# Copyright 2016-2020 The GPflow Contributors. All Rights Reserved.
#
# Licensed under the Apache License, Version 2.0 (the "License");
# you may not use this file except in compliance with the License.
# You may obtain a copy of the License at
#
# http://www.apache.org/licenses/LICENSE-2.0
#
# Unless required by applicable law or agreed to in writing, software
# distributed under the License is distributed on an "AS IS" BASIS,
# WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
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from typing import Optional, Tuple

import tensorflow as tf

import gpflow

from ..kernels import Kernel
from ..logdensities import multivariate_normal
from ..mean_functions import MeanFunction
from .model import GPModel, InputData, MeanAndVariance, RegressionData
from .training_mixins import InternalDataTrainingLossMixin
from .util import data_input_to_tensor


class GPR(GPModel, InternalDataTrainingLossMixin):
    r"""
    Gaussian Process Regression.

    This is a vanilla implementation of GP regression with a Gaussian
    likelihood.  Multiple columns of Y are treated independently.

    The log likelihood of this model is given by

    .. math::
       \log p(Y \,|\, \mathbf f) =
            \mathcal N(Y \,|\, 0, \sigma_n^2 \mathbf{I})
            
    To train the model, we maximise the log _marginal_ likelihood
    w.r.t. the likelihood variance and kernel hyperparameters theta.
    The marginal likelihood is found by integrating the likelihood
    over the prior, and has the form
    
    .. math::
       \log p(Y \,|\, \sigma_n, \theta) =
            \mathcal N(Y \,|\, 0, \mathbf{K} + \sigma_n^2 \mathbf{I})
    """

    def __init__(
        self,
        data: RegressionData,
        kernel: Kernel,
        mean_function: Optional[MeanFunction] = None,
        noise_variance: float = 1.0,
    ):
        likelihood = gpflow.likelihoods.Gaussian(noise_variance)
        _, Y_data = data
        super().__init__(kernel, likelihood, mean_function, num_latent_gps=Y_data.shape[-1])
        self.data = data_input_to_tensor(data)

    def maximum_log_likelihood_objective(self) -> tf.Tensor:
        return self.log_marginal_likelihood()

    def log_marginal_likelihood(self) -> tf.Tensor:
        r"""
        Computes the log marginal likelihood.

        .. math::
            \log p(Y | \theta).

        """
        X, Y = self.data
        K = self.kernel(X)
        num_data = tf.shape(X)[0]
        k_diag = tf.linalg.diag_part(K)
        s_diag = tf.fill([num_data], self.likelihood.variance)
        ks = tf.linalg.set_diag(K, k_diag + s_diag)
        L = tf.linalg.cholesky(ks)
        m = self.mean_function(X)

        # [R,] log-likelihoods for each independent dimension of Y
        log_prob = multivariate_normal(Y, m, L)
        return tf.reduce_sum(log_prob)

    def predict_f(
        self, Xnew: InputData, full_cov: bool = False, full_output_cov: bool = False
    ) -> MeanAndVariance:
        r"""
        This method computes predictions at X \in R^{N \x D} input points

        .. math::
            p(F* | Y)

        where F* are points on the GP at new data points, Y are noisy observations at training data points.
        """
        X_data, Y_data = self.data
        err = Y_data - self.mean_function(X_data)

        kmm = self.kernel(X_data)
        knn = self.kernel(Xnew, full_cov=full_cov)
        kmn = self.kernel(X_data, Xnew)

        num_data = X_data.shape[0]
        s = tf.linalg.diag(tf.fill([num_data], self.likelihood.variance))

        conditional = gpflow.conditionals.base_conditional
        f_mean_zero, f_var = conditional(
            kmn, kmm + s, knn, err, full_cov=full_cov, white=False
        )  # [N, P], [N, P] or [P, N, N]
        f_mean = f_mean_zero + self.mean_function(Xnew)
        return f_mean, f_var