# 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.
# See the License for the specific language governing permissions and
# limitations under the License.

import enum
from abc import ABC, abstractmethod
from typing import Optional, Tuple, Union, cast

import tensorflow as tf

from . import covariances, kernels, mean_functions
from .base import Module, Parameter, TensorType
from .conditionals.util import (
    base_conditional,
    base_conditional_with_lm,
    expand_independent_outputs,
    fully_correlated_conditional,
    independent_interdomain_conditional,
    mix_latent_gp,
    separate_independent_conditional_implementation,
)
from .config import default_float, default_jitter
from .covariances import Kuf, Kuu
from .inducing_variables import (
    FallbackSeparateIndependentInducingVariables,
    FallbackSharedIndependentInducingVariables,
    InducingPoints,
    InducingVariables,
    SeparateIndependentInducingVariables,
    SharedIndependentInducingVariables,
)
from .types import MeanAndVariance
from .utilities import Dispatcher, add_noise_cov


class _QDistribution(Module):
    """
    Base class for our parametrization of q(u) in the `AbstractPosterior`.
    Internal - do not rely on this outside of GPflow.
    """


class _DeltaDist(_QDistribution):
    def __init__(self, q_mu) -> None:
        self.q_mu = q_mu  # [M, L]

    @property
    def q_sqrt(self):
        return None


class _DiagNormal(_QDistribution):
    def __init__(self, q_mu, q_sqrt) -> None:
        self.q_mu = q_mu  # [M, L]
        self.q_sqrt = q_sqrt  # [M, L]


class _MvNormal(_QDistribution):
    def __init__(self, q_mu, q_sqrt) -> None:
        self.q_mu = q_mu  # [M, L]
        self.q_sqrt = q_sqrt  # [L, M, M], lower-triangular


class PrecomputeCacheType(enum.Enum):
    """
    - `PrecomputeCacheType.TENSOR` (or `"tensor"`): Precomputes the cached
      quantities and stores them as tensors (which allows differentiating
      through the prediction). This is the default.
    - `PrecomputeCacheType.VARIABLE` (or `"variable"`): Precomputes the cached
      quantities and stores them as variables, which allows for updating
      their values without changing the compute graph (relevant for AOT
      compilation).
    - `PrecomputeCacheType.NOCACHE` (or `"nocache"` or `None`): Avoids
      immediate cache computation. This is useful for avoiding extraneous
      computations when you only want to call the posterior's
      `fused_predict_f` method.
    """

    TENSOR = "tensor"
    VARIABLE = "variable"
    NOCACHE = "nocache"


def _validate_precompute_cache_type(value) -> PrecomputeCacheType:
    if value is None:
        return PrecomputeCacheType.NOCACHE
    elif isinstance(value, PrecomputeCacheType):
        return value
    elif isinstance(value, str):
        return PrecomputeCacheType(value.lower())
    else:
        raise ValueError(
            f"{value} is not a valid PrecomputeCacheType. Valid options: 'tensor', 'variable', 'nocache' (or None)."
        )


class AbstractPosterior(Module, ABC):
    def __init__(
        self,
        kernel,
        X_data: Union[tf.Tensor, InducingVariables],
        alpha: Optional[TensorType] = None,
        Qinv: Optional[TensorType] = None,
        mean_function: Optional[mean_functions.MeanFunction] = None,
    ) -> None:
        """
        Users should use `create_posterior` to create instances of concrete
        subclasses of this AbstractPosterior class instead of calling this
        constructor directly. For `create_posterior` to be able to correctly
        instantiate subclasses, developers need to ensure their subclasses
        don't change the constructor signature.
        """
        super().__init__()

        self.kernel = kernel
        self.X_data = X_data
        self.alpha = alpha
        self.Qinv = Qinv
        self.mean_function = mean_function

    def _add_mean_function(self, Xnew, mean) -> tf.Tensor:
        if self.mean_function is None:
            return mean
        else:
            return mean + self.mean_function(Xnew)

    def fused_predict_f(
        self, Xnew, full_cov: bool = False, full_output_cov: bool = False
    ) -> MeanAndVariance:
        """
        Computes predictive mean and (co)variance at Xnew, including mean_function
        Does not make use of caching
        """
        mean, cov = self._conditional_fused(
            Xnew, full_cov=full_cov, full_output_cov=full_output_cov
        )
        return self._add_mean_function(Xnew, mean), cov

