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Alan Saul 2013-10-04 16:32:04 +01:00
parent 4925d8a0d9
commit 91f194cd29
4 changed files with 110 additions and 48 deletions
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9
GPy/likelihoods/laplace.py
View file

@ -203,8 +203,9 @@ class Laplace(likelihood):
"""
The laplace approximation algorithm, find K and expand hessian
For nomenclature see Rasmussen & Williams 2006 - modified for numerical stability
:param K: Covariance matrix evaluated at locations X
:type K: NxD matrix
:param K: Prior covariance matrix evaluated at locations X
:type K: NxN matrix
"""
self.K = K.copy()
@ -236,8 +237,8 @@ class Laplace(likelihood):
Rasmussen suggests the use of a numerically stable positive definite matrix B
Which has a positive diagonal element and can be easyily inverted
:param K: Covariance matrix evaluated at locations X
:type K: NxD matrix
:param K: Prior covariance matrix evaluated at locations X
:type K: NxN matrix
:param W: Negative hessian at a point (diagonal matrix)
:type W: Vector of diagonal values of hessian (1xN)
:param a: Matrix to calculate W12BiW12a

11
GPy/likelihoods/noise_model_constructors.py
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@ -90,7 +90,9 @@ def gaussian(gp_link=None, variance=2, D=None, N=None):
Construct a Gaussian likelihood
:param gp_link: a GPy gp_link function
:param variance: scalar, variance
:param variance: variance
:type variance: scalar
:returns: Gaussian noise model:
"""
if gp_link is None:
gp_link = noise_models.gp_transformations.Identity()
@ -104,8 +106,11 @@ def student_t(gp_link=None, deg_free=5, sigma2=2):
Construct a Student t likelihood
:param gp_link: a GPy gp_link function
:param deg_free: scalar, degrees of freedom
:param sigma2: scalar, variance
:param deg_free: degrees of freedom of student-t
:type deg_free: scalar
:param sigma2: variance
:type sigma2: scalar
:returns: Student-T noise model
"""
if gp_link is None:
gp_link = noise_models.gp_transformations.Identity()

