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72 lines
2.5 KiB
Python
72 lines
2.5 KiB
Python
from scipy.special import gammaln
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import numpy as np
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from GPy.likelihoods.likelihood_functions import likelihood_function
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class student_t(likelihood_function):
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"""Student t likelihood distribution
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For nomanclature see Bayesian Data Analysis 2003 p576
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$$\ln p(y_{i}|f_{i}) = \ln \Gamma(\frac{v+1}{2}) - \ln \Gamma(\frac{v}{2})\sqrt{v \pi}\sigma - \frac{v+1}{2}\ln (1 + \frac{1}{v}\left(\frac{y_{i} - f_{i}}{\sigma}\right)^2$$
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Laplace:
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Needs functions to calculate
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ln p(yi|fi)
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dln p(yi|fi)_dfi
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d2ln p(yi|fi)_d2fifj
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"""
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def __init__(self, deg_free, sigma=1):
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self.v = deg_free
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self.sigma = 1
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def link_function(self, y, f):
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"""link_function $\ln p(y|f)$
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$$\ln p(y_{i}|f_{i}) = \ln \Gamma(\frac{v+1}{2}) - \ln \Gamma(\frac{v}{2})\sqrt{v \pi}\sigma - \frac{v+1}{2}\ln (1 + \frac{1}{v}\left(\frac{y_{i} - f_{i}}{\sigma}\right)^2$$
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:y: datum number i
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:f: latent variable f
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:returns: float(likelihood evaluated for this point)
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"""
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e = y - f
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#print "Link ", y.shape, f.shape, e.shape
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objective = (gammaln((self.v + 1) * 0.5)
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- gammaln(self.v * 0.5)
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+ np.log(self.sigma * np.sqrt(self.v * np.pi))
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- (self.v + 1) * 0.5
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* np.log(1 + ((e**2 / self.sigma**2) / self.v))
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)
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return np.sum(objective)
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def link_grad(self, y, f):
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"""
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Gradient of the link function at y, given f w.r.t f
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$$\frac{d}{df}p(y_{i}|f_{i}) = \frac{(v + 1)(y - f)}{v \sigma^{2} + (y_{i} - f_{i})^{2}}$$
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:y: datum number i
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:f: latent variable f
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:returns: float(gradient of likelihood evaluated at this point)
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"""
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e = y - f
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#print "Grad ", y.shape, f.shape, e.shape
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grad = ((self.v + 1) * e) / (self.v * (self.sigma**2) + (e**2))
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return grad
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def link_hess(self, y, f):
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"""
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Hessian at this point (if we are only looking at the link function not the prior) the hessian will be 0 unless i == j
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i.e. second derivative link_function at y given f f_j w.r.t f and f_j
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Will return diaganol of hessian, since every where else it is 0
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$$\frac{d^{2}p(y_{i}|f_{i})}{df^{2}} = \frac{(v + 1)(y - f)}{v \sigma^{2} + (y_{i} - f_{i})^{2}}$$
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:y: datum number i
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:f: latent variable f
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:returns: float(second derivative of likelihood evaluated at this point)
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"""
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e = y - f
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hess = ((self.v + 1) * e) / ((((self.sigma**2)*self.v) + e**2)**2)
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return hess
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