    @abstractmethod
    def _conditional_fused(
        self, Xnew, full_cov: bool = False, full_output_cov: bool = False
    ) -> MeanAndVariance:
        """
        Computes predictive mean and (co)variance at Xnew, *excluding* mean_function
        Does not make use of caching
        """

    def predict_f(
        self, Xnew, full_cov: bool = False, full_output_cov: bool = False
    ) -> MeanAndVariance:
        """
        Computes predictive mean and (co)variance at Xnew, including mean_function.
        Relies on precomputed alpha and Qinv (see _precompute method)
        """
        if self.alpha is None or self.Qinv is None:
            raise ValueError(
                "Cache has not been precomputed yet. Call update_cache first or use fused_predict_f"
            )
        mean, cov = self._conditional_with_precompute(
            Xnew, full_cov=full_cov, full_output_cov=full_output_cov
        )
        return self._add_mean_function(Xnew, mean), cov

    @abstractmethod
    def _conditional_with_precompute(
        self, Xnew, full_cov: bool = False, full_output_cov: bool = False
    ) -> MeanAndVariance:
        """
        Computes predictive mean and (co)variance at Xnew, *excluding* mean_function.
        Relies on cached alpha and Qinv.
        """

    def update_cache(self, precompute_cache: Optional[PrecomputeCacheType] = None) -> None:
        """
        Sets the cache depending on the value of `precompute_cache` to a
        `tf.Tensor`, `tf.Variable`, or clears the cache. If `precompute_cache`
        is not given, the setting defaults to the most-recently-used one.
        """
        if precompute_cache is None:
            try:
                precompute_cache = cast(
                    PrecomputeCacheType,
                    self._precompute_cache,  # type: ignore
                )
            except AttributeError:
                raise ValueError(
                    "You must pass precompute_cache explicitly (the cache had not been updated before)."
                )
        else:
            self._precompute_cache = precompute_cache

        if precompute_cache is PrecomputeCacheType.NOCACHE:
            self.alpha = self.Qinv = None

        elif precompute_cache is PrecomputeCacheType.TENSOR:
            self.alpha, self.Qinv = self._precompute()

        elif precompute_cache is PrecomputeCacheType.VARIABLE:
            alpha, Qinv = self._precompute()
            if isinstance(self.alpha, tf.Variable) and isinstance(self.Qinv, tf.Variable):
                # re-use existing variables
                self.alpha.assign(alpha)
                self.Qinv.assign(Qinv)
            else:  # create variables
                self.alpha = tf.Variable(alpha, trainable=False)
                self.Qinv = tf.Variable(Qinv, trainable=False)


class GPRPosterior(AbstractPosterior):
    def __init__(
        self,
        kernel,
        data: Tuple[tf.Tensor, tf.Tensor],
        likelihood_variance: Parameter,
        mean_function: Optional[mean_functions.MeanFunction] = None,
        *,
        precompute_cache: Optional[PrecomputeCacheType],
    ) -> None:
        X, Y = data
        super().__init__(kernel, X, mean_function=mean_function)
        self.mean_function = mean_function
        self.Y_data = Y
        self.likelihood_variance = likelihood_variance

        if precompute_cache is not None:
            self.update_cache(precompute_cache)

    def _conditional_with_precompute(
        self, Xnew, full_cov: bool = False, full_output_cov: bool = False
    ) -> MeanAndVariance:
        """
        Computes predictive mean and (co)variance at Xnew, *excluding* mean_function.
        Relies on cached alpha and Qinv.
        """
        Kmn = self.kernel(self.X_data, Xnew)
        Knn = self.kernel(Xnew, full_cov=full_cov)

        return base_conditional_with_lm(Kmn, self.Qinv, Knn, self.alpha, full_cov=full_cov)

    def _precompute(self) -> Tuple[tf.Tensor, tf.Tensor]:

        """
        Precomputes the cholesky decomposition of Kmm_plus_s for later reuse we will call
        base_conditional_with_lm implementation ('Qinv' in the Abstract Posterior class). We also
        precompute the less compute intensive error term ('alpha' in the Abstract Posterior class)
        """

        Kmm = self.kernel(self.X_data)
        Kmm_plus_s = add_noise_cov(Kmm, self.likelihood_variance)

