The following quotation is from *[Machine Learning: a Probabilistic Perspective][1]* Chapter 7, page 234
> In machine learning, we often care more about predictive accuracy than about interpreting the parameters. Using Equation 4.126, we can easily show that the posterior predictive distribution at a test
point $\mathbf{x}$ is also Gaussian:
>$$
\begin{aligned}
p(y|\mathbf{x}, \mathcal{D}, \sigma^2) &= \int\mathcal{N}\left(y|\mathbf{x}^T\mathbf{w}, \sigma^2\right)\mathcal{N}\left(\mathbf{w}|\mathbf{w}_N, \mathbf{V}_N\right)d\mathbf{w}\\
&=\mathcal{N}\left(y|\mathbf{w}^T_N\mathbf{x}, \sigma^2_N(\mathbf{x})\right)\\
\sigma^2_N(\mathbf{x}) &= \sigma^2 + \mathbf{x}^T\mathbf{V}_N\mathbf{x}
\end{aligned}
$$
>The variance in this prediction, $\sigma^2_N(x)$, depends on two terms: the variance of the observation noise, $\sigma^2$, and the variance in the parameters, $\mathbf{V}_N$. The latter translates into variance about observations in a way which depends on how close $\mathbf{x}$ is to the training data $\mathcal{D}$. 

My question is in which way does "$\mathbf{V}_N$  translates into variance about observations in a way which depends on how close $\mathbf{x}$ is to the training data $\mathcal{D}$"? I mean, OK intuitively, but how can i prove it in a formal way?


  [1]: https://probml.github.io/pml-book/book0.html