# Copyright (c) 2012-2014 The GPy authors (see AUTHORS.txt)
# Licensed under the BSD 3-clause license (see LICENSE.txt)
import numpy as np
from ..util.univariate_Gaussian import std_norm_pdf, std_norm_cdf
from . import link_functions
from .likelihood import Likelihood
from scipy import special
[docs]class Binomial(Likelihood):
"""
Binomial likelihood
.. math::
p(y_{i}|\\lambda(f_{i})) = \\lambda(f_{i})^{y_{i}}(1-f_{i})^{1-y_{i}}
.. Note::
Y takes values in either {-1, 1} or {0, 1}.
link function should have the domain [0, 1], e.g. probit (default) or Heaviside
.. See also::
likelihood.py, for the parent class
"""
def __init__(self, gp_link=None):
if gp_link is None:
gp_link = link_functions.Probit()
super(Binomial, self).__init__(gp_link, 'Binomial')
[docs] def pdf_link(self, inv_link_f, y, Y_metadata):
"""
Likelihood function given inverse link of f.
.. math::
p(y_{i}|\\lambda(f_{i})) = \\lambda(f_{i})^{y_{i}}(1-f_{i})^{1-y_{i}}
:param inv_link_f: latent variables inverse link of f.
:type inv_link_f: Nx1 array
:param y: data
:type y: Nx1 array
:param Y_metadata: Y_metadata must contain 'trials'
:returns: likelihood evaluated for this point
:rtype: float
.. Note:
Each y_i must be in {0, 1}
"""
return np.exp(self.logpdf_link(inv_link_f, y, Y_metadata))
[docs] def logpdf_link(self, inv_link_f, y, Y_metadata=None):
"""
Log Likelihood function given inverse link of f.
.. math::
\\ln p(y_{i}|\\lambda(f_{i})) = y_{i}\\log\\lambda(f_{i}) + (1-y_{i})\\log (1-f_{i})
:param inv_link_f: latent variables inverse link of f.
:type inv_link_f: Nx1 array
:param y: data
:type y: Nx1 array
:param Y_metadata: Y_metadata must contain 'trials'
:returns: log likelihood evaluated at points inverse link of f.
:rtype: float
"""
N = Y_metadata['trials']
np.testing.assert_array_equal(N.shape, y.shape)
nchoosey = special.gammaln(N+1) - special.gammaln(y+1) - special.gammaln(N-y+1)
Ny = N-y
t1 = np.zeros(y.shape)
t2 = np.zeros(y.shape)
t1[y>0] = y[y>0]*np.log(inv_link_f[y>0])
t2[Ny>0] = Ny[Ny>0]*np.log(1.-inv_link_f[Ny>0])
return nchoosey + t1 + t2
[docs] def dlogpdf_dlink(self, inv_link_f, y, Y_metadata=None):
"""
Gradient of the pdf at y, given inverse link of f w.r.t inverse link of f.
.. math::
\\frac{d^{2}\\ln p(y_{i}|\\lambda(f_{i}))}{d\\lambda(f)^{2}} = \\frac{y_{i}}{\\lambda(f)} - \\frac{(N-y_{i})}{(1-\\lambda(f))}
:param inv_link_f: latent variables inverse link of f.
:type inv_link_f: Nx1 array
:param y: data
:type y: Nx1 array
:param Y_metadata: Y_metadata must contain 'trials'
:returns: gradient of log likelihood evaluated at points inverse link of f.
:rtype: Nx1 array
"""
N = Y_metadata['trials']
np.testing.assert_array_equal(N.shape, y.shape)
Ny = N-y
t1 = np.zeros(y.shape)
t2 = np.zeros(y.shape)
t1[y>0] = y[y>0]/inv_link_f[y>0]
t2[Ny>0] = (Ny[Ny>0])/(1.-inv_link_f[Ny>0])
return t1 - t2
[docs] def d2logpdf_dlink2(self, inv_link_f, y, Y_metadata=None):
"""
Hessian at y, given inv_link_f, w.r.t inv_link_f the hessian will be 0 unless i == j
i.e. second derivative logpdf at y given inverse link of f_i and inverse link of f_j w.r.t inverse link of f_i and inverse link of f_j.
.. math::
\\frac{d^{2}\\ln p(y_{i}|\\lambda(f_{i}))}{d\\lambda(f)^{2}} = \\frac{-y_{i}}{\\lambda(f)^{2}} - \\frac{(N-y_{i})}{(1-\\lambda(f))^{2}}
:param inv_link_f: latent variables inverse link of f.
:type inv_link_f: Nx1 array
:param y: data
:type y: Nx1 array
:param Y_metadata: Y_metadata not used in binomial
:returns: Diagonal of log hessian matrix (second derivative of log likelihood evaluated at points inverse link of f.
:rtype: Nx1 array
.. Note::
Will return diagonal of hessian, since every where else it is 0, as the likelihood factorizes over cases
(the distribution for y_i depends only on inverse link of f_i not on inverse link of f_(j!=i)
"""
N = Y_metadata['trials']
np.testing.assert_array_equal(N.shape, y.shape)
Ny = N-y
t1 = np.zeros(y.shape)
t2 = np.zeros(y.shape)
t1[y>0] = -y[y>0]/np.square(inv_link_f[y>0])
t2[Ny>0] = -(Ny[Ny>0])/np.square(1.-inv_link_f[Ny>0])
return t1+t2
[docs] def d3logpdf_dlink3(self, inv_link_f, y, Y_metadata=None):
"""
Third order derivative log-likelihood function at y given inverse link of f w.r.t inverse link of f
.. math::
\\frac{d^{2}\\ln p(y_{i}|\\lambda(f_{i}))}{d\\lambda(f)^{2}} = \\frac{2y_{i}}{\\lambda(f)^{3}} - \\frac{2(N-y_{i})}{(1-\\lambda(f))^{3}}
:param inv_link_f: latent variables inverse link of f.
