Revision b08f3062c96677de266af26767634fd7c6e6611d authored by Alexander G. de G. Matthews on 09 September 2016, 10:59:46 UTC, committed by James Hensman on 09 September 2016, 10:59:46 UTC
* Renaming tf_hacks to tf_wraps

* Changing tf_hacks to tf_wraps in code.

* adding a tf_hacks file that raises deprecationwarnings

* release notes

* bumpng version on docs

* importing tf_hacks, tf_wraps

1 parent 61b0659
likelihoods.py
# Copyright 2016 Valentine Svensson, James Hensman, alexggmatthews, Alexis Boukouvalas
#
# 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

from __future__ import absolute_import
from . import densities, transforms
from .param import Parameterized, Param
import tensorflow as tf
import numpy as np

hermgauss = np.polynomial.hermite.hermgauss

class Likelihood(Parameterized):
def __init__(self):
Parameterized.__init__(self)
self.scoped_keys.extend(['logp', 'variational_expectations', 'predict_mean_and_var', 'predict_density'])
self.num_gauss_hermite_points = 20

def logp(self, F, Y):
"""
Return the log density of the data given the function values.
"""
raise NotImplementedError("implement the logp function\
for this likelihood")

def conditional_mean(self, F):
"""
Given a value of the latent function, compute the mean of the data

If this object represents

p(y|f)

then this method computes

\int y p(y|f) dy
"""
raise NotImplementedError

def conditional_variance(self, F):
"""
Given a value of the latent function, compute the variance of the data

If this object represents

p(y|f)

then this method computes

\int y^2 p(y|f) dy  - [\int y p(y|f) dy] ^ 2

"""
raise NotImplementedError

def predict_mean_and_var(self, Fmu, Fvar):
"""
Given a Normal distribution for the latent function,
return the mean of Y

if
q(f) = N(Fmu, Fvar)

and this object represents

p(y|f)

then this method computes the predictive mean

\int\int y p(y|f)q(f) df dy

and the predictive variance

\int\int y^2 p(y|f)q(f) df dy  - [ \int\int y^2 p(y|f)q(f) df dy ]^2

Here, we implement a default Gauss-Hermite quadrature routine, but some
likelihoods (e.g. Gaussian) will implement specific cases.
"""
gh_x, gh_w = hermgauss(self.num_gauss_hermite_points)
gh_w /= np.sqrt(np.pi)
gh_w = gh_w.reshape(-1, 1)
shape = tf.shape(Fmu)
Fmu, Fvar = [tf.reshape(e, (-1, 1)) for e in (Fmu, Fvar)]
X = gh_x[None, :] * tf.sqrt(2.0 * Fvar) + Fmu

# here's the quadrature for the mean
E_y = tf.reshape(tf.matmul(self.conditional_mean(X), gh_w), shape)

# here's the quadrature for the variance
integrand = self.conditional_variance(X) \
+ tf.square(self.conditional_mean(X))
V_y = tf.reshape(tf.matmul(integrand, gh_w), shape) - tf.square(E_y)

return E_y, V_y

def predict_density(self, Fmu, Fvar, Y):
"""
Given a Normal distribution for the latent function, and a datum Y,
compute the (log) predictive density of Y.

i.e. if
q(f) = N(Fmu, Fvar)

and this object represents

p(y|f)

then this method computes the predictive density

\int p(y=Y|f)q(f) df

Here, we implement a default Gauss-Hermite quadrature routine, but some
likelihoods (Gaussian, Poisson) will implement specific cases.
"""
gh_x, gh_w = hermgauss(self.num_gauss_hermite_points)

gh_w = gh_w.reshape(-1, 1) / np.sqrt(np.pi)
shape = tf.shape(Fmu)
Fmu, Fvar, Y = [tf.reshape(e, (-1, 1)) for e in (Fmu, Fvar, Y)]
X = gh_x[None, :] * tf.sqrt(2.0 * Fvar) + Fmu

Y = tf.tile(Y, [1, self.num_gauss_hermite_points])  # broadcast Y to match X

logp = self.logp(X, Y)
return tf.reshape(tf.log(tf.matmul(tf.exp(logp), gh_w)), shape)

def variational_expectations(self, Fmu, Fvar, Y):
"""
Compute the expected log density of the data, given a Gaussian
distribution for the function values.

if
q(f) = N(Fmu, Fvar)

and this object represents

p(y|f)

then this method computes

\int (\log p(y|f)) q(f) df.

