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Alan Saul 2013-10-22 13:37:12 +01:00
parent ceb1f7490d
commit a3422eae21
6 changed files with 61 additions and 50 deletions
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26
GPy/likelihoods/noise_models/bernoulli_noise.py
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@ -11,12 +11,14 @@ from noise_distributions import NoiseDistribution
class Bernoulli(NoiseDistribution):
"""
Probit likelihood
Y is expected to take values in {-1,1}
-----
$$
L(x) = \\Phi (Y_i*f_i)
$$
Bernoulli likelihood
.. math::
p(y_{i}|\\lambda(f_{i})) = \\lambda(f_{i})^{y_{i}}(1-f_{i})^{1-y_{i}}
.. Note::
Y is expected to take values in {-1,1}
Probit likelihood usually used
"""
def __init__(self,gp_link=None,analytical_mean=False,analytical_variance=False):
super(Bernoulli, self).__init__(gp_link,analytical_mean,analytical_variance)
@ -82,7 +84,7 @@ class Bernoulli(NoiseDistribution):
Likelihood function given link(f)
.. math::
\\p(y_{i}|\\lambda(f_{i})) = \\lambda(f_{i})^{y_{i}}(1-f_{i})^{1-y_{i}}
p(y_{i}|\\lambda(f_{i})) = \\lambda(f_{i})^{y_{i}}(1-f_{i})^{1-y_{i}}
:param link_f: latent variables link(f)
:type link_f: Nx1 array
@ -111,7 +113,7 @@ class Bernoulli(NoiseDistribution):
:param y: data
:type y: Nx1 array
:param extra_data: extra_data not used in bernoulli
:returns: log likelihood evaluated for this point
:returns: log likelihood evaluated at points link(f)
:rtype: float
"""
assert np.asarray(link_f).shape == np.asarray(y).shape
@ -130,8 +132,8 @@ class Bernoulli(NoiseDistribution):
:type link_f: Nx1 array
:param y: data
:type y: Nx1 array
:param extra_data: extra_data not used in gaussian
:returns: gradient of log likelihood evaluated at points
:param extra_data: extra_data not used in bernoulli
:returns: gradient of log likelihood evaluated at points link(f)
:rtype: Nx1 array
"""
assert np.asarray(link_f).shape == np.asarray(y).shape
@ -151,7 +153,7 @@ class Bernoulli(NoiseDistribution):
:type link_f: Nx1 array
:param y: data
:type y: Nx1 array
:param extra_data: extra_data not used in gaussian
:param extra_data: extra_data not used in bernoulli
:returns: Diagonal of log hessian matrix (second derivative of log likelihood evaluated at points link(f))
:rtype: Nx1 array
@ -174,7 +176,7 @@ class Bernoulli(NoiseDistribution):
:type link_f: Nx1 array
:param y: data
:type y: Nx1 array
:param extra_data: extra_data not used in gaussian
:param extra_data: extra_data not used in bernoulli
:returns: third derivative of log likelihood evaluated at points link(f)
:rtype: Nx1 array
"""

25
GPy/likelihoods/noise_models/gaussian_noise.py
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@ -12,12 +12,15 @@ class Gaussian(NoiseDistribution):
"""
Gaussian likelihood
:param mean: mean value of the Gaussian distribution
:param variance: mean value of the Gaussian distribution
.. math::
\\ln p(y_{i}|\\lambda(f_{i})) = -\\frac{N \\ln 2\\pi}{2} - \\frac{\\ln |K|}{2} - \\frac{(y_{i} - \\lambda(f_{i}))^{T}\\sigma^{-2}(y_{i} - \\lambda(f_{i}))}{2}
:param variance: variance value of the Gaussian distribution
:param N: Number of data points
:type N: int
"""
def __init__(self,gp_link=None,analytical_mean=False,analytical_variance=False,variance=1., D=None, N=None):
self.variance = variance
self.D = D
self.N = N
self._set_params(np.asarray(variance))
super(Gaussian, self).__init__(gp_link,analytical_mean,analytical_variance)
@ -109,7 +112,6 @@ class Gaussian(NoiseDistribution):
#Assumes no covariance, exp, sum, log for numerical stability
return np.exp(np.sum(np.log(stats.norm.pdf(y, link_f, np.sqrt(self.variance)))))
def logpdf_link(self, link_f, y, extra_data=None):
"""
Log likelihood function given link(f)
@ -150,9 +152,11 @@ class Gaussian(NoiseDistribution):
def d2logpdf_dlink2(self, link_f, y, extra_data=None):
"""
Hessian at y, given link_f, w.r.t link_f the hessian will be 0 unless i == j
Hessian at y, given link_f, w.r.t link_f.
