likelihoods.py
# Copyright 2016 Valentine Svensson, James Hensman, alexggmatthews, Alexis Boukouvalas
#
# 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 __future__ import absolute_import
from . import densities, transforms
import tensorflow as tf
import numpy as np
from .param import Parameterized, Param, ParamList
from ._settings import settings
from .quadrature import hermgauss
float_type = settings.dtypes.float_type
np_float_type = np.float32 if float_type is tf.float32 else np.float64
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):
def __init__(self, invlink=tf.exp):
Likelihood.__init__(self)
self.invlink = invlink
def logp(self, F, Y):
return densities.poisson(self.invlink(F), Y)
def conditional_variance(self, F):
return self.invlink(F)
def conditional_mean(self, F):
return self.invlink(F)
def variational_expectations(self, Fmu, Fvar, Y):
if self.invlink is tf.exp:
return Y * Fmu - tf.exp(Fmu + Fvar / 2) - tf.lgamma(Y + 1)
else:
return Likelihood.variational_expectations(self, Fmu, Fvar, Y)
class Exponential(Likelihood):
def __init__(self, invlink=tf.exp):
Likelihood.__init__(self)
self.invlink = invlink
def logp(self, F, Y):
return densities.exponential(self.invlink(F), Y)
def conditional_mean(self, F):
return self.invlink(F)
def conditional_variance(self, F):
return tf.square(self.invlink(F))
def variational_expectations(self, Fmu, Fvar, Y):
if self.invlink is tf.exp:
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):
def __init__(self, invlink=probit):
Likelihood.__init__(self)
self.invlink = invlink
def logp(self, F, Y):
return densities.bernoulli(self.invlink(F), Y)
def predict_mean_and_var(self, Fmu, Fvar):
if self.invlink is probit:
p = probit(Fmu / tf.sqrt(1 + Fvar))
return p, p - tf.square(p)
else:
# for other invlink, use quadrature
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):
return self.invlink(F)
def conditional_variance(self, F):
p = self.invlink(F)
return p - tf.square(p)
class Gamma(Likelihood):
"""
Use the transformed GP to give the *scale* (inverse rate) of the Gamma
"""
def __init__(self, invlink=tf.exp):
Likelihood.__init__(self)
self.invlink = invlink
self.shape = Param(1.0, transforms.positive)
def logp(self, F, Y):
return densities.gamma(self.shape, self.invlink(F), Y)
def conditional_mean(self, F):
return self.shape * self.invlink(F)
def conditional_variance(self, F):
scale = self.invlink(F)
return self.shape * tf.square(scale)
def variational_expectations(self, Fmu, Fvar, Y):
if self.invlink is tf.exp:
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)
"""
def __init__(self, invlink=probit, scale=1.0):
Likelihood.__init__(self)
self.scale = Param(scale, transforms.positive)
self.invlink = invlink
def logp(self, F, Y):
mean = self.invlink(F)
alpha = mean * self.scale
beta = self.scale - alpha
return densities.beta(alpha, beta, Y)
def conditional_mean(self, F):
return self.invlink(F)
def conditional_variance(self, F):
mean = self.invlink(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.), float_type)
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.), float_type)
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):
def __init__(self, num_classes, invlink=None):
"""
A likelihood that can do multi-way classification.
Currently the only valid choice
of inverse-link function (invlink) is an instance of RobustMax.
"""
Likelihood.__init__(self)
self.num_classes = num_classes
if invlink is None:
invlink = RobustMax(self.num_classes)
elif not isinstance(invlink, RobustMax):
raise NotImplementedError
self.invlink = invlink
def logp(self, F, Y):
if isinstance(self.invlink, RobustMax):
hits = tf.equal(tf.expand_dims(tf.argmax(F, 1), 1), Y)
yes = tf.ones(tf.shape(Y), dtype=float_type) - self.invlink.epsilon
no = tf.zeros(tf.shape(Y), dtype=float_type) + self.invlink._eps_K1
p = tf.where(hits, yes, no)
return tf.log(p)
else:
raise NotImplementedError
def variational_expectations(self, Fmu, Fvar, Y):
if isinstance(self.invlink, RobustMax):
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):
if isinstance(self.invlink, RobustMax):
# To compute this, we'll compute the density for each possible output
possible_outputs = [tf.fill(tf.stack([tf.shape(Fmu)[0], 1]), np.array(i, dtype=np.int64)) for i in
range(self.num_classes)]
ps = [self._predict_non_logged_density(Fmu, Fvar, po) for po in possible_outputs]
ps = tf.transpose(tf.stack([tf.reshape(p, (-1,)) for p in ps]))
return ps, ps - tf.square(ps)
else:
raise NotImplementedError
def predict_density(self, Fmu, Fvar, Y):
return tf.log(self._predict_non_logged_density(Fmu, Fvar, Y))
def _predict_non_logged_density(self, Fmu, Fvar, Y):
if isinstance(self.invlink, RobustMax):
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):
return self.invlink(F)
def conditional_variance(self, F):
p = self.conditional_mean(F)
return p - tf.square(p)
class SwitchedLikelihood(Likelihood):
def __init__(self, likelihood_list):
"""
In this likelihood, we assume at extra column of Y, which contains
integers that specify a likelihood from the list of likelihoods.
