# 贝叶斯统计学的主题 |MA40189 Topics in Bayesian statistics代写

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Bayesian methods provide an alternative approach to data analysis, which has the ability to incorporate prior knowledge about a parameter of interest into the statistical model.

to $\sigma^{2}$ as $d_{i j}$ tends to zero. In some cases, there will further baseline variability and the covariance may be defined as
$$\Sigma=\tau^{2} \quad \sigma^{2} R(d)$$
with the limiting variance as $d_{i j}$ tends to zero being $\tau^{2} \quad \sigma^{2}$ instead of $\sigma^{2}$. Another formulation in this case involves a discontinuity when $d_{i i}=0$, so that
\begin{aligned} &\Sigma_{i j}=\sigma^{2} r_{i j}\left(d_{i j}\right) \quad i \neq j \ &\Sigma_{i i j}=\tau^{2} \end{aligned}
Whatever the model adopted for the correlation between errors, the problem reduces to simultaneously estimating the regression coefficients and the parameters of the distance decay function.

## MA40189 COURSE NOTES ：

A technique often used to estimate the covariance structure focuses on functions of dissimilarity between errors or observations. Let $\Sigma(d)$ denote the covariance matrix at distance $d$ between points. Whereas
$$\Sigma(d)=\sigma^{2} R(d)$$
diminishes to zero for widely separated points and attains its maximum as $d_{i j}$ tends to zero, the variogram function
$$\gamma(d)=\sigma^{2}-\Sigma(d)=\sigma^{2}(I-R(d))$$
has value zero when $d_{i j}=0$, and reaches its maximum at $\sigma^{2}$ as spatial covariation in $\Sigma(d)$ disappears; hence $\sigma^{2}$ is known as the sill in geostatistics. For instance, the variogram for the exponential model is
$$\gamma\left(d_{i j}\right)=\sigma^{2}\left(1-e^{-3 d_{i} / h}\right)$$
As above, an extra variance parameter to allow for measurement error at $d=0$ (the ‘nugget’ error) may be added to give
$$\gamma(d)=\nu^{2} \quad \sigma^{2}[\mathrm{I}-R(d)]$$
and the sill is now $v^{2} \quad \sigma^{2}$. An alternative version of a nugget error model is
$$\gamma(d)=v^{2} \quad\left(\sigma^{2}-v^{2}\right)[I-R(d)]$$