        # obtain the cholesky decomposition of Kmm_plus_s
        Lm = tf.linalg.cholesky(Kmm_plus_s)

        alpha = self.Y_data - self.mean_function(self.X_data)  # type: ignore
        tf.debugging.assert_shapes(
            [
                (Lm, ["M", "M"]),
                (Kmm, ["M", "M"]),
            ]
        )
        return alpha, Lm

    def _conditional_fused(
        self, Xnew, full_cov: bool = False, full_output_cov: bool = False
    ) -> MeanAndVariance:
        """
        Computes predictive mean and (co)variance at Xnew, *excluding* mean_function
        Does not make use of caching
        """

        # taken directly from the deprecated GPR implementation
        err = self.Y_data - self.mean_function(self.X_data)  # type: ignore

        Kmm = self.kernel(self.X_data)
        Knn = self.kernel(Xnew, full_cov=full_cov)
        Kmn = self.kernel(self.X_data, Xnew)
        Kmm_plus_s = add_noise_cov(Kmm, self.likelihood_variance)

        return base_conditional(
            Kmn, Kmm_plus_s, Knn, err, full_cov=full_cov, white=False
        )  # [N, P], [N, P] or [P, N, N]


class SGPRPosterior(AbstractPosterior):
    """
    This class represents posteriors which can be derived from SGPR
    models to compute faster predictions on unseen points.
    """

    def __init__(
        self,
        kernel,
        data: Tuple[tf.Tensor, tf.Tensor],
        inducing_variable: InducingPoints,
        likelihood_variance: Parameter,
        num_latent_gps: int,
        mean_function: Optional[mean_functions.MeanFunction] = None,
        *,
        precompute_cache: Optional[PrecomputeCacheType],
    ) -> None:
        X, Y = data
        super().__init__(kernel, X, mean_function=mean_function)
        self.mean_function = mean_function
        self.Y_data = Y
        self.likelihood_variance = likelihood_variance
        self.inducing_variable = inducing_variable
        self.num_latent_gps = num_latent_gps

        if precompute_cache is not None:
            self.update_cache(precompute_cache)

    def _precompute(self) -> Tuple[tf.Tensor, tf.Tensor]:

        """
        Precomputes the cholesky decomposition of Kmm_plus_s for later reuse we will call
        base_conditional_with_lm implementation ('Qinv' in the Abstract Posterior class). We also
        precompute the less compute intensive error term ('alpha' in the Abstract Posterior class)
        """

        # taken directly from the deprecated SGPR implementation
        num_inducing = self.inducing_variable.num_inducing
        err = self.Y_data - self.mean_function(self.X_data)  # type:ignore
        kuf = Kuf(self.inducing_variable, self.kernel, self.X_data)
        kuu = Kuu(self.inducing_variable, self.kernel, jitter=default_jitter())
        sigma = tf.sqrt(self.likelihood_variance)
        L = tf.linalg.cholesky(kuu)  # cache alpha, qinv
        A = tf.linalg.triangular_solve(L, kuf, lower=True) / sigma
        B = tf.linalg.matmul(A, A, transpose_b=True) + tf.eye(
            num_inducing, dtype=default_float()
        )  # cache qinv
        LB = tf.linalg.cholesky(B)  # cache alpha
        Aerr = tf.linalg.matmul(A, err)
        c = tf.linalg.triangular_solve(LB, Aerr, lower=True) / sigma  # cache alpha

        # get intermediate variables
        Linv = tf.linalg.triangular_solve(L, tf.eye(num_inducing, dtype=default_float()))
        LBinv = tf.linalg.triangular_solve(LB, tf.eye(num_inducing, dtype=default_float()))
        Binv = tf.linalg.inv(B)  # naive...can do better?
        tmp = tf.eye(num_inducing, dtype=default_float()) - Binv

        # calculate cached values
        LinvT = tf.transpose(Linv)
        alpha = LinvT @ tf.transpose(LBinv) @ c
        Qinv = LinvT @ tmp @ Linv

        return alpha, Qinv

    def _conditional_with_precompute(
        self, Xnew, full_cov: bool = False, full_output_cov: bool = False
    ) -> MeanAndVariance:
        """
        Computes predictive mean and (co)variance at Xnew, *excluding* mean_function.
        Relies on cached alpha and Qinv.
        """
        Kus = Kuf(self.inducing_variable, self.kernel, Xnew)
        Knn = self.kernel(Xnew, full_cov=full_cov)

