这是一份durham杜伦大学MATH43220-WE01/MATH3341-WE01/MATH4031-WE01作业代写的成功案例
Higher order difference priors may be used for seasonal effects. For example, for quarterly data, a possible smoothness prior is
$$
h(t)=s(t) \quad s(t-1) \quad s(t-2) \quad s(t-3) \sim \mathrm{N}(0, \tau s)
$$
For monthly data, the analogous scheme is
$$
h(t)=s(t) \quad s(t-1) \quad s(t-2) \quad \ldots . \quad s(t-11) \sim \mathrm{N}(0, \tau s)
$$
Instead of simple random walk priors, autoregressive priors involving lag coefficients $\phi_{1}, \ldots, \phi_{k}$ may be specified as smoothness priors. For example, an $\operatorname{AR}(2)$ prior in the true series would be
$$
f(t) \sim \mathrm{N}\left(\phi_{1} f(t-1) \quad \phi_{2} f(t-2), \tau^{2}\right)
$$
illustrate the use of such priors (with high order $k$ ) to estimate the spectral distribution of a stationary time series.
MATH43220-WE01/MATH3341-WE01/MATH4031-WE01 COURSE NOTES ：
The corresponding conditional ; Bernardinelli et al.,is
$$
P\left(e_{i} \mid e_{j}, j \neq i\right) \sim \mathrm{N}\left(M_{i}, \sigma_{i}^{2}\right)
$$
with
$$
M_{i}=\sum_{j \neq i} c_{i j} e_{j} / \sum_{j \neq i} c_{i j}=\sum_{j \neq i} w_{i j} e_{j}
$$
and $c_{i j}$ being spatial interactions as above. The variances differ by area with
$$
\sigma_{i}^{2}=\kappa^{2} / \sum_{j \neq i} c_{i j}
$$