##### https://github.com/GPflow/GPflow
Tip revision: 47e788a
``````# Copyright 2017-2018 the GPflow authors.
#
# you may not use this file except in compliance with the License.
# You may obtain a copy of the License at
#
#
# Unless required by applicable law or agreed to in writing, software
# WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
# See the License for the specific language governing permissions and

import itertools
from collections.abc import Iterable

import numpy as np
import tensorflow as tf

from .config import default_float
from .utilities import to_default_float

def hermgauss(n: int):
x, w = np.polynomial.hermite.hermgauss(n)
x, w = x.astype(default_float()), w.astype(default_float())
return x, w

def mvhermgauss(H: int, D: int):
"""
Return the evaluation locations 'xn', and weights 'wn' for a multivariate

The outputs can be used to approximate the following type of integral:
int exp(-x)*f(x) dx ~ sum_i w[i,:]*f(x[i,:])

:param H: Number of Gauss-Hermite evaluation points.
:param D: Number of input dimensions. Needs to be known at call-time.
:return: eval_locations 'x' (H**DxD), weights 'w' (H**D)
"""
gh_x, gh_w = hermgauss(H)
x = np.array(list(itertools.product(*(gh_x,) * D)))  # H**DxD
w = np.prod(np.array(list(itertools.product(*(gh_w,) * D))), 1)  # H**D
return x, w

def mvnquad(func, means, covs, H: int, Din: int = None, Dout=None):
"""
Computes N Gaussian expectation integrals of a single function 'f'
:param f: integrand function. Takes one input of shape ?xD.
:param means: NxD
:param covs: NxDxD
:param H: Number of Gauss-Hermite evaluation points.
:param Din: Number of input dimensions. Needs to be known at call-time.
:param Dout: Number of output dimensions. Defaults to (). Dout is assumed
to leave out the item index, i.e. f actually maps (?xD)->(?x*Dout).
"""
# Figure out input shape information
if Din is None:
Din = means.shape[1]

if Din is None:
raise ValueError(
"If `Din` is passed as `None`, `means` must have a known shape. "
"Running mvnquad in `autoflow` without specifying `Din` and `Dout` "
"is problematic. Consider using your own session."
)  # pragma: no cover

xn, wn = mvhermgauss(H, Din)
N = means.shape[0]

# transform points based on Gaussian parameters
cholXcov = tf.linalg.cholesky(covs)  # NxDxD
Xt = tf.linalg.matmul(
cholXcov, tf.tile(xn[None, :, :], (N, 1, 1)), transpose_b=True
)  # NxDxH**D
X = 2.0 ** 0.5 * Xt + tf.expand_dims(means, 2)  # NxDxH**D
Xr = tf.reshape(tf.transpose(X, [2, 0, 1]), (-1, Din))  # (H**D*N)xD

fevals = func(Xr)
if Dout is None:
Dout = tuple((d if type(d) is int else d.value) for d in fevals.shape[1:])

if any([d is None for d in Dout]):
raise ValueError(
"If `Dout` is passed as `None`, the output of `func` must have known "
"shape. Running mvnquad in `autoflow` without specifying `Din` and `Dout` "
"is problematic. Consider using your own session."
)  # pragma: no cover
fX = tf.reshape(fevals, (H ** Din, N,) + Dout)
wr = np.reshape(wn * np.pi ** (-Din * 0.5), (-1,) + (1,) * (1 + len(Dout)))
return tf.reduce_sum(fX * wr, 0)

def ndiagquad(funcs, H: int, Fmu, Fvar, logspace: bool = False, **Ys):
"""
Computes N Gaussian expectation integrals of one or more functions
using Gauss-Hermite quadrature. The Gaussians must be independent.

The means and variances of the Gaussians are specified by Fmu and Fvar.
The N-integrals are assumed to be taken wrt the last dimensions of Fmu, Fvar.

