Gaussian Process Log Likelihood Gradient Incorrect

### Describe the bug

The gradient function of in the GaussianProcessRegressor Class is incorrect. This leads to inefficiencies fitting kernel (hyper) parameters.
The root of the issue is in that the gradient of the kernel function is made with respect to the log of the kernel parameter.

See the plot on the right showcasing the incorrect gradient currently being used in the optimization step:

![Image](https://github.com/user-attachments/assets/ec7f5582-b928-4738-9371-02ad1391c7c4)


To fix this apply the chain rule giving:

$\frac{\partial k(x)}{\partial \ln(x)} \frac{\partial \ln(x)}{\partial x} = \frac{\partial k(x)}{\partial x}$
$\frac{\partial k(x)}{\partial \ln(x)} \frac{1}{x} = \frac{\partial k(x)}{\partial x}$

When implementing this the correct gradient is given (middle plot).


TO REPRODUCE THIS BUG - see [my branch](https://github.com/conradstevens/scikit-learn/tree/gpr-log-likelihood-check) 
specifically: [sklearn/gaussian_process/tests/test_gpr_loglikelihood_check.py](https://github.com/conradstevens/scikit-learn/blob/gpr-log-likelihood-check/sklearn/gaussian_process/tests/test_gpr_loglikelihood_check.py)

I am happy to fix this bug by modifying the `_gpr` class's `log_marginal_likelihood` function. Keeping in mind things may get trickier when kernel functions have multiple hyper parameters (likely resulting in multiple iterations of the chain rule). 

### Steps/Code to Reproduce

TO REPRODUCE THIS BUG - see [my branch](https://github.com/conradstevens/scikit-learn/tree/gpr-log-likelihood-check) 
specifically: [sklearn/gaussian_process/tests/test_gpr_loglikelihood_check.py](https://github.com/conradstevens/scikit-learn/blob/gpr-log-likelihood-check/sklearn/gaussian_process/tests/test_gpr_loglikelihood_check.py)

### Expected Results

CRRECTION AND GROUND TRUTH TANGENTS CAN BE FOUND IN [my branch](https://github.com/conradstevens/scikit-learn/tree/gpr-log-likelihood-check) 
specifically: [sklearn/gaussian_process/tests/test_gpr_loglikelihood_check.py](https://github.com/conradstevens/scikit-learn/blob/gpr-log-likelihood-check/sklearn/gaussian_process/tests/test_gpr_loglikelihood_check.py)

### Actual Results

Logic error giving incorrect results.

### Versions

```shell
System:
    python: 3.12.5 | packaged by conda-forge | (main, Aug  8 2024, 18:31:54) [Clang 16.0.6 ]
executable: /opt/homebrew/Caskroom/miniconda/base/envs/sklearn-dev-5/bin/python
   machine: macOS-14.4.1-x86_64-i386-64bit

Python dependencies:
      sklearn: 1.8.dev0
          pip: 24.2
   setuptools: 72.2.0
        numpy: 2.1.0
        scipy: 1.14.1
       Cython: 3.0.11
       pandas: 2.2.2
   matplotlib: 3.9.2
       joblib: 1.4.2
threadpoolctl: 3.5.0

Built with OpenMP: True

threadpoolctl info:
       user_api: blas
   internal_api: openblas
    num_threads: 10
         prefix: libopenblas
       filepath: /opt/homebrew/Caskroom/miniconda/base/envs/sklearn-dev-5/lib/libopenblasp-r0.3.27.dylib
        version: 0.3.27
threading_layer: openmp
   architecture: Nehalem

       user_api: openmp
   internal_api: openmp
    num_threads: 10
         prefix: libomp
       filepath: /opt/homebrew/Caskroom/miniconda/base/envs/sklearn-dev-5/lib/libomp.dylib
        version: None
```

COMMENT:
This corrected derivative would be something as follows (repurposing your code):

```python
def CORRECTED_LogLikelihoodDerivCal(length_scale: float) -> float:
    """Return the correct log likelihood derivative calculation"""
    kernel_ = RBF(length_scale=length_scale.copy())
    
    y_train = y.copy()
    if y_train.ndim == 1:
        y_train = y_train[:, np.newaxis]
    K, K_gradient = kernel_(X.copy(), eval_gradient=True)
    ####################################################################################
    # The gradient of the kernel function is made with respect to the log of the
    # hyper parameter (length scale). To correct for this the chain rule shows:
    #
    # df(x)/dln(x) * dln(x)/dx = df(x)/dx
    # df(x)/dln(x) * 1 / x = df(x)/dx
    #
    # Giving correction:

    K_gradient = (
       K_gradient/length_scale
    )  # THIS IS THE ADDED STEP CORRECTING THE LOG LIKELIHOOD
    ####################################################################################

    K[np.diag_indices_from(K)] += ALPHA_NOISE
    L = cholesky(K, lower=GPR_CHOLESKY_LOWER, check_finite=False)

    # Compute log-marginal-likelihood Z and also store some temporaries
    # which can be reused for computing Z's gradient
    Z, (pi, W_sr, L, b, a) = gpc._posterior_mode(K, return_temporaries=True)
    # Compute gradient based on Algorithm 5.1 of GPML
    d_Z = []
    # XXX: Get rid of the np.diag() in the next line
    R = W_sr[:, np.newaxis] * cho_solve((L, True), np.diag(W_sr))  # Line 7
    C = solve(L, W_sr[:, np.newaxis] * K)  # Line 8
    # Line 9: (use einsum to compute np.diag(C.T.dot(C))))
    s_2 = (
        -0.5
        * (np.diag(K) - np.einsum("ij, ij -> j", C, C))
        * (pi * (1 - pi) * (1 - 2 * pi))
    )  # third derivative

    for j in range(K_gradient.shape[-1]):
        C = K_gradient[:, :, j]  # Line 11
        # Line 12: (R.T.ravel().dot(C.ravel()) = np.trace(R.dot(C)))
        s_1 = 0.5 * a.T.dot(C).dot(a) - 0.5 * R.T.ravel().dot(C.ravel())

        b = C.dot(y_train - pi)  # Line 13
        # Chain rule applies here again
        s_3 = b - K.dot(R.dot(b))/length_scale  # Line 14

        d_Z.append(s_1 + s_2.T.dot(s_3))  # Line 15

    return np.sum(np.stack(d_Z),axis=0)

Labels: Bug, Needs Investigation