:type inv_link_f: Nx1 array
:param y: data
:type y: Nx1 array
:param Y_metadata: Y_metadata not used in binomial
:returns: Diagonal of log hessian matrix (second derivative of log likelihood evaluated at points inverse link of f.
:rtype: Nx1 array
.. Note::
Will return diagonal of hessian, since every where else it is 0, as the likelihood factorizes over cases
(the distribution for y_i depends only on inverse link of f_i not on inverse link of f_(j!=i)
"""
N = Y_metadata['trials']
np.testing.assert_array_equal(N.shape, y.shape)
#inv_link_f2 = np.square(inv_link_f) #TODO Remove. Why is this here?
Ny = N-y
t1 = np.zeros(y.shape)
t2 = np.zeros(y.shape)
t1[y>0] = 2*y[y>0]/inv_link_f[y>0]**3
t2[Ny>0] = - 2*(Ny[Ny>0])/(1.-inv_link_f[Ny>0])**3
return t1 + t2
[docs] def samples(self, gp, Y_metadata=None, **kw):
"""
Returns a set of samples of observations based on a given value of the latent variable.
:param gp: latent variable
"""
orig_shape = gp.shape
gp = gp.flatten()
N = Y_metadata['trials']
Ysim = np.random.binomial(N, self.gp_link.transf(gp))
return Ysim.reshape(orig_shape)
[docs] def exact_inference_gradients(self, dL_dKdiag,Y_metadata=None):
pass
[docs] def moments_match_ep(self,obs,tau,v,Y_metadata_i=None):
"""
Calculation of moments using quadrature
:param obs: observed output
:param tau: cavity distribution 1st natural parameter (precision)
:param v: cavity distribution 2nd natural paramenter (mu*precision)
"""
#Compute first integral for zeroth moment.
#NOTE constant np.sqrt(2*pi/tau) added at the end of the function
if (isinstance(self.gp_link, link_functions.Probit) or isinstance(self.gp_link, link_functions.ScaledProbit)) and (Y_metadata_i is None or int(Y_metadata_i.get('trials', 1)) == int(1)): #Special case for probit likelihood. Can be found from Riihimaki et Vehtari 2010
if isinstance(self.gp_link, link_functions.ScaledProbit):
nu = self.gp_link.nu
else:
nu = 1.0
nu = self.gp_link.nu
mu = v/tau
sigma2 = 1./tau
t = np.asarray(1 + sigma2*(nu**2))
t[t<1e-20] = 1e-20
a = np.sqrt(t)
z = obs*mu/a
normc_z = max(self.gp_link.transf(z), 1e-20)
m0 = normc_z
normp_z = self.gp_link.dtransf_df(z)
m1 = mu + (obs*sigma2*normp_z)/(normc_z*a)
#print('tau: {}, v: {}, nu: {}, z: {}, normc_z: {}, normp_z: {}'.format(tau, v, nu.values, z, normc_z, normp_z))
m2 = sigma2 - ((sigma2**2)*normp_z)/((1./(nu**2)+sigma2)*normc_z)*(z + normp_z/(nu**2)/normc_z)
#print("m0: {}, m1: {}, m2: {}".format(m0,m1,m2))
#m0a, m1a, m2a = super(Binomial, self).moments_match_ep(obs,tau,v,Y_metadata_i)
#print("m0a: {}, m1a: {}, m2a: {}".format(m0a,m1a,m2a))
return m0, m1, m2
else:
return super(Binomial, self).moments_match_ep(obs,tau,v,Y_metadata_i)
[docs] def variational_expectations(self, Y, m, v, gh_points=None, Y_metadata=None):
if isinstance(self.gp_link, link_functions.Probit):
if gh_points is None:
gh_x, gh_w = self._gh_points()
else:
gh_x, gh_w = gh_points
gh_w = gh_w / np.sqrt(np.pi)
shape = m.shape
C = np.atleast_1d(Y_metadata['trials'])
m,v,Y, C = m.flatten(), v.flatten(), Y.flatten()[:,None], C.flatten()[:,None]
X = gh_x[None,:]*np.sqrt(2.*v[:,None]) + m[:,None]
p = std_norm_cdf(X)
p = np.clip(p, 1e-9, 1.-1e-9) # for numerical stability
N = std_norm_pdf(X)
#TODO: missing nchoosek coefficient! use gammaln?
F = (Y*np.log(p) + (C-Y)*np.log(1.-p)).dot(gh_w)
NoverP = N/p
NoverP_ = N/(1.-p)
dF_dm = (Y*NoverP - (C-Y)*NoverP_).dot(gh_w)
dF_dv = -0.5* ( Y*(NoverP**2 + NoverP*X) + (C-Y)*(NoverP_**2 - NoverP_*X) ).dot(gh_w)
return F.reshape(*shape), dF_dm.reshape(*shape), dF_dv.reshape(*shape), None
else:
raise NotImplementedError