        Ksu = tf.transpose(Kus)
        mean = Ksu @ self.alpha

        if full_cov:
            var = Knn - Ksu @ self.Qinv @ Kus
            var = tf.tile(var[None, ...], [self.num_latent_gps, 1, 1])  # [P, N, N]
        else:
            Kfu_Qinv_Kuf = tf.reduce_sum(Kus * tf.matmul(self.Qinv, Kus), axis=-2)
            var = Knn - Kfu_Qinv_Kuf
            var = tf.tile(var[:, None], [1, self.num_latent_gps])

        return mean, var

    def _conditional_fused(
        self, Xnew, full_cov: bool = False, full_output_cov: bool = False
    ) -> MeanAndVariance:
        """
        Compute the mean and variance of the latent function at some new points
        Xnew. Does not make use of caching
        """

        # taken directly from the deprecated SGPR implementation
        num_inducing = self.inducing_variable.num_inducing
        err = self.Y_data - self.mean_function(self.X_data)  # type: ignore
        kuf = Kuf(self.inducing_variable, self.kernel, self.X_data)
        kuu = Kuu(self.inducing_variable, self.kernel, jitter=default_jitter())
        Kus = Kuf(self.inducing_variable, self.kernel, Xnew)
        sigma = tf.sqrt(self.likelihood_variance)  # type: ignore
        L = tf.linalg.cholesky(kuu)  # cache alpha, qinv
        A = tf.linalg.triangular_solve(L, kuf, lower=True) / sigma
        B = tf.linalg.matmul(A, A, transpose_b=True) + tf.eye(
            num_inducing, dtype=default_float()
        )  # cache qinv
        LB = tf.linalg.cholesky(B)  # cache alpha
        Aerr = tf.linalg.matmul(A, err)
        c = tf.linalg.triangular_solve(LB, Aerr, lower=True) / sigma  # cache alpha
        tmp1 = tf.linalg.triangular_solve(L, Kus, lower=True)
        tmp2 = tf.linalg.triangular_solve(LB, tmp1, lower=True)
        mean = tf.linalg.matmul(tmp2, c, transpose_a=True)
        if full_cov:
            var = (
                self.kernel(Xnew)
                + tf.linalg.matmul(tmp2, tmp2, transpose_a=True)
                - tf.linalg.matmul(tmp1, tmp1, transpose_a=True)
            )
            var = tf.tile(var[None, ...], [self.num_latent_gps, 1, 1])  # [P, N, N]
        else:
            var = (
                self.kernel(Xnew, full_cov=False)
                + tf.reduce_sum(tf.square(tmp2), 0)
                - tf.reduce_sum(tf.square(tmp1), 0)
            )
            var = tf.tile(var[:, None], [1, self.num_latent_gps])

        return mean, var


class BasePosterior(AbstractPosterior):
    def __init__(
        self,
        kernel,
        inducing_variable,
        q_mu: tf.Tensor,
        q_sqrt: tf.Tensor,
        whiten: bool = True,
        mean_function: Optional[mean_functions.MeanFunction] = None,
        *,
        precompute_cache: Optional[PrecomputeCacheType],
    ):

        super().__init__(kernel, inducing_variable, mean_function=mean_function)
        self.whiten = whiten
        self._set_qdist(q_mu, q_sqrt)

        if precompute_cache is not None:
            self.update_cache(precompute_cache)

    @property
    def q_mu(self):
        return self._q_dist.q_mu

    @property
    def q_sqrt(self):
        return self._q_dist.q_sqrt

    def _set_qdist(self, q_mu, q_sqrt):
        if q_sqrt is None:
            self._q_dist = _DeltaDist(q_mu)
        elif len(q_sqrt.shape) == 2:  # q_diag
            self._q_dist = _DiagNormal(q_mu, q_sqrt)
        else:
            self._q_dist = _MvNormal(q_mu, q_sqrt)

    def _precompute(self):
        Kuu = covariances.Kuu(self.X_data, self.kernel, jitter=default_jitter())  # [(R), M, M]
        q_mu = self._q_dist.q_mu