i.e. second derivative logpdf at y given link(f_i) link(f_j) w.r.t link(f_i) and link(f_j)
The hessian will be 0 unless i == j
.. math::
\\frac{d^{2} \\ln p(y_{i}|\\lambda(f_{i}))}{d^{2}f} = -\\frac{1}{\\sigma^{2}}
@ -193,10 +197,10 @@ class Gaussian(NoiseDistribution):
def dlogpdf_link_dvar(self, link_f, y, extra_data=None):
"""
Gradient of the negative log-likelihood function at y given link(f), w.r.t variance parameter (noise_variance)
Gradient of the log-likelihood function at y given link(f), w.r.t variance parameter (noise_variance)
.. math::
\\frac{d \\ln p(y_{i}|\\lambda(f_{i}))}{d\\sigma^{2}} = \\frac{N}{2\\sigma^{2}} + \\frac{(y_{i} - \\lambda(f_{i}))^{2}}{2\\sigma^{4}}
\\frac{d \\ln p(y_{i}|\\lambda(f_{i}))}{d\\sigma^{2}} = -\\frac{N}{2\\sigma^{2}} + \\frac{(y_{i} - \\lambda(f_{i}))^{2}}{2\\sigma^{4}}
:param link_f: latent variables link(f)
:type link_f: Nx1 array
@ -209,7 +213,7 @@ class Gaussian(NoiseDistribution):
assert np.asarray(link_f).shape == np.asarray(y).shape
e = y - link_f
s_4 = 1.0/(self.variance**2)
dlik_dsigma = -0.5*self.N/self.variance + 0.5*s_4*np.dot(e.T, e)
dlik_dsigma = -0.5*self.N/self.variance + 0.5*s_4*np.square(e)
return np.sum(dlik_dsigma) # Sure about this sum?
def dlogpdf_dlink_dvar(self, link_f, y, extra_data=None):
@ -228,8 +232,9 @@ class Gaussian(NoiseDistribution):
:rtype: Nx1 array
"""
assert np.asarray(link_f).shape == np.asarray(y).shape
s_4 = 1.0/(self.variance**2)
dlik_grad_dsigma = -np.dot(s_4*self.I, y) + np.dot(s_4*self.I, link_f)
s_4 = 1./(self.variance**2)
#dlik_grad_dsigma = -np.dot(s_4*self.I, y) + np.dot(s_4*self.I, link_f)
dlik_grad_dsigma = -s_4*y + s_4*link_f
return dlik_grad_dsigma
def d2logpdf_dlink2_dvar(self, link_f, y, extra_data=None):

7
GPy/likelihoods/noise_models/noise_distributions.py
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@ -12,14 +12,9 @@ import gp_transformations
from GPy.util.misc import chain_1, chain_2, chain_3
from scipy.integrate import quad
class NoiseDistribution(object):
"""
Likelihood class for doing Expectation propagation
:param Y: observed output (Nx1 numpy.darray)
.. note:: Y values allowed depend on the LikelihoodFunction used
Likelihood class for doing approximations
"""
def __init__(self,gp_link,analytical_mean=False,analytical_variance=False):
assert isinstance(gp_link,gp_transformations.GPTransformation), "gp_link is not a valid GPTransformation."

