//  [python]
def compute_expected_coordinates4D(grid, x0, x1, x2, x3, return_2Dgrids=False):
    """
    Computes expected coordinates of a 4D grid

    :param grid: 4D grid to integrate
    :param x0: coordinates of all points along axis=0
    :param x1: coordinates of all points along axis=1
    :param x2: coordinates of all points along axis=2
    :param x3: coordinates of all points along axis=3
    :param return_2Dgrids: if ``True`` returns the 1D and 2D marginals in a
        dictionary.

    :type grid: numpy array
    :type x0: numpy array
    :type x1: numpy array
    :type x2: numpy array
    :type x3: numpy array
    :type return_2Dgrids: boolean

    :rtype: numpy array
    :returns:
        * exp_x0 : expected value along axis=0
        * exp_x1 : expected value along axis=1
        * exp_x2 : expected value along axis=2
        * exp_x3 : expected value along axis=2
        * cov_matrix : covariance matrix
        * grid_dict : dictionary containing 1D and 2D marginals (optional)

    """

    # normalize grid
    grid_integral = compute_integral4D(grid, x0, x1, x2, x3)
    grid = grid / grid_integral

    # get 1D marginals, expected values and variances
    prob_x0 = si.trapezoid(si.trapezoid(si.trapezoid(grid, x=x1, axis=1), x=x2, axis=1),
                       x=x3, axis=1)
    exp_x0 = si.trapezoid(x0*prob_x0, x=x0)
    var_x0 = si.trapezoid((x0-exp_x0)*(x0-exp_x0)*prob_x0, x=x0)

    prob_x1 = si.trapezoid(si.trapezoid(si.trapezoid(grid, x=x0, axis=0), x=x2, axis=1),
                       x=x3, axis=1)
    exp_x1 = si.trapezoid(x1*prob_x1, x=x1)
    var_x1 = si.trapezoid((x1-exp_x1)*(x1-exp_x1)*prob_x1, x=x1)

    prob_x2 = si.trapezoid(si.trapezoid(si.trapezoid(grid, x=x0, axis=0), x=x1, axis=0),
                       x=x3, axis=1)
    exp_x2 = si.trapezoid(x2*prob_x2, x=x2)
    var_x2 = si.trapezoid((x2-exp_x2)*(x2-exp_x2)*prob_x2, x=x2)

    prob_x3 = si.trapezoid(si.trapezoid(si.trapezoid(grid, x=x0, axis=0), x=x1, axis=0),
                       x=x2, axis=0)
    exp_x3 = si.trapezoid(x3*prob_x3, x=x3)
    var_x3 = si.trapezoid((x3-exp_x3)*(x3-exp_x3)*prob_x3, x=x3)

    # get 2D marginals and covariances
    prob_x0_x1 = si.trapezoid(si.trapezoid(grid, x=x2, axis=2), x=x3, axis=2)
    cov_x0_x1 = si.trapezoid((x0-exp_x0)*si.trapezoid((x1-exp_x1)*prob_x0_x1,
                                              x=x1, axis=1), x=x0, axis=0)
    prob_x0_x2 = si.trapezoid(si.trapezoid(grid, x=x1, axis=1), x=x3, axis=2)
    cov_x0_x2 = si.trapezoid((x0-exp_x0)*si.trapezoid((x2-exp_x2)*prob_x0_x2,
                                              x=x2, axis=1), x=x0, axis=0)
    prob_x0_x3 = si.trapezoid(si.trapezoid(grid, x=x1, axis=1), x=x2, axis=1)
    cov_x0_x3 = si.trapezoid((x0-exp_x0)*si.trapezoid((x3-exp_x3)*prob_x0_x3,
                                              x=x3, axis=1), x=x0, axis=0)
    prob_x1_x2 = si.trapezoid(si.trapezoid(grid, x=x0, axis=0), x=x3, axis=2)
    cov_x1_x2 = si.trapezoid((x1-exp_x1)*si.trapezoid((x2-exp_x2)*prob_x1_x2,
                                              x=x2, axis=1), x=x1, axis=0)
    prob_x1_x3 = si.trapezoid(si.trapezoid(grid, x=x0, axis=0), x=x2, axis=1)
    cov_x1_x3 = si.trapezoid((x1-exp_x1)*si.trapezoid((x3-exp_x3)*prob_x1_x3,
                                              x=x3, axis=1), x=x1, axis=0)
    prob_x2_x3 = si.trapezoid(si.trapezoid(grid, x=x0, axis=0), x=x1, axis=0)
    cov_x2_x3 = si.trapezoid((x2-exp_x2)*si.trapezoid((x3-exp_x3)*prob_x2_x3,
                                              x=x3, axis=1), x=x2, axis=0)

    # construct covariance matrix
    cov_matrix = np.eye(4)
    cov_matrix[0, 0] = var_x0
    cov_matrix[1, 1] = var_x1
    cov_matrix[2, 2] = var_x2
    cov_matrix[3, 3] = var_x3

    cov_matrix[0, 1] = cov_x0_x1
    cov_matrix[1, 0] = cov_x0_x1

    cov_matrix[0, 2] = cov_x0_x2
    cov_matrix[2, 0] = cov_x0_x2

    cov_matrix[0, 3] = cov_x0_x3
    cov_matrix[3, 0] = cov_x0_x3

    cov_matrix[1, 2] = cov_x1_x2
    cov_matrix[2, 1] = cov_x1_x2

    cov_matrix[1, 3] = cov_x1_x3
    cov_matrix[3, 1] = cov_x1_x3

    cov_matrix[2, 3] = cov_x2_x3
    cov_matrix[3, 2] = cov_x2_x3

    if return_2Dgrids:
        grid_dict = {}
        grid_dict['prob_x0'] = prob_x0
        grid_dict['prob_x1'] = prob_x1
        grid_dict['prob_x2'] = prob_x2
        grid_dict['prob_x3'] = prob_x3
        grid_dict['prob_x0_x1'] = prob_x0_x1
        grid_dict['prob_x0_x2'] = prob_x0_x2
        grid_dict['prob_x0_x3'] = prob_x0_x3
        grid_dict['prob_x1_x2'] = prob_x1_x2
        grid_dict['prob_x1_x3'] = prob_x1_x3
        grid_dict['prob_x2_x3'] = prob_x2_x3

        return exp_x0, exp_x1, exp_x2, exp_x3, cov_matrix, grid_dict
    else:
        return exp_x0, exp_x1, exp_x2, exp_x3, cov_matrix