Here, we implement a default Gauss-Hermite quadrature routine, but some
likelihoods (Gaussian, Poisson) will implement specific cases.
"""

gh_x, gh_w = hermgauss(self.num_gauss_hermite_points)
gh_x = gh_x.reshape(1, -1)
gh_w = gh_w.reshape(-1, 1) / np.sqrt(np.pi)
shape = tf.shape(Fmu)
Fmu, Fvar, Y = [tf.reshape(e, (-1, 1)) for e in (Fmu, Fvar, Y)]
X = gh_x * tf.sqrt(2.0 * Fvar) + Fmu
Y = tf.tile(Y, [1, self.num_gauss_hermite_points])  # broadcast Y to match X

logp = self.logp(X, Y)
return tf.reshape(tf.matmul(logp, gh_w), shape)

class Gaussian(Likelihood):
def __init__(self):
Likelihood.__init__(self)
self.variance = Param(1.0, transforms.positive)

def logp(self, F, Y):
return densities.gaussian(F, Y, self.variance)

def conditional_mean(self, F):
return tf.identity(F)

def conditional_variance(self, F):
return tf.fill(tf.shape(F), tf.squeeze(self.variance))

def predict_mean_and_var(self, Fmu, Fvar):
return tf.identity(Fmu), Fvar + self.variance

def predict_density(self, Fmu, Fvar, Y):
return densities.gaussian(Fmu, Y, Fvar + self.variance)

def variational_expectations(self, Fmu, Fvar, Y):
return -0.5 * np.log(2 * np.pi) - 0.5 * tf.log(self.variance) \
- 0.5 * (tf.square(Y - Fmu) + Fvar) / self.variance

class Poisson(Likelihood):
Likelihood.__init__(self)

def logp(self, F, Y):

def conditional_variance(self, F):

def conditional_mean(self, F):

def variational_expectations(self, Fmu, Fvar, Y):
return Y * Fmu - tf.exp(Fmu + Fvar / 2) - tf.lgamma(Y + 1)
else:
return Likelihood.variational_expectations(self, Fmu, Fvar, Y)

class Exponential(Likelihood):
Likelihood.__init__(self)

def logp(self, F, Y):

def conditional_mean(self, F):

def conditional_variance(self, F):

def variational_expectations(self, Fmu, Fvar, Y):
return -tf.exp(-Fmu + Fvar / 2) * Y - Fmu
else:
return Likelihood.variational_expectations(self, Fmu, Fvar, Y)

class StudentT(Likelihood):
def __init__(self, deg_free=3.0):
Likelihood.__init__(self)
self.deg_free = deg_free
self.scale = Param(1.0, transforms.positive)

def logp(self, F, Y):
return densities.student_t(Y, F, self.scale, self.deg_free)

def conditional_mean(self, F):
return tf.identity(F)

def conditional_variance(self, F):
return F * 0.0 + (self.deg_free / (self.deg_free - 2.0))

def probit(x):
return 0.5 * (1.0 + tf.erf(x / np.sqrt(2.0))) * (1 - 2e-3) + 1e-3

class Bernoulli(Likelihood):
Likelihood.__init__(self)

def logp(self, F, Y):

def predict_mean_and_var(self, Fmu, Fvar):
p = probit(Fmu / tf.sqrt(1 + Fvar))
return p, p - tf.square(p)
else:
return Likelihood.predict_mean_and_var(self, Fmu, Fvar)

def predict_density(self, Fmu, Fvar, Y):
p = self.predict_mean_and_var(Fmu, Fvar)[0]
return densities.bernoulli(p, Y)

def conditional_mean(self, F):

def conditional_variance(self, F):
return p - tf.square(p)

class Gamma(Likelihood):
"""
Use the transformed GP to give the *scale* (inverse rate) of the Gamma
"""

Likelihood.__init__(self)
self.shape = Param(1.0, transforms.positive)

def logp(self, F, Y):

def conditional_mean(self, F):

def conditional_variance(self, F):
return self.shape * tf.square(scale)

def variational_expectations(self, Fmu, Fvar, Y):
return -self.shape * Fmu - tf.lgamma(self.shape) \
+ (self.shape - 1.) * tf.log(Y) - Y * tf.exp(-Fmu + Fvar / 2.)
else:
return Likelihood.variational_expectations(self, Fmu, Fvar, Y)

class Beta(Likelihood):
"""
This uses a reparameterisation of the Beta density. We have the mean of the
Beta distribution given by the transformed process:

m = sigma(f)

and a scale parameter. The familiar alpha, beta parameters are given by

m     = alpha / (alpha + beta)
scale = alpha + beta

so:
alpha = scale * m
beta  = scale * (1-m)
"""

Likelihood.__init__(self)
self.scale = Param(scale, transforms.positive)

def logp(self, F, Y):
alpha = mean * self.scale
beta = self.scale - alpha
return densities.beta(alpha, beta, Y)

def conditional_mean(self, F):

def conditional_variance(self, F):
return (mean - tf.square(mean)) / (self.scale + 1.)

class RobustMax(object):
"""
This class represent a multi-class inverse-link function. Given a vector
f=[f_1, f_2, ... f_k], the result of the mapping is

y = [y_1 ... y_k]

with

y_i = (1-eps)  i == argmax(f)
eps/(k-1)  otherwise.

