Problem 1

The model I used to is a standard hierarchical spatial model \[Z | \beta, Y, \tau^2_{\epsilon} \sim Gau(X\beta + Y, \tau^2_{\epsilon}I)\] where \(Z\) is the response variable, \(\beta\) is the fixed effect for location (trend) , \(Y\) is the random effect for spatial dependency and \(\tau^2_{\epsilon}\) is the error. Moreover, a Multivariate normal distribution is assumed for \(Y\) with mean \(0\) and a gaussian covaiance structure \(\rho(h) = \sigma_0^2 \exp(-h/\phi))\). The parameters are \(\beta, \tau^2_{\epsilon}, \phi, \sigma_0^2\).

The priors are: \[p(\beta) \propto N(\mu_{\beta}, \Sigma_{\beta})\] \[\sigma_0^2 \propto 1/\sigma_0^2\] \[\phi \propto 1\] \[p(\tau^2_{\epsilon} / \sigma_0^2) \propto 1\]

Problem 2

The prior for \(\beta\) is set to be a multivariate normal distribution with mean and covariance matrix obtained from the MLE.

The prior for \(tau^2\) and \(phi\) are discrete uniform. The discrete values are chosen to centered at the MLE.

The choice of prior for \(\sigma_0^2\) is the Jeffrey’s prior.

Since we don’t have any other prior information, the priors are chosen to be as non-informative as possible.

Problem 3

The posterior distribution for \(\beta\) is normal with mean 620.9362645, -1.3380109, -1.1636347. See the plots for the posterior distribution for \(\beta\).

The following two plots showed evidence of bayesian learning: the posterior is a compromise between data and the prior.

Problem 4

Problem 5

I changed to a flat prior on \(\beta\)s. This is dramatically different because the previous one used the mean and variance from the MLE and that a lot of prior information. This led to a smaller estimator of \(\tau^2\) and smaller variance in the estimator of \(\phi\) (see the last plot).