39
GPy/likelihoods/noise_models/student_t_noise.py
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@ -16,7 +16,7 @@ class StudentT(NoiseDistribution):
For nomanclature see Bayesian Data Analysis 2003 p576
.. math::
\\ln p(y_{i}|f_{i}) = \\ln \\Gamma(\\frac{v+1}{2}) - \\ln \\Gamma(\\frac{v}{2})\\sqrt{v \\pi}\\sigma - \\frac{v+1}{2}\\ln (1 + \\frac{1}{v}\\left(\\frac{y_{i} - f_{i}}{\\sigma}\\right)^2)
p(y_{i}|\\lambda(f_{i})) = \\frac{\\Gamma\\left(\\frac{v+1}{2}\\right)}{\\Gamma\\left(\\frac{v}{2}\\right)\\sqrt{v\\pi\\sigma^{2}}}\\left(1 + \\frac{1}{v}\\left(\\frac{(y_{i} - f_{i})^{2}}{\\sigma^{2}}\\right)\\right)^{\\frac{-v+1}{2}}
"""
def __init__(self,gp_link=None,analytical_mean=True,analytical_variance=True, deg_free=5, sigma2=2):
@ -45,13 +45,13 @@ class StudentT(NoiseDistribution):
Likelihood function given link(f)
.. math::
\\ln p(y_{i}|\\lambda(f_{i})) = \\frac{\\Gamma\\left(\\frac{v+1}{2}\\right)}{\\Gamma\\left(\\frac{v}{2}\\right)\\sqrt{v\\pi\\sigma^{2}}}\\left(1 + \\frac{1}{v}\\left(\\frac{(y_{i} - f_{i})^{2}}{\\sigma^{2}}\\right)\\right)^{\\frac{-v+1}{2}}
p(y_{i}|\\lambda(f_{i})) = \\frac{\\Gamma\\left(\\frac{v+1}{2}\\right)}{\\Gamma\\left(\\frac{v}{2}\\right)\\sqrt{v\\pi\\sigma^{2}}}\\left(1 + \\frac{1}{v}\\left(\\frac{(y_{i} - \\lambda(f_{i}))^{2}}{\\sigma^{2}}\\right)\\right)^{\\frac{-v+1}{2}}
:param link_f: latent variables link(f)
:type link_f: Nx1 array
:param y: data
:type y: Nx1 array
:param extra_data: extra_data which is not used in student t distribution - not used
:param extra_data: extra_data which is not used in student t distribution
:returns: likelihood evaluated for this point
:rtype: float
"""
@ -69,13 +69,13 @@ class StudentT(NoiseDistribution):
Log Likelihood Function given link(f)
.. math::
\\ln p(y_{i}|f_{i}) = \\ln \\Gamma\\left(\\frac{v+1}{2}\\right) - \\ln \\Gamma\\left(\\frac{v}{2}\\right) - \\ln \\sqrt{v \\pi\\sigma^{2}} - \\frac{v+1}{2}\\ln \\left(1 + \\frac{1}{v}\\left(\\frac{(y_{i} - f_{i})^{2}}{\\sigma^{2}}\\right)\\right)
\\ln p(y_{i}|\lambda(f_{i})) = \\ln \\Gamma\\left(\\frac{v+1}{2}\\right) - \\ln \\Gamma\\left(\\frac{v}{2}\\right) - \\ln \\sqrt{v \\pi\\sigma^{2}} - \\frac{v+1}{2}\\ln \\left(1 + \\frac{1}{v}\\left(\\frac{(y_{i} - \lambda(f_{i}))^{2}}{\\sigma^{2}}\\right)\\right)
:param link_f: latent variables (link(f))
:type link_f: Nx1 array
:param y: data
:type y: Nx1 array
:param extra_data: extra_data which is not used in student t distribution - not used
:param extra_data: extra_data which is not used in student t distribution
:returns: likelihood evaluated for this point
:rtype: float
@ -94,13 +94,13 @@ class StudentT(NoiseDistribution):
Gradient of the log likelihood function at y, given link(f) w.r.t link(f)
.. math::
\\frac{d \\ln p(y_{i}|f_{i})}{df} = \\frac{(v+1)(y_{i}-f_{i})}{(y_{i}-f_{i})^{2} + \\sigma^{2}v}
\\frac{d \\ln p(y_{i}|\lambda(f_{i}))}{d\\lambda(f)} = \\frac{(v+1)(y_{i}-\lambda(f_{i}))}{(y_{i}-\lambda(f_{i}))^{2} + \\sigma^{2}v}
