Expression for Derivative of Hyperparameter of Kernel with respect to New Data I would like to determine how the hyperparameter will change when a new data is observed and the GP is updated with this new data.
Considering the following predictive distribution of the GP:
$$\mu(x^*)=K(x^*,X)^\top[K(X,X)+\sigma_n^2\mathbf{I}]^{-1} \mathbf{y_n}$$
and the squared exponential kernel:
$$K(x,x') = \sigma^2\exp\left(\frac{-(x-x')^T(x-x')}{2l^2}\right)$$
How can I compute $\frac{\partial l}{\partial \mu}$?
Secondly is it right metric to compute to deremine the change in hyperparameter especially the lengthscale with respect to new data?
 A: As written, the question doesn't really make sense: $l$ is a hyperparameter of the problem and doesn't depend on the data at all, so $\frac{\mathrm d l}{\mathrm d \mu}$ or any similar derivative is always just zero. I'm also not really sure what you mean by taking the derivative by $\mu$.
What it seems like you want is something like: "how should I locally update my hyperparameters given new data?" I'll answer a slightly different question: "how should I locally update my hyperparameters as my training dataset changes"? This question, though less natural, is much more amenable to taking derivatives.
I'm going to use $\theta$ to refer to the general set of parameters, both because $l$ is a bad variable name ($\ell$ is better...) and because you probably want to optimize more than one. One way to formalize this would be:


*

*Decide on a rule for picking $\theta$ given data: perhaps it's the $l$ that maximizes the marginal likelihood $p_\theta(\mathbf y \mid \mathbf X)$. You can call this
\begin{align}
\hat \theta
  &= \operatorname{argmax}_{\theta} \log p_\theta(\mathbf y \mid \mathbf X)
\\&= \operatorname{argmax}_{\theta} - \mathbf y^T K_\theta(\mathbf X, \mathbf X)^{-1} \mathbf y - \tfrac12 \log\det K_\theta(\mathbf X, \mathbf X)
.\end{align}

*You can then ask about how this $\hat \theta$ changes as your dataset $\mathbf X, \mathbf y$ changes. For instance,
\begin{align}
     \frac{\mathrm d \hat\theta}{\mathrm d \mathbf y}
  &= \frac{\mathrm d}{\mathrm d \mathbf y} \operatorname{argmax}_{\theta} \log p_\theta(\mathbf y \mid \mathbf X)
.\end{align}
Now, this is well-defined. The problem is that taking that derivative is not trivial. (If there were a closed-form expression for $\hat\theta$, we could just write that out and differentiate it. But there almost never is.)
Instead, we can turn to the implicit function theorem to see that
$$
\frac{\mathrm d \hat\theta}{\mathrm d \mathbf y} \Bigg\rvert_{\mathbf y = \mathbf y_0}
= - \left[ \frac{\partial^2 \log p_\theta(\mathbf y \mid \mathbf X)}{\partial \theta \, \partial\theta}  \right]^{-1} \frac{\partial^2 \log p_\theta(\mathbf y \mid \mathbf X)}{\partial \theta \, \partial \mathbf y} \Bigg\rvert_{\substack{\mathbf y = \mathbf y_0 \\\theta = \hat\theta(\mathbf y_0)}}
$$
or the same for $\mathbf X$.
Now, taking these derivates by hand will be somewhat unpleasant, but probably doable; there are closed forms for the derivative of the matrix inverse and logdet, for instance.
If $\theta$ is more than a few parameters, this will also be quite computationally expensive in an automatic differentiation library like PyTorch, TensorFlow, or JAX – but for a handful of $\theta$, it shouldn't be too much work to implement. (For moderate-sized $\theta$, JAX might be most efficient, since unlike the other two it allows for forward-mode autodiff.)
(Double-check my application of the theorem here; I copied it from a blog post rather than work it out again.)