102
GPy/likelihoods/noise_models/gaussian_noise.py
View file

@ -117,14 +117,19 @@ class Gaussian(NoiseDistribution):
return 0
def lik_function(self, y, f, extra_data=None):
"""lik_function $\ln p(y|f)$
$$\ln p(y_{i}|f_{i}) = \ln $$
"""
Log likelihood function
:y: data
:f: latent variables f
:extra_data: extra_data which is not used in student t distribution
:returns: float(likelihood evaluated for this point)
.. math::
\\ln p(y_{i}|f_{i}) = -\\frac{D \\ln 2\\pi}{2} - \\frac{\\ln |K|}{2} - \\frac{(y_{i} - f_{i})^{T}\\sigma^{-2}(y_{i} - f_{i})}{2}
:param y: data
:type y: Nx1 array
:param f: latent variables f
:type f: Nx1 array
:param extra_data: extra_data which is not used in student t distribution - not used
:returns: likelihood evaluated for this point
:rtype: float
"""
assert y.shape == f.shape
e = y - f
@ -138,10 +143,16 @@ class Gaussian(NoiseDistribution):
"""
Gradient of the link function at y, given f w.r.t f
:y: data
:f: latent variables f
:extra_data: extra_data which is not used in student t distribution
.. math::
\\frac{d \\ln p(y_{i}|f_{i})}{df} = \\frac{1}{\\sigma^{2}}(y_{i} - f_{i})
:param y: data
:type y: Nx1 array
:param f: latent variables f
:type f: Nx1 array
:param extra_data: extra_data which is not used in student t distribution - not used
:returns: gradient of likelihood evaluated at points
:rtype: Nx1 array
"""
assert y.shape == f.shape
@ -151,16 +162,23 @@ class Gaussian(NoiseDistribution):
def d2lik_d2f(self, y, f, extra_data=None):
"""
Hessian at this point (if we are only looking at the link function not the prior) the hessian will be 0 unless i == j
i.e. second derivative lik_function at y given f f_j w.r.t f and f_j
Hessian at y, given f, w.r.t f the hessian will be 0 unless i == j
i.e. second derivative lik_function at y given f_{i} f_{j} w.r.t f_{i} and f_{j}
.. math::
\\frac{d^{2} \\ln p(y_{i}|f_{i})}{d^{2}f} = -\\frac{1}{\\sigma^{2}}
:param y: data
:type y: Nx1 array
:param f: latent variables f
:type f: Nx1 array
:param extra_data: extra_data which is not used in student t distribution - not used
:returns: Diagonal of hessian matrix (second derivative of likelihood evaluated at points 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 f_{i} not on f_{j!=i}
:y: data
:f: latent variables f
:extra_data: extra_data which is not used in student t distribution
:returns: array which is diagonal of covariance matrix (second derivative of likelihood evaluated at points)
"""
assert y.shape == f.shape
hess = -(1.0/self.variance)*np.ones((self.N, 1))
@ -168,9 +186,18 @@ class Gaussian(NoiseDistribution):
def d3lik_d3f(self, y, f, extra_data=None):
"""
Third order derivative lik_function (log-likelihood ) at y given f f_j w.r.t f and f_j
Third order derivative log-likelihood function at y given f w.r.t f
$$\frac{d^{3}p(y_{i}|f_{i})}{d^{3}f} = \frac{-2(v+1)((y_{i} - f_{i})^3 - 3(y_{i} - f_{i}) \sigma^{2} v))}{((y_{i} - f_{i}) + \sigma^{2} v)^3}$$
.. math::
\\frac{d^{3} \\ln p(y_{i}|f_{i})}{d^{3}f} = 0
:param y: data
:type y: Nx1 array
:param f: latent variables f
:type f: Nx1 array
:param extra_data: extra_data which is not used in student t distribution - not used
:returns: third derivative of likelihood evaluated at points f
:rtype: Nx1 array
"""
assert y.shape == f.shape
d3lik_d3f = np.diagonal(0*self.I)[:, None] # FIXME: CAREFUL THIS MAY NOT WORK WITH MULTIDIMENSIONS?
@ -178,7 +205,18 @@ class Gaussian(NoiseDistribution):
def dlik_dvar(self, y, f, extra_data=None):
"""
Gradient of the likelihood (lik) w.r.t sigma parameter (standard deviation)
Gradient of the log-likelihood function at y given f, w.r.t variance parameter (noise_variance)
.. math::
\\frac{d \\ln p(y_{i}|f_{i})}{d\\sigma^{2}} = \\frac{N}{2\\sigma^{2}} + \\frac{(y_{i} - f_{i})^{2}}{2\\sigma^{4}}
:param y: data
:type y: Nx1 array
:param f: latent variables f
:type f: Nx1 array
:param extra_data: extra_data which is not used in student t distribution - not used
:returns: derivative of likelihood evaluated at points f w.r.t variance parameter
:rtype: float
"""
assert y.shape == f.shape
e = y - f
@ -188,7 +226,18 @@ class Gaussian(NoiseDistribution):
def dlik_df_dvar(self, y, f, extra_data=None):
"""
Gradient of the dlik_df w.r.t sigma parameter (standard deviation)
Derivative of the dlik_df w.r.t variance parameter (noise_variance)
.. math::
\\frac{d}{d\\sigma^{2}}(\\frac{d \\ln p(y_{i}|f_{i})}{df}) = \\frac{1}{\\sigma^{4}}(-y_{i} + f_{i})
:param y: data
:type y: Nx1 array
:param f: latent variables f
:type f: Nx1 array
:param extra_data: extra_data which is not used in student t distribution - not used
:returns: derivative of likelihood evaluated at points f w.r.t variance parameter
:rtype: Nx1 array
"""
assert y.shape == f.shape
s_4 = 1.0/(self.variance**2)
@ -197,9 +246,18 @@ class Gaussian(NoiseDistribution):
def d2lik_d2f_dvar(self, y, f, extra_data=None):
"""
Gradient of the hessian (d2lik_d2f) w.r.t sigma parameter (standard deviation)
Gradient of the hessian (d2lik_d2f) w.r.t variance parameter (noise_variance)
$$\frac{d}{d\sigma}(\frac{d^{2}p(y_{i}|f_{i})}{d^{2}f}) = \frac{2\sigma v(v + 1)(\sigma^2 v - 3(y-f)^2)}{((y-f)^2 + \sigma^2 v)^3}$$
.. math::
\\frac{d}{d\\sigma^{2}}(\\frac{d^{2} \\ln p(y_{i}|f_{i})}{d^{2}f}) = \\frac{1}{\\sigma^{4}}
:param y: data
:type y: Nx1 array
:param f: latent variables f
:type f: Nx1 array
:param extra_data: extra_data which is not used in student t distribution - not used
:returns: derivative of hessian evaluated at points f and f_j w.r.t variance parameter
:rtype: Nx1 array
"""
assert y.shape == f.shape
dlik_hess_dsigma = np.diag((1.0/(self.variance**2))*self.I)[:, None]

