# Copyright 2016 James Hensman, alexggmatthews
#
# 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.
# See the License for the specific language governing permissions and
# limitations under the License.
from typing import Optional

import numpy as np
import tensorflow as tf
import tensorflow_probability as tfp

from ..base import Parameter
from ..conditionals import conditional
from ..inducing_variables import InducingPoints
from ..kernels import Kernel
from ..likelihoods import Likelihood
from ..mean_functions import MeanFunction
from ..models.model import Data, GPModel, MeanAndVariance
from ..utilities import to_default_float
from .util import inducingpoint_wrapper


class SGPMC(GPModel):
    """
    This is the Sparse Variational GP using MCMC (SGPMC). The key reference is

    ::

      @inproceedings{hensman2015mcmc,
        title={MCMC for Variatinoally Sparse Gaussian Processes},
        author={Hensman, James and Matthews, Alexander G. de G.
                and Filippone, Maurizio and Ghahramani, Zoubin},
        booktitle={Proceedings of NIPS},
        year={2015}
      }

    The latent function values are represented by centered
    (whitened) variables, so

    .. math::
       :nowrap:

       \\begin{align}
       \\mathbf v & \\sim N(0, \\mathbf I) \\\\
       \\mathbf u &= \\mathbf L\\mathbf v
       \\end{align}

    with

    .. math::
        \\mathbf L \\mathbf L^\\top = \\mathbf K


    """
    def __init__(self,
                 data: Data,
                 kernel: Kernel,
                 likelihood: Likelihood,
                 mean_function: Optional[MeanFunction] = None,
                 num_latent: int = 1,
                 inducing_variable: Optional[InducingPoints] = None):
        """
        data is a tuple of X, Y with X, a data matrix, size [N, D] and Y, a data matrix, size [N, R]
        Z is a data matrix, of inducing inputs, size [M, D]
        kernel, likelihood, mean_function are appropriate GPflow objects
        """
        super().__init__(kernel, likelihood, mean_function, num_latent=num_latent)
        self.data = data
        self.num_data = data[0].shape[0]
        self.inducing_variable = inducingpoint_wrapper(inducing_variable)
        self.V = Parameter(np.zeros((len(self.inducing_variable), self.num_latent)))
        self.V.prior = tfp.distributions.Normal(loc=to_default_float(0.), scale=to_default_float(1.))

    def log_likelihood(self, *args, **kwargs) -> tf.Tensor:
        """
        This function computes the optimal density for v, q*(v), up to a constant
        """
        # get the (marginals of) q(f): exactly predicting!
        x_data, y_data = self.data
        fmean, fvar = self.predict_f(x_data, full_cov=False)
        return tf.reduce_sum(self.likelihood.variational_expectations(fmean, fvar, y_data))

    def predict_f(self, X: tf.Tensor, full_cov=False, full_output_cov=False) -> MeanAndVariance:
        """
        Xnew is a data matrix, point at which we want to predict

        This method computes

            p(F* | (U=LV) )

        where F* are points on the GP at Xnew, F=LV are points on the GP at Z,

        """
        mu, var = conditional(X,
                              self.inducing_variable,
                              self.kernel,
                              self.V,
                              full_cov=full_cov,
                              q_sqrt=None,
                              white=True,
                              full_output_cov=full_output_cov)
        return mu + self.mean_function(X), var