Method elbo

gpflow/models/vgp.py:111–143

r""" This method computes the variational lower bound on the likelihood, which is: E_{q(F)} [ \log p(Y|F) ] - KL[ q(F) || p(F)] with q(\mathbf f) = N(\mathbf f \,|\, \boldsymbol \mu, \boldsymbol \Sigma)

(self)

        "return: []",
    )
    def elbo(self) -> tf.Tensor:
        r"""
        This method computes the variational lower bound on the likelihood,
        which is:

            E_{q(F)} [ \log p(Y|F) ] - KL[ q(F) || p(F)]

        with

            q(\mathbf f) = N(\mathbf f \,|\, \boldsymbol \mu, \boldsymbol \Sigma)

        """
        X_data, Y_data = self.data
        num_data = tf.convert_to_tensor(self.num_data)

        # Get prior KL.
        KL = gauss_kl(self.q_mu, self.q_sqrt)

        # Get conditionals
        K = self.kernel(X_data) + tf.eye(num_data, dtype=default_float()) * default_jitter()
        L = tf.linalg.cholesky(K)
        fmean = tf.linalg.matmul(L, self.q_mu) + self.mean_function(X_data)  # [NN, ND] -> ND
        q_sqrt_dnn = tf.linalg.band_part(self.q_sqrt, -1, 0)  # [D, N, N]
        L_tiled = tf.tile(tf.expand_dims(L, 0), tf.stack([self.num_latent_gps, 1, 1]))
        LTA = tf.linalg.matmul(L_tiled, q_sqrt_dnn)  # [D, N, N]
        fvar = tf.reduce_sum(tf.square(LTA), 2)

        fvar = tf.transpose(fvar)

        # Get variational expectations.
        var_exp = self.likelihood.variational_expectations(X_data, fmean, fvar, Y_data)

        return tf.reduce_sum(var_exp) - KL

    @inherit_check_shapes
    def predict_f(