result.sim$sims.list$beta1
print(result.sim, digits=3)

plot(as.mcmc(result.sim$sims.matrix))