1. (a)

(b). \(alpha\) and \(beta\) being ”shape” parameters of the model, a Uniform or Gaussian prior is ideal (unless advise from archaeologists is available).

- 1.
alpha: Chosen a Gaussian prior with mean = 0.5 and precision = 0.0001. Low precision means a ”vague” prior in absence of collective opinions from archaeologists. The specified mean is chosen after some exploratory analysis. It’s also obvious that the we want a mean \(\neq 0\). Gamma prior (with \(\alpha=\beta=f\)) is skipped because it tends to ”pull” parameter estimates towards zero.

- 2.
beta: Chosen a Gaussian prior with mean = 0.3 and precision = 0.0001. Low precision means a ”vague” prior in absence of collective opinions from archaeologists. The specified mean is chosen after some exploratory analysis. It’s also obvious that the we want a mean \(\neq 0\). Gamma prior (with \(\alpha=\beta=f\)) is skipped because it tends to ”pull” parameter estimates towards zero.

- 3.
tau: It’s a scale parameter. For normal likelihood models (as with the case of log-linear regression here), it’s a standard practice (for conjugacy reasons) to choose inverse-gamma for \(\sigma^{2}\) or gamma for the \(\tau\) parameter. Jeffrey’s prior for \(\tau\) is given by \(p(\tau)\propto\frac{1}{\tau}\) which is a improper and ”diffused” prior. To make it ”just proper” we can model it as \(\tau\sim Gamma(\varepsilon,\varepsilon)\) where \(\varepsilon\) is a small number.

(d)

model {

for(i in 1:N) {

w[i] ~ dnorm(z[i], tau)

w[i] <- log(y[i])

z[i] <- lalpha + beta*log(x[i])

}

alpha ~ dnorm(0.5, 0.0001)

beta ~ dnorm(0.3, 0.0001)

tau ~ dgamma(0.1, 0.001)

sigma <- 1/sqrt(tau)

lalpha <- log(alpha)

var <- sigma*sigma

}

(e)