deb-bose-MATH5960-assign2

1. (a)

DAG

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

1. 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. 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. 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)