        if Kuu.shape.ndims == 4:
            ML = tf.reduce_prod(tf.shape(Kuu)[:2])
            Kuu = tf.reshape(Kuu, [ML, ML])
        if Kuu.shape.ndims == 3:
            q_mu = tf.linalg.adjoint(self._q_dist.q_mu)[..., None]  # [..., R, M, 1]
        L = tf.linalg.cholesky(Kuu)

        if not self.whiten:
            # alpha = Kuu⁻¹ q_mu
            alpha = tf.linalg.cholesky_solve(L, q_mu)
        else:
            # alpha = L⁻ᵀ q_mu
            alpha = tf.linalg.triangular_solve(L, q_mu, adjoint=True)
        # predictive mean = Kfu alpha
        # predictive variance = Kff - Kfu Qinv Kuf
        # S = q_sqrt q_sqrtᵀ
        I = tf.eye(tf.shape(L)[-1], dtype=L.dtype)
        if isinstance(self._q_dist, _DeltaDist):
            B = I
        else:
            if not self.whiten:
                # Qinv = Kuu⁻¹ - Kuu⁻¹ S Kuu⁻¹
                #      = Kuu⁻¹ - L⁻ᵀ L⁻¹ S L⁻ᵀ L⁻¹
                #      = L⁻ᵀ (I - L⁻¹ S L⁻ᵀ) L⁻¹
                #      = L⁻ᵀ B L⁻¹
                if isinstance(self._q_dist, _DiagNormal):
                    q_sqrt = tf.linalg.diag(tf.linalg.adjoint(self._q_dist.q_sqrt))
                elif isinstance(self._q_dist, _MvNormal):
                    q_sqrt = self._q_dist.q_sqrt
                Linv_qsqrt = tf.linalg.triangular_solve(L, q_sqrt)
                Linv_cov_u_LinvT = tf.matmul(Linv_qsqrt, Linv_qsqrt, transpose_b=True)
            else:
                if isinstance(self._q_dist, _DiagNormal):
                    Linv_cov_u_LinvT = tf.linalg.diag(tf.linalg.adjoint(self._q_dist.q_sqrt ** 2))
                elif isinstance(self._q_dist, _MvNormal):
                    q_sqrt = self._q_dist.q_sqrt
                    Linv_cov_u_LinvT = tf.matmul(q_sqrt, q_sqrt, transpose_b=True)
                # Qinv = Kuu⁻¹ - L⁻ᵀ S L⁻¹
                # Linv = (L⁻¹ I) = solve(L, I)
                # Kinv = Linvᵀ @ Linv
            B = I - Linv_cov_u_LinvT
        LinvT_B = tf.linalg.triangular_solve(L, B, adjoint=True)
        B_Linv = tf.linalg.adjoint(LinvT_B)
        Qinv = tf.linalg.triangular_solve(L, B_Linv, adjoint=True)

        M, L = tf.unstack(tf.shape(self._q_dist.q_mu), num=2)
        Qinv = tf.broadcast_to(Qinv, [L, M, M])

        tf.debugging.assert_shapes(
            [
                (Qinv, ["L", "M", "M"]),
            ]
        )

        return alpha, Qinv


class IndependentPosterior(BasePosterior):
    def _post_process_mean_and_cov(self, mean, cov, full_cov, full_output_cov):
        return mean, expand_independent_outputs(cov, full_cov, full_output_cov)

    def _get_Kff(self, Xnew, full_cov):

        # TODO: this assumes that Xnew has shape [N, D] and no leading dims

        if isinstance(self.kernel, (kernels.SeparateIndependent, kernels.IndependentLatent)):
            # NOTE calling kernel(Xnew, full_cov=full_cov, full_output_cov=False) directly would return
            # if full_cov: [P, N, N] -- this is what we want
            # else: [N, P] instead of [P, N] as we get from the explicit stack below
            Kff = tf.stack([k(Xnew, full_cov=full_cov) for k in self.kernel.kernels], axis=0)
        elif isinstance(self.kernel, kernels.MultioutputKernel):
            # effectively, SharedIndependent path
            Kff = self.kernel.kernel(Xnew, full_cov=full_cov)
            # NOTE calling kernel(Xnew, full_cov=full_cov, full_output_cov=False) directly would return
            # if full_cov: [P, N, N] instead of [N, N]
            # else: [N, P] instead of [N]
        else:
            # standard ("single-output") kernels
            Kff = self.kernel(Xnew, full_cov=full_cov)  # [N, N] if full_cov else [N]

        return Kff

    def _conditional_with_precompute(
        self, Xnew, full_cov: bool = False, full_output_cov: bool = False
    ) -> MeanAndVariance:
        # Qinv: [L, M, M]
        # alpha: [M, L]

