human
1. Distribution regression
Kernel mean embedding
Instead of fitting a model to the instances, the idea of distribution regression is to find a regression on the underlying probability distributions the instances come from. It is based on the assumption that the data is ${(x_i, y_i)}_{i=1}^{n}$ with:

$n$ the number of bags in the dataset ;
$x_i$ the probability distribution of bag $i$ ;
$y_i$ is the aerosol optical depth of bag $i$.

However, $x_i$ is not observed: for each bag $i$, the $100$ instances $x_{i,l}$, $l=1,...,100$, are samples from the distribution $x_i$. Our dataset is thus  ${({x_{i,l}}{l=1}^{100}, y_i)}{i=1}^{n}$ and we want to find a mapping $\hat{f}$ that will best predict unseen bags.
The mapping  $\hat{f}$ on ${({x_{i,l}}{l=1}^{100}, y_i)}{i=1}^{n}$ will try to learn the relationship between the true distributions ${x_i}{i=1}^{n}$ and the target values ${y_i}{i=1}^{n}$. To achieve that, the information of the 100 instances in each bag has to be summarised whilst losing the less information possible. The aggregated approach that simply computes the mean of the features for each bag is an example of information summary, yet plenty of data is lost that way.
A better way to represent each bag is via kernel mean embedding:
$$\mu_{\hat{x}i} = \frac{1}{100}\sum{l=1}^{100} k(\cdot, x_{i,l})$$
Each bag is represented as a linear combination of kernels, and with the right choice of kernel, the lost information can be very negligible.
Kernel Ridge Regression
We now want to find $\hat{f}$ that minimises the following regularised least square problem:
$$ \underset{f}{arg min} \sum_{i=1}^{n} (f(\mu_{\hat{x}_i}) - y_i)^2 + \lambda \Vert f \Vert^2$$
with $\lambda>0$ the L2 regularisation parameter.
In kernel ridge regression, $f$ is interpreted as a linear combination of feature space mappings $\phi$ of the data points $\mu_{\hat{x}i}$:
$$ f = \sum{i=1}^{n} \alpha_i \phi(\mu_{\hat{x}_i} ) $$
The equation thus becomes:
$$ \underset{\alpha}{arg min} (\Vert y -K\alpha \Vert^2 + \lambda \alpha^T K..."
Please write code following the instructions in jupyter notebook style.
 Write me a code to 1?
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#Here is code to 1.

class Kernel(object):
    """ Kernel class from Zoltan Szabo
        giving the kernel mean embedding.
    """

    def __init__(self, par=None):
        """ Initialization.

        Parameters
        ----------
        par : dictionary, optional
              Name of the kernel and its parameters (default is
              {"name": "RBF", "sigma": 1}). The name of the kernel comes
              from "RBF", "exponential", "Cauchy", "student", "Matern3p2",
              "Matern5p2", "polynomial", "ratquadr" (rational quadratic),
              "invmquadr" (inverse multiquadr).
        """
        if par is None:
            par = {"name": "RBF", "sigma": 1}

        name = par["name"]
        self.name = name

        # other attributes:
        if name == "RBF" or name == "exponential" or name == "Cauchy":
            self.sigma = par["sigma"]
        elif name == "student":
            self.d = par["d"]
        elif name == "Matern3p2" or name == "Matern5p2":
            self.l = par["l"]
        elif name == "polynomial":
            self.c = par["c"]
            self.exponent = par["exponent"]
        elif name == "ratquadr" or name == "invmquadr":
            self.c = par["c"]
        else:
            raise Exception("kernel=?")

    def gram_matrix(self, y1, y2):
        """  Compute the Gram matrix = [k(y1[i,:], y2[j,:])]; i, j: running.

        Parameters
        ----------
        y1 : (number of samples1, dimension)-ndarray
             One row of y1 corresponds to one sample.
        y2 : (number of samples2, dimension)-ndarray
             One row of y2 corresponds to one sample.

        Returns
        -------
        g : ndarray.
            Gram matrix of y1 and y2.
        """

        if self.name == "RBF":
            sigma = self.sigma
            g = cdist(y1, y2)
            g = exp(-g ** 2 / (2 * sigma ** 2))
        elif self.name == "exponential":
            sigma = self.sigma
            g = cdist(y1, y2)
            g = exp(-g / (2 * sigma ** 2))
        elif self.name ==...