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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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34
GPy/likelihoods/noise_models/student_t_noise.py
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@ -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