        Kuf = covariances.Kuf(self.X_data, self.kernel, Xnew)  # [(R), M, N]
        Kff = self._get_Kff(Xnew, full_cov)

        mean = tf.matmul(Kuf, self.alpha, transpose_a=True)
        if Kuf.shape.ndims == 3:
            mean = tf.linalg.adjoint(tf.squeeze(mean, axis=-1))

        if full_cov:
            Kfu_Qinv_Kuf = tf.matmul(Kuf, self.Qinv @ Kuf, transpose_a=True)
            cov = Kff - Kfu_Qinv_Kuf
        else:
            # [Aᵀ B]_ij = Aᵀ_ik B_kj = A_ki B_kj
            # TODO check whether einsum is faster now?
            Kfu_Qinv_Kuf = tf.reduce_sum(Kuf * tf.matmul(self.Qinv, Kuf), axis=-2)
            cov = Kff - Kfu_Qinv_Kuf
            cov = tf.linalg.adjoint(cov)

        return self._post_process_mean_and_cov(mean, cov, full_cov, full_output_cov)


class IndependentPosteriorSingleOutput(IndependentPosterior):
    # could almost be the same as IndependentPosteriorMultiOutput ...
    def _conditional_fused(
        self, Xnew, full_cov: bool = False, full_output_cov: bool = False
    ) -> MeanAndVariance:
        # same as IndependentPosteriorMultiOutput, Shared~/Shared~ branch, except for following line:
        Knn = self.kernel(Xnew, full_cov=full_cov)

        Kmm = covariances.Kuu(self.X_data, self.kernel, jitter=default_jitter())  # [M, M]
        Kmn = covariances.Kuf(self.X_data, self.kernel, Xnew)  # [M, N]

        fmean, fvar = base_conditional(
            Kmn, Kmm, Knn, self.q_mu, full_cov=full_cov, q_sqrt=self.q_sqrt, white=self.whiten
        )  # [N, P],  [P, N, N] or [N, P]
        return self._post_process_mean_and_cov(fmean, fvar, full_cov, full_output_cov)


class IndependentPosteriorMultiOutput(IndependentPosterior):
    def _conditional_fused(
        self, Xnew, full_cov: bool = False, full_output_cov: bool = False
    ) -> MeanAndVariance:
        if isinstance(self.X_data, SharedIndependentInducingVariables) and isinstance(
            self.kernel, kernels.SharedIndependent
        ):
            # same as IndependentPosteriorSingleOutput except for following line
            Knn = self.kernel.kernel(Xnew, full_cov=full_cov)
            # we don't call self.kernel() directly as that would do unnecessary tiling

            Kmm = covariances.Kuu(self.X_data, self.kernel, jitter=default_jitter())  # [M, M]
            Kmn = covariances.Kuf(self.X_data, self.kernel, Xnew)  # [M, N]

            fmean, fvar = base_conditional(
                Kmn, Kmm, Knn, self.q_mu, full_cov=full_cov, q_sqrt=self.q_sqrt, white=self.whiten
            )  # [N, P],  [P, N, N] or [N, P]
        else:
            # this is the messy thing with tf.map_fn, cleaned up by the st/clean_up_broadcasting_conditionals branch

            # Following are: [P, M, M]  -  [P, M, N]  -  [P, N](x N)
            Kmms = covariances.Kuu(self.X_data, self.kernel, jitter=default_jitter())  # [P, M, M]
            Kmns = covariances.Kuf(self.X_data, self.kernel, Xnew)  # [P, M, N]
            if isinstance(self.kernel, kernels.Combination):
                kernel_list = self.kernel.kernels
            else:
                kernel_list = [self.kernel.kernel] * len(self.X_data.inducing_variable_list)
            Knns = tf.stack(
                [k.K(Xnew) if full_cov else k.K_diag(Xnew) for k in kernel_list], axis=0
            )