"""

def __init__(self, num_classes, epsilon=1e-3):
self.epsilon = epsilon
self.num_classes = num_classes
self._eps_K1 = self.epsilon / (self.num_classes - 1)

def __call__(self, F):
i = tf.argmax(F, 1)
return tf.one_hot(i, self.num_classes, 1. - self.epsilon, self._eps_K1)

def prob_is_largest(self, Y, mu, var, gh_x, gh_w):
# work out what the mean and variance is of the indicated latent function.
oh_on = tf.cast(tf.one_hot(tf.reshape(Y, (-1,)), self.num_classes, 1., 0.), tf.float64)
mu_selected = tf.reduce_sum(oh_on * mu, 1)
var_selected = tf.reduce_sum(oh_on * var, 1)

# generate Gauss Hermite grid
X = tf.reshape(mu_selected, (-1, 1)) + gh_x * tf.reshape(
tf.sqrt(tf.clip_by_value(2. * var_selected, 1e-10, np.inf)), (-1, 1))

# compute the CDF of the Gaussian between the latent functions and the grid (including the selected function)
dist = (tf.expand_dims(X, 1) - tf.expand_dims(mu, 2)) / tf.expand_dims(
tf.sqrt(tf.clip_by_value(var, 1e-10, np.inf)), 2)
cdfs = 0.5 * (1.0 + tf.erf(dist / np.sqrt(2.0)))

cdfs = cdfs * (1 - 2e-4) + 1e-4

# blank out all the distances on the selected latent function
oh_off = tf.cast(tf.one_hot(tf.reshape(Y, (-1,)), self.num_classes, 0., 1.), tf.float64)
cdfs = cdfs * tf.expand_dims(oh_off, 2) + tf.expand_dims(oh_on, 2)

# take the product over the latent functions, and the sum over the GH grid.
return tf.matmul(tf.reduce_prod(cdfs, reduction_indices=[1]), tf.reshape(gh_w / np.sqrt(np.pi), (-1, 1)))

class MultiClass(Likelihood):
"""
A likelihood that can do multi-way classification.
Currently the only valid choice
"""
Likelihood.__init__(self)
self.num_classes = num_classes
raise NotImplementedError

def logp(self, F, Y):
hits = tf.equal(tf.expand_dims(tf.argmax(F, 1), 1), Y)
yes = tf.ones(tf.shape(Y), dtype=tf.float64) - self.invlink.epsilon
no = tf.zeros(tf.shape(Y), dtype=tf.float64) + self.invlink._eps_K1
p = tf.select(hits, yes, no)
return tf.log(p)
else:
raise NotImplementedError

def variational_expectations(self, Fmu, Fvar, Y):
gh_x, gh_w = hermgauss(self.num_gauss_hermite_points)
p = self.invlink.prob_is_largest(Y, Fmu, Fvar, gh_x, gh_w)
return p * np.log(1 - self.invlink.epsilon) + (1. - p) * np.log(self.invlink._eps_K1)
else:
raise NotImplementedError

def predict_mean_and_var(self, Fmu, Fvar):
# To compute this, we'll compute the density for each possible output
possible_outputs = [tf.fill(tf.pack([tf.shape(Fmu)[0], 1]), np.array(i, dtype=np.int64)) for i in
range(self.num_classes)]
ps = [self.predict_density(Fmu, Fvar, po) for po in possible_outputs]
ps = tf.transpose(tf.pack([tf.reshape(p, (-1,)) for p in ps]))
return ps, ps - tf.square(ps)
else:
raise NotImplementedError

def predict_density(self, Fmu, Fvar, Y):
gh_x, gh_w = hermgauss(self.num_gauss_hermite_points)
p = self.invlink.prob_is_largest(Y, Fmu, Fvar, gh_x, gh_w)
return p * (1. - self.invlink.epsilon) + (1. - p) * self.invlink._eps_K1
else:
raise NotImplementedError

def conditional_mean(self, F):