def logpdf_link(self, link_f, y, Y_metadata=None):
        """
        Log Likelihood Function given link(f)

        .. math::
            \\ln p(y_{i}|\lambda(f_{i})) = \\alpha_{i}\\log \\beta - \\log \\Gamma(\\alpha_{i}) + (\\alpha_{i} - 1)\\log y_{i} - \\beta y_{i}\\\\
            \\alpha_{i} = \\beta y_{i}

        :param link_f: latent variables (link(f))
        :type link_f: Nx1 array
        :param y: data
        :type y: Nx1 array
        :param Y_metadata: Y_metadata which is not used in poisson distribution
        :returns: likelihood evaluated for this point
        :rtype: float

        """
        # alpha = self.gp_link.transf(gp)*self.beta    sum(log(a) + (a-1).*log(y)- f - exp(-f).*y.^a)
        # return (1. - alpha)*np.log(obs) + self.beta*obs - alpha * np.log(self.beta) + np.log(special.gamma(alpha))
        assert np.atleast_1d(link_f).shape == np.atleast_1d(y).shape
        c = np.zeros_like(y)
        if Y_metadata is not None and 'censored' in Y_metadata.keys():
            c = Y_metadata['censored']

        # uncensored = (1-c)* (np.log(self.r) + (self.r - 1) * np.log(y) - link_f - (np.exp(-link_f) * (y ** self.r)))
        # censored = (-c)*np.exp(-link_f)*(y**self.r)
        uncensored = (1-c)*( np.log(self.r)-np.log(link_f)+(self.r-1)*np.log(y) - y**self.r/link_f)
        censored = -c*y**self.r/link_f

        log_objective = uncensored + censored
        return log_objective