            fmean, fvar = separate_independent_conditional_implementation(
                Kmns,
                Kmms,
                Knns,
                self.q_mu,
                q_sqrt=self.q_sqrt,
                full_cov=full_cov,
                white=self.whiten,
            )

        return self._post_process_mean_and_cov(fmean, fvar, full_cov, full_output_cov)


class LinearCoregionalizationPosterior(IndependentPosteriorMultiOutput):
    def _post_process_mean_and_cov(self, mean, cov, full_cov, full_output_cov):
        """
        mean: [N, L]
        cov: [L, N, N] or [N, L]
        """
        cov = expand_independent_outputs(cov, full_cov, full_output_cov=False)
        mean, cov = mix_latent_gp(self.kernel.W, mean, cov, full_cov, full_output_cov)
        return mean, cov


class FullyCorrelatedPosterior(BasePosterior):
    def _conditional_with_precompute(
        self, Xnew, full_cov: bool = False, full_output_cov: bool = False
    ) -> MeanAndVariance:
        # TODO: this assumes that Xnew has shape [N, D] and no leading dims

        # Qinv: [L, M, M]
        # alpha: [M, L]

        Kuf = covariances.Kuf(self.X_data, self.kernel, Xnew)
        assert Kuf.shape.ndims == 4
        M, L, N, K = tf.unstack(tf.shape(Kuf), num=Kuf.shape.ndims, axis=0)
        Kuf = tf.reshape(Kuf, (M * L, N * K))

        Kff = self.kernel(Xnew, full_cov=full_cov, full_output_cov=full_output_cov)
        # full_cov=True and full_output_cov=True: [N, P, N, P]
        # full_cov=True and full_output_cov=False: [P, N, N]
        # full_cov=False and full_output_cov=True: [N, P, P]
        # full_cov=False and full_output_cov=False: [N, P]
        if full_cov == full_output_cov:
            new_shape = (N * K, N * K) if full_cov else (N * K,)
            Kff = tf.reshape(Kff, new_shape)

        N = tf.shape(Xnew)[0]
        K = tf.shape(Kuf)[-1] // N

        mean = tf.matmul(Kuf, self.alpha, transpose_a=True)
        if Kuf.shape.ndims == 3:
            mean = tf.linalg.adjoint(tf.squeeze(mean, axis=-1))

        if not full_cov and not full_output_cov:
            # fully diagonal case in both inputs and outputs
            # [Aᵀ B]_ij = Aᵀ_ik B_kj = A_ki B_kj
            # TODO check whether einsum is faster now?
            Kfu_Qinv_Kuf = tf.reduce_sum(Kuf * tf.matmul(self.Qinv, Kuf), axis=-2)
        else:
            Kfu_Qinv_Kuf = tf.matmul(Kuf, self.Qinv @ Kuf, transpose_a=True)
            if not (full_cov and full_output_cov):
                # diagonal in either inputs or outputs
                new_shape = tf.concat([tf.shape(Kfu_Qinv_Kuf)[:-2], (N, K, N, K)], axis=0)
                Kfu_Qinv_Kuf = tf.reshape(Kfu_Qinv_Kuf, new_shape)
                if full_cov:
                    # diagonal in outputs: move outputs to end
                    tmp = tf.linalg.diag_part(tf.einsum("...ijkl->...ikjl", Kfu_Qinv_Kuf))
                elif full_output_cov:
                    # diagonal in inputs: move inputs to end
                    tmp = tf.linalg.diag_part(tf.einsum("...ijkl->...jlik", Kfu_Qinv_Kuf))
                Kfu_Qinv_Kuf = tf.einsum("...ijk->...kij", tmp)  # move diagonal dim to [-3]
        cov = Kff - Kfu_Qinv_Kuf

        if not full_cov and not full_output_cov:
            cov = tf.linalg.adjoint(cov)

        mean = tf.reshape(mean, (N, K))
        if full_cov == full_output_cov:
            cov_shape = (N, K, N, K) if full_cov else (N, K)
        else:
            cov_shape = (K, N, N) if full_cov else (N, K, K)
        cov = tf.reshape(cov, cov_shape)

        return mean, cov

    def _conditional_fused(
        self, Xnew, full_cov: bool = False, full_output_cov: bool = False
    ) -> MeanAndVariance:
        Kmm = covariances.Kuu(self.X_data, self.kernel, jitter=default_jitter())  # [M, L, M, L]
        Kmn = covariances.Kuf(self.X_data, self.kernel, Xnew)  # [M, L, N, P]
        Knn = self.kernel(
            Xnew, full_cov=full_cov, full_output_cov=full_output_cov
        )  # [N, P](x N)x P  or  [N, P](x P)

