In a classical noisy regression setting, let $\big(f(x)\big)_{x\in\cal X}$ be a centered Gaussian process of covariance $k$ on a compact $\cal X$, and $\mathcal{F}_n$ be the filtration generated by some noisy measurements $\big(y_i = f(x_i)+\epsilon_i\big)_{i\leq n}$ where $\epsilon_i \overset{iid}{\sim} \mathcal{N}(0,\eta^2)$. Then, let $\mu_n(x) = \mathbb{E}[f(x) \mid \mathcal{F_n}]$ be the posterior mean of the GP given the noisy observations.
**Is it possible to prove tail bound for the distribution of $\lVert \mu_n \rVert_{\cal H_k}$ ?**

We know that $\lVert \mu_n \rVert_{\cal H_k}^2=\rm Y^T(\rm K + \eta^2 \rm I)^{-1 T}(\rm K+\eta^2\rm I)^{-1}\rm Y$, where $\rm Y$ is the Gaussian vector of the observations $y_i$ and $\rm K$ is the kernel matrix of the $x_i$, but the inverse terms make the upper bound difficult to grasp...

We further know that $\mu_n$ solves the following regression in the corresponding RKHS $\cal H_k$:
$$\arg\!\min_{\hat{f} \in \cal H_k} \frac 1 2 \lVert \hat{f} \rVert_{\cal H_k}^2 + \frac 1 {2\eta^2} \sum_{i=1}^n \big(y_i - \hat{f}(x_i)\big)^2.$$