"""
Likelihood.__init__(self)
for l in likelihood_list:
assert isinstance(l, Likelihood)
self.likelihood_list = ParamList(likelihood_list)
self.num_likelihoods = len(self.likelihood_list)
def _partition_and_stitch(self, args, func_name):
"""
args is a list of tensors, to be passed to self.likelihoods.<func_name>
args[-1] is the 'Y' argument, which contains the indexes to self.likelihoods.
This function splits up the args using dynamic_partition, calls the
relevant function on the likelihoods, and re-combines the result.
"""
# get the index from Y
Y = args[-1]
ind = tf.gather(tf.transpose(Y), tf.shape(Y)[1]-1) # ind = Y[:,-1]
ind = tf.cast(ind, tf.int32)
Y = tf.transpose(tf.gather(tf.transpose(Y), tf.range(0, tf.shape(Y)[1]-1))) # Y = Y[:,:-1]
args[-1] = Y
# split up the arguments into chunks corresponding to the relevant likelihoods
args = zip(*[tf.dynamic_partition(X, ind, self.num_likelihoods) for X in args])
# apply the likelihood-function to each section of the data
funcs = [getattr(lik, func_name) for lik in self.likelihood_list]
results = [f(*args_i) for f, args_i in zip(funcs, args)]
# stitch the results back together
partitions = tf.dynamic_partition(tf.range(0, tf.size(ind)), ind, self.num_likelihoods)
results = tf.dynamic_stitch(partitions, results)
return results
def logp(self, F, Y):
return self._partition_and_stitch([F, Y], 'logp')
def predict_density(self, Fmu, Fvar, Y):
return self._partition_and_stitch([Fmu, Fvar, Y], 'predict_density')
def variational_expectations(self, Fmu, Fvar, Y):
return self._partition_and_stitch([Fmu, Fvar, Y], 'variational_expectations')
def predict_mean_and_var(self, Fmu, Fvar):
mu_list, var_list = zip(*[lik.predict_mean_and_var(Fmu, Fvar) for lik in self.likelihood_list])
mu = tf.concat(mu_list, 1)
var = tf.concat(var_list, 1)
return mu, var
class Ordinal(Likelihood):
"""
A likelihood for doing ordinal regression.
The data are integer values from 0 to K, and the user must specify (K-1)
'bin edges' which define the points at which the labels switch. Let the bin
edges be [a_0, a_1, ... a_{K-1}], then the likelihood is
p(Y=0|F) = phi((a_0 - F) / sigma)
p(Y=1|F) = phi((a_1 - F) / sigma) - phi((a_0 - F) / sigma)
p(Y=2|F) = phi((a_2 - F) / sigma) - phi((a_1 - F) / sigma)
...
p(Y=K|F) = 1 - phi((a_{K-1} - F) / sigma)
where phi is the cumulative density function of a Gaussian (the probit
function) and sigma is a parameter to be learned. A reference is:
@article{chu2005gaussian,
title={Gaussian processes for ordinal regression},
author={Chu, Wei and Ghahramani, Zoubin},
journal={Journal of Machine Learning Research},
volume={6},
number={Jul},
pages={1019--1041},
year={2005}
}
"""
def __init__(self, bin_edges):
"""
bin_edges is a numpy array specifying at which function value the
output label should switch. In the possible Y values are 0...K, then
the size of bin_edges should be (K-1).
"""
Likelihood.__init__(self)
self.bin_edges = bin_edges
self.num_bins = bin_edges.size + 1
self.sigma = Param(1.0, transforms.positive)
def logp(self, F, Y):
Y = tf.cast(Y, tf.int32)
scaled_bins_left = tf.concat([self.bin_edges/self.sigma, np.array([np.inf])], 0)
scaled_bins_right = tf.concat([np.array([-np.inf]), self.bin_edges/self.sigma], 0)
selected_bins_left = tf.gather(scaled_bins_left, Y)
selected_bins_right = tf.gather(scaled_bins_right, Y)
return tf.log(probit(selected_bins_left - F / self.sigma) -
probit(selected_bins_right - F / self.sigma) + 1e-6)
def _make_phi(self, F):
"""
A helper function for making predictions. Constructs a probability
matrix where each row output the probability of the corresponding
label, and the rows match the entries of F.
Note that a matrix of F values is flattened.
"""
scaled_bins_left = tf.concat([self.bin_edges/self.sigma, np.array([np.inf])], 0)
scaled_bins_right = tf.concat([np.array([-np.inf]), self.bin_edges/self.sigma], 0)
return probit(scaled_bins_left - tf.reshape(F, (-1, 1)) / self.sigma)\
- probit(scaled_bins_right - tf.reshape(F, (-1, 1)) / self.sigma)
def conditional_mean(self, F):
phi = self._make_phi(F)
Ys = tf.reshape(np.arange(self.num_bins, dtype=np.float64), (-1, 1))
return tf.reshape(tf.matmul(phi, Ys), tf.shape(F))
def conditional_variance(self, F):
phi = self._make_phi(F)
Ys = tf.reshape(np.arange(self.num_bins, dtype=np.float64), (-1, 1))
E_y = tf.matmul(phi, Ys)
E_y2 = tf.matmul(phi, tf.square(Ys))
return tf.reshape(E_y2 - tf.square(E_y), tf.shape(F))