:param funcs: the integrand(s):
Callable or Iterable of Callables that operates elementwise
:param H: number of Gauss-Hermite quadrature points
:param Fmu: array/tensor or `Din`-tuple/list thereof
:param Fvar: array/tensor or `Din`-tuple/list thereof
:param logspace: if True, funcs are the log-integrands and this calculates
the log-expectation of exp(funcs)
:param **Ys: arrays/tensors; deterministic arguments to be passed by name

Fmu, Fvar, Ys should all have same shape, with overall size `N`
:return: shape is the same as that of the first Fmu
"""

def unify(f_list):
"""
Stack a list of means/vars into a full block
"""
return tf.reshape(tf.concat([tf.reshape(f, (-1, 1)) for f in f_list], axis=1), (-1, 1, Din))

if isinstance(Fmu, (tuple, list)):
Din = len(Fmu)
shape = tf.shape(Fmu[0])
Fmu, Fvar = map(unify, [Fmu, Fvar])  # both [N, 1, Din]
else:
Din = 1
shape = tf.shape(Fmu)
Fmu, Fvar = [tf.reshape(f, (-1, 1, 1)) for f in [Fmu, Fvar]]

xn, wn = mvhermgauss(H, Din)
# xn: H**Din x Din, wn: H**Din

gh_x = xn.reshape(1, -1, Din)  # [1, H]**Din x Din
Xall = gh_x * tf.sqrt(2.0 * Fvar) + Fmu  # [N, H]**Din x Din
Xs = [Xall[:, :, i] for i in range(Din)]  # [N, H]**Din  each

gh_w = wn * np.pi ** (-0.5 * Din)  # H**Din x 1

for name, Y in Ys.items():
Y = tf.reshape(Y, (-1, 1))
Y = tf.tile(Y, [1, H ** Din])  # broadcast Y to match X
# without the tiling, some calls such as tf.where() (in bernoulli) fail
Ys[name] = Y  # now [N, H]**Din

def eval_func(f):
feval = f(*Xs, **Ys)  # f should be elementwise: return shape [N, H]**Din
if logspace:
log_gh_w = np.log(gh_w.reshape(1, -1))
result = tf.reduce_logsumexp(feval + log_gh_w, axis=1)
else:
result = tf.linalg.matmul(feval, gh_w.reshape(-1, 1))
return tf.reshape(result, shape)

if isinstance(funcs, Iterable):
return [eval_func(f) for f in funcs]

return eval_func(funcs)

def ndiag_mc(funcs, S: int, Fmu, Fvar, logspace: bool = False, epsilon=None, **Ys):
"""
Computes N Gaussian expectation integrals of one or more functions
using Monte Carlo samples. The Gaussians must be independent.

:param funcs: the integrand(s):
Callable or Iterable of Callables that operates elementwise
:param S: number of Monte Carlo sampling points
:param Fmu: array/tensor
:param Fvar: array/tensor
:param logspace: if True, funcs are the log-integrands and this calculates
the log-expectation of exp(funcs)
:param **Ys: arrays/tensors; deterministic arguments to be passed by name

Fmu, Fvar, Ys should all have same shape, with overall size `N`
:return: shape is the same as that of the first Fmu
"""
N, D = Fmu.shape[0], Fvar.shape[1]

if epsilon is None:
epsilon = tf.random.normal((S, N, D), dtype=default_float())

mc_x = Fmu[None, :, :] + tf.sqrt(Fvar[None, :, :]) * epsilon
mc_Xr = tf.reshape(mc_x, (S * N, D))

for name, Y in Ys.items():
D_out = Y.shape[1]
# we can't rely on broadcasting and need tiling
mc_Yr = tf.tile(Y[None, ...], [S, 1, 1])  # [S, N, D]_out
Ys[name] = tf.reshape(mc_Yr, (S * N, D_out))  # S * [N, _]out

def eval_func(func):
feval = func(mc_Xr, **Ys)
feval = tf.reshape(feval, (S, N, -1))
if logspace:
log_S = tf.math.log(to_default_float(S))
return tf.reduce_logsumexp(feval, axis=0) - log_S  # [N, D]
else:
return tf.reduce_mean(feval, axis=0)

if isinstance(funcs, Iterable):
return [eval_func(f) for f in funcs]
else:
return eval_func(funcs)
``````