:param link_f: latent variables (f)
:type link_f: Nx1 array
:param y: data
:type y: Nx1 array
:param extra_data: extra_data which is not used in student t distribution - not used
:param extra_data: extra_data which is not used in student t distribution
:returns: gradient of likelihood evaluated at points
:rtype: Nx1 array
@ -112,17 +112,18 @@ class StudentT(NoiseDistribution):
def d2logpdf_dlink2(self, link_f, y, extra_data=None):
"""
Hessian at y, given link(f), w.r.t link(f) the hessian will be 0 unless i == j
Hessian at y, given link(f), w.r.t link(f)
i.e. second derivative logpdf at y given link(f_i) and link(f_j) w.r.t link(f_i) and link(f_j)
The hessian will be 0 unless i == j
.. math::
\\frac{d^{2} \\ln p(y_{i}|f_{i})}{d^{2}f} = \\frac{(v+1)((y_{i}-f_{i})^{2} - \\sigma^{2}v)}{((y_{i}-f_{i})^{2} + \\sigma^{2}v)^{2}}
\\frac{d^{2} \\ln p(y_{i}|\lambda(f_{i}))}{d^{2}\\lambda(f)} = \\frac{(v+1)((y_{i}-\lambda(f_{i}))^{2} - \\sigma^{2}v)}{((y_{i}-\lambda(f_{i}))^{2} + \\sigma^{2}v)^{2}}
:param link_f: latent variables link(f)
:type link_f: Nx1 array
:param y: data
:type y: Nx1 array
:param extra_data: extra_data which is not used in student t distribution - not used
:param extra_data: extra_data which is not used in student t distribution
:returns: Diagonal of hessian matrix (second derivative of likelihood evaluated at points f)
:rtype: Nx1 array
@ -137,16 +138,16 @@ class StudentT(NoiseDistribution):
def d3logpdf_dlink3(self, link_f, y, extra_data=None):
"""
Third order derivative log-likelihood function at y given f w.r.t f
Third order derivative log-likelihood function at y given link(f) w.r.t link(f)
.. math::
\\frac{d^{3} \\ln p(y_{i}|f_{i})}{d^{3}f} = \\frac{-2(v+1)((y_{i} - f_{i})^3 - 3(y_{i} - f_{i}) \\sigma^{2} v))}{((y_{i} - f_{i}) + \\sigma^{2} v)^3}
\\frac{d^{3} \\ln p(y_{i}|\lambda(f_{i}))}{d^{3}\\lambda(f)} = \\frac{-2(v+1)((y_{i} - \lambda(f_{i}))^3 - 3(y_{i} - \lambda(f_{i})) \\sigma^{2} v))}{((y_{i} - \lambda(f_{i})) + \\sigma^{2} v)^3}
:param link_f: latent variables link(f)
:type link_f: Nx1 array
:param y: data
:type y: Nx1 array
:param extra_data: extra_data which is not used in student t distribution - not used
:param extra_data: extra_data which is not used in student t distribution
:returns: third derivative of likelihood evaluated at points f
:rtype: Nx1 array
"""
@ -162,13 +163,13 @@ class StudentT(NoiseDistribution):
Gradient of the log-likelihood function at y given f, w.r.t variance parameter (t_noise)
.. math::
\\frac{d \\ln p(y_{i}|f_{i})}{d\\sigma^{2}} = \\frac{v((y_{i} - f_{i})^{2} - \\sigma^{2})}{2\\sigma^{2}(\\sigma^{2}v + (y_{i} - f_{i})^{2})}
\\frac{d \\ln p(y_{i}|\lambda(f_{i}))}{d\\sigma^{2}} = \\frac{v((y_{i} - \lambda(f_{i}))^{2} - \\sigma^{2})}{2\\sigma^{2}(\\sigma^{2}v + (y_{i} - \lambda(f_{i}))^{2})}
:param link_f: latent variables link(f)
:type link_f: Nx1 array