34
GPy/likelihoods/noise_models/student_t_noise.py
View file

@ -48,9 +48,9 @@ class StudentT(NoiseDistribution):
\\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
:param y: data
:type y: Nx1 matrix
:type y: Nx1 array
:param f: latent variables f
:type f: Nx1 matrix
:type f: Nx1 array
:param extra_data: extra_data which is not used in student t distribution - not used
:returns: likelihood evaluated for this point
:rtype: float
@ -73,9 +73,9 @@ class StudentT(NoiseDistribution):
\\frac{d \\ln p(y_{i}|f_{i})}{df} = \\frac{(v+1)(y_{i}-f_{i})}{(y_{i}-f_{i})^{2} + \\sigma^{2}v}
:param y: data
:type y: Nx1 matrix
:type y: Nx1 array
:param f: latent variables f
:type f: Nx1 matrix
:type f: Nx1 array
:param extra_data: extra_data which is not used in student t distribution - not used
:returns: gradient of likelihood evaluated at points
:rtype: Nx1 array
@ -95,9 +95,9 @@ class StudentT(NoiseDistribution):
\\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}}
:param y: data
:type y: Nx1 matrix
:type y: Nx1 array
:param f: latent variables f
:type f: Nx1 matrix
:type f: Nx1 array
:param extra_data: extra_data which is not used in student t distribution - not used
:returns: Diagonal of hessian matrix (second derivative of likelihood evaluated at points f)
:rtype: Nx1 array
@ -119,9 +119,9 @@ class StudentT(NoiseDistribution):
\\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}
:param y: data
:type y: Nx1 matrix
:type y: Nx1 array
:param f: latent variables f
:type f: Nx1 matrix
:type f: Nx1 array
:param extra_data: extra_data which is not used in student t distribution - not used
:returns: third derivative of likelihood evaluated at points f
:rtype: Nx1 array
@ -140,12 +140,10 @@ class StudentT(NoiseDistribution):
.. 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{1}{\\sigma} + \\frac{(1+v)(y_{i}-f_{i})^2}{\\sigma^3 v(1 + \\frac{1}{v}(\\frac{(y_{i} - f_{i})}{\\sigma^2})^2)}
:param y: data
:type y: Nx1 matrix
:type y: Nx1 array
:param f: latent variables f
:type f: Nx1 matrix
:type f: Nx1 array
:param extra_data: extra_data which is not used in student t distribution - not used
:returns: derivative of likelihood evaluated at points f w.r.t variance parameter
:rtype: float
@ -164,9 +162,9 @@ class StudentT(NoiseDistribution):
\\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}
:param y: data
:type y: Nx1 matrix
:type y: Nx1 array
:param f: latent variables f
:type f: Nx1 matrix
:type f: Nx1 array
:param extra_data: extra_data which is not used in student t distribution - not used
:returns: derivative of likelihood evaluated at points f w.r.t variance parameter
:rtype: Nx1 array
@ -178,15 +176,15 @@ class StudentT(NoiseDistribution):
def d2lik_d2f_dvar(self, y, f, extra_data=None):
"""
Gradient of the hessian (d2lik_d2f) w.r.t sigma parameter (standard deviation)
Gradient of the hessian (d2lik_d2f) 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{2\\sigma v(v + 1)(\\sigma^2 v - 3(y-f)^2)}{((y-f)^2 + \\sigma^2 v)^3}
\\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}}
:param y: data
:type y: Nx1 matrix
:type y: Nx1 array
:param f: latent variables f
:type f: Nx1 matrix
:type f: Nx1 array
:param extra_data: extra_data which is not used in student t distribution - not used
:returns: derivative of hessian evaluated at points f and f_j w.r.t variance parameter
:rtype: Nx1 array