Revision

**aeb0ab472296c0298c2b007c30af2705a75a89f8**authored by ST John on**18 June 2019, 09:46:26 UTC**, committed by ST John on**18 June 2019, 09:48:10 UTC****1 parent**4ad6260

quadrature.py

```
# Copyright 2017-2018 the GPflow authors.
#
# 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 itertools
from collections import Iterable
import numpy as np
import tensorflow as tf
from . import settings
from .core.errors import GPflowError
def hermgauss(n: int):
x, w = np.polynomial.hermite.hermgauss(n)
x, w = x.astype(settings.float_type), w.astype(settings.float_type)
return x, w
def mvhermgauss(H: int, D: int):
"""
Return the evaluation locations 'xn', and weights 'wn' for a multivariate
Gauss-Hermite quadrature.
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'
using Gauss-Hermite quadrature.
: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).
:return: quadratures (N,*Dout)
"""
# Figure out input shape information
if Din is None:
Din = means.shape[1] if type(means.shape) is tuple else means.shape[1].value
if Din is None:
raise GPflowError("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 = tf.shape(means)[0]
# transform points based on Gaussian parameters
cholXcov = tf.cholesky(covs) # NxDxD
Xt = tf.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
# perform quadrature
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 GPflowError("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.
:param funcs: the integrand(s):
Callable or Iterable of Callables that operates elementwise, on the following arguments:
- `Din` positional arguments to match Fmu and Fvar; i.e., 1 if Fmu and Fvar are tensors;
otherwise len(Fmu) (== len(Fvar)) positional arguments F1, F2, ...
- the same keyword arguments as given by **Ys
All arguments will be tensors of shape (N, 1)
: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` (i.e., shape (N,) or (N, 1))
:return: shape is the same as that of the first Fmu
Example use-cases:
Fmu, Fvar are mean and variance of the latent GP, can be shape (N, 1) or (N,)
m1, m2 are 'scalar' functions of a single argument F, broadcasting over arrays
Em1, Em2 = ndiagquad([m1, m2], 50, Fmu, Fvar)
calculates Em1 = ∫ m1(F) N(F; Fmu, Fvar) dF and Em2 = ∫ m2(F) N(F; Fmu, Fvar) dF
for each of the elements of Fmu and Fvar. Em1 and Em2 have the same shape as Fmu.
logp is a 'scalar' function of F and Y
Y are the observations, with shape (N,) or (N, 1) with same length as Fmu and Fvar
Ev = ndiagquad(logp, 50, Fmu, Fvar, Y=Y)
calculates Ev = ∫ logp(F, Y) N(F; Fmu, Fvar) dF (variational expectations)
for each of the elements of Y, Fmu and Fvar. Ev has the same shape as Fmu.
Ep = ndiagquad(logp, 50, Fmu, Fvar, logspace=True, Y=Y)
calculates Ep = log ∫ exp(logp(F, Y)) N(F; Fmu, Fvar) dF (predictive density)
for each of the elements of Y, Fmu and Fvar. Ep has the same shape as Fmu.
Heteroskedastic likelihoods:
g1, g2 are now functions of both F and G
logp is a function of F, G and Y
Gmu, Gvar are mean and variance of a different GP controlling the variance
Em = ndiagquad(m1, 50, Fmu, Fvar)
-> Em1 = ∫∫ m1(F, G) N(F; Fmu, Fvar) N(G; Gmu, Gvar) dF dG
Ev = ndiagquad(logp, 50, Fmu, Fvar, Y=Y)
-> Ev = ∫∫ logp(F, G, Y) N(F; Fmu, Fvar) N(G; Gmu, Gvar) dF dG
(variational expectations)
Ep = ndiagquad(logp, 50, Fmu, Fvar, logspace=True, Y=Y)
-> Ep = log ∫∫ exp(logp(F, G, Y)) N(F; Fmu, Fvar) N(G; Gmu, Gvar) dF dG
(predictive density)
"""
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 x 1 x 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 x H**Din x Din
Xall = gh_x * tf.sqrt(2.0 * Fvar) + Fmu # N x H**Din x Din
Xs = [Xall[:, :, i] for i in range(Din)] # N x 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 x H**Din
def eval_func(f):
feval = f(*Xs, **Ys) # f should be elementwise: return shape N x 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.matmul(feval, gh_w.reshape(-1, 1))
return tf.reshape(result, shape)
if isinstance(funcs, Iterable):
return [eval_func(f) for f in funcs]
else:
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 = tf.shape(Fmu)[0], tf.shape(Fvar)[1]
if epsilon is None:
epsilon = tf.random_normal((S, N, D), dtype=settings.float_type)
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 = tf.shape(Y)[1]
# we can't rely on broadcasting and need tiling
mc_Yr = tf.tile(Y[None, ...], [S, 1, 1]) # S x N x D_out
Ys[name] = tf.reshape(mc_Yr, (S * N, D_out)) # S * N x D_out
def eval_func(func):
feval = func(mc_Xr, **Ys)
feval = tf.reshape(feval, (S, N, -1))
if logspace:
log_S = tf.log(tf.cast(S, settings.float_type))
return tf.reduce_logsumexp(feval, axis=0) - log_S # N x 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)
```

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