:param y: data
:type y: Nx1 array
:param extra_data: extra_data which is not used in student t distribution - not used
:param extra_data: extra_data which is not used in student t distribution
:returns: derivative of likelihood evaluated at points f w.r.t variance parameter
:rtype: float
"""
@ -182,13 +183,13 @@ class StudentT(NoiseDistribution):
Derivative of the dlogpdf_dlink w.r.t variance parameter (t_noise)
.. math::
\\frac{d}{d\\sigma^{2}}(\\frac{d \\ln p(y_{i}|f_{i})}{df}) = \\frac{-2\\sigma v(v + 1)(y_{i}-f_{i})}{(y_{i}-f_{i})^2 + \\sigma^2 v)^2}
\\frac{d}{d\\sigma^{2}}(\\frac{d \\ln p(y_{i}|\lambda(f_{i}))}{df}) = \\frac{-2\\sigma v(v + 1)(y_{i}-\lambda(f_{i}))}{(y_{i}-\lambda(f_{i}))^2 + \\sigma^2 v)^2}
:param link_f: latent variables link_f
:type link_f: Nx1 array
:param y: data
:type y: Nx1 array
:param extra_data: extra_data which is not used in student t distribution - not used
:param extra_data: extra_data which is not used in student t distribution
:returns: derivative of likelihood evaluated at points f w.r.t variance parameter
:rtype: Nx1 array
"""
@ -202,13 +203,13 @@ class StudentT(NoiseDistribution):
Gradient of the hessian (d2logpdf_dlink2) w.r.t variance parameter (t_noise)
.. math::
\\frac{d}{d\\sigma^{2}}(\\frac{d^{2} \\ln p(y_{i}|f_{i})}{d^{2}f}) = \\frac{v(v+1)(\\sigma^{2}v - 3(y_{i} - f_{i})^{2})}{(\\sigma^{2}v + (y_{i} - f_{i})^{2})^{3}}
\\frac{d}{d\\sigma^{2}}(\\frac{d^{2} \\ln p(y_{i}|\lambda(f_{i}))}{d^{2}f}) = \\frac{v(v+1)(\\sigma^{2}v - 3(y_{i} - \lambda(f_{i}))^{2})}{(\\sigma^{2}v + (y_{i} - \lambda(f_{i}))^{2})^{3}}
:param link_f: latent variables link(f)
:type link_f: Nx1 array
:param y: data
:type y: Nx1 array
:param extra_data: extra_data which is not used in student t distribution - not used
:param extra_data: extra_data which is not used in student t distribution
:returns: derivative of hessian evaluated at points f and f_j w.r.t variance parameter
:rtype: Nx1 array
"""

6
doc/GPy.likelihoods.noise_models.rst
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@ -4,10 +4,10 @@ GPy.likelihoods.noise_models package
Submodules
----------
GPy.likelihoods.noise_models.binomial_noise module
--------------------------------------------------
GPy.likelihoods.noise_models.bernoulli_noise module
---------------------------------------------------
.. automodule:: GPy.likelihoods.noise_models.binomial_noise
.. automodule:: GPy.likelihoods.noise_models.bernoulli_noise
:members:
:undoc-members:
:show-inheritance:

8
doc/GPy.testing.rst
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@ -36,6 +36,14 @@ GPy.testing.examples_tests module
:undoc-members:
:show-inheritance:
GPy.testing.gp_transformation_tests module
------------------------------------------
.. automodule:: GPy.testing.gp_transformation_tests
:members:
:undoc-members:
:show-inheritance:
GPy.testing.gplvm_tests module
------------------------------