        M, L, N, K = tf.unstack(tf.shape(Kmn), num=Kmn.shape.ndims, axis=0)
        Kmm = tf.reshape(Kmm, (M * L, M * L))

        if full_cov == full_output_cov:
            Kmn = tf.reshape(Kmn, (M * L, N * K))
            Knn = tf.reshape(Knn, (N * K, N * K)) if full_cov else tf.reshape(Knn, (N * K,))
            mean, cov = base_conditional(
                Kmn, Kmm, Knn, self.q_mu, full_cov=full_cov, q_sqrt=self.q_sqrt, white=self.whiten
            )  # [K, 1], [1, K](x NK)
            mean = tf.reshape(mean, (N, K))
            cov = tf.reshape(cov, (N, K, N, K) if full_cov else (N, K))
        else:
            Kmn = tf.reshape(Kmn, (M * L, N, K))
            mean, cov = fully_correlated_conditional(
                Kmn,
                Kmm,
                Knn,
                self.q_mu,
                full_cov=full_cov,
                full_output_cov=full_output_cov,
                q_sqrt=self.q_sqrt,
                white=self.whiten,
            )
        return mean, cov


class FallbackIndependentLatentPosterior(FullyCorrelatedPosterior):  # XXX
    def _conditional_fused(
        self, Xnew, full_cov: bool = False, full_output_cov: bool = False
    ) -> MeanAndVariance:
        Kmm = covariances.Kuu(self.X_data, self.kernel, jitter=default_jitter())  # [L, M, M]
        Kmn = covariances.Kuf(self.X_data, self.kernel, Xnew)  # [M, L, N, P]
        Knn = self.kernel(
            Xnew, full_cov=full_cov, full_output_cov=full_output_cov
        )  # [N, P](x N)x P  or  [N, P](x P)

        return independent_interdomain_conditional(
            Kmn,
            Kmm,
            Knn,
            self.q_mu,
            full_cov=full_cov,
            full_output_cov=full_output_cov,
            q_sqrt=self.q_sqrt,
            white=self.whiten,
        )


get_posterior_class = Dispatcher("get_posterior_class")


@get_posterior_class.register(kernels.Kernel, InducingVariables)
def _get_posterior_base_case(kernel, inducing_variable):
    # independent single output
    return IndependentPosteriorSingleOutput


@get_posterior_class.register(kernels.MultioutputKernel, InducingPoints)
def _get_posterior_fully_correlated_mo(kernel, inducing_variable):
    return FullyCorrelatedPosterior


@get_posterior_class.register(
    (kernels.SharedIndependent, kernels.SeparateIndependent),
    (SeparateIndependentInducingVariables, SharedIndependentInducingVariables),
)
def _get_posterior_independent_mo(kernel, inducing_variable):
    # independent multi-output
    return IndependentPosteriorMultiOutput


@get_posterior_class.register(
    kernels.IndependentLatent,
    (FallbackSeparateIndependentInducingVariables, FallbackSharedIndependentInducingVariables),
)
def _get_posterior_independentlatent_mo_fallback(kernel, inducing_variable):
    return FallbackIndependentLatentPosterior


@get_posterior_class.register(
    kernels.LinearCoregionalization,
    (SeparateIndependentInducingVariables, SharedIndependentInducingVariables),
)
def _get_posterior_linearcoregionalization_mo_efficient(kernel, inducing_variable):
    # Linear mixing---efficient multi-output
    return LinearCoregionalizationPosterior


def create_posterior(
    kernel,
    inducing_variable,
    q_mu,
    q_sqrt,
    whiten,
    mean_function=None,
    precompute_cache: Union[PrecomputeCacheType, str, None] = PrecomputeCacheType.TENSOR,
):
    posterior_class = get_posterior_class(kernel, inducing_variable)
    precompute_cache = _validate_precompute_cache_type(precompute_cache)
    return posterior_class(
        kernel,
        inducing_variable,
        q_mu,
        q_sqrt,
        whiten,
        mean_function,
        precompute_cache=precompute_cache,
    )