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Got most of laplace approximation working

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Alan Saul 2013-03-13 17:55:41 +00:00
parent ad2c266c65
commit 3f114aa020
9 changed files with 124 additions and 45 deletions
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80
python/likelihoods/likelihood_function.py
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@ -1,62 +1,72 @@
import GPy
from scipy.special import gamma, gammaln
from scipy.special import gammaln
import numpy as np
from GPy.likelihoods.likelihood_functions import likelihood_function
class student_t(GPy.likelihoods.likelihood_function):
class student_t(likelihood_function):
"""Student t likelihood distribution
For nomanclature see Bayesian Data Analysis 2003 p576
$$\ln(\frac{\Gamma(\frac{(v+1)}{2})}{\Gamma(\sqrt(v \pi \Gamma(\frac{v}{2}))})+ \ln(1+\frac{(y_i-f_i)^2}{\sigma v})^{-\frac{(v+1)}{2}}$$
TODO:Double check this
$$\ln p(y_{i}|f_{i}) = \ln \Gamma(\frac{v+1}{2}) - \ln \Gamma(\frac{v}{2})\sqrt{v \pi}\sigma - \frac{v+1}{2}\ln (1 + \frac{1}{v}\left(\frac{y_{i} - f_{i}}{\sigma}\right)^2$$
Laplace:
Needs functions to calculate
ln p(yi|fi)
dln p(yi|fi)_dfi
d2ln p(yi|fi)_d2fi
d2ln p(yi|fi)_d2fifj
"""
def __init__(self, deg_free, sigma=1):
self.v = deg_free
self.sigma = 1
def link_function(self, y_i, f_i):
"""link_function $\ln p(y_i|f_i)$
$$\ln \Gamma(\frac{v+1}{2}) - \ln \Gamma(\frac{v}{2}) - \ln \frac{v \pi \sigma}{2} - \frac{v+1}{2}\ln (1 + \frac{(y_{i} - f_{i})^{2}}{v\sigma})$$
TODO: Double check this
def link_function(self, y, f):
"""link_function $\ln p(y|f)$
$$\ln p(y_{i}|f_{i}) = \ln \Gamma(\frac{v+1}{2}) - \ln \Gamma(\frac{v}{2})\sqrt{v \pi}\sigma - \frac{v+1}{2}\ln (1 + \frac{1}{v}\left(\frac{y_{i} - f_{i}}{\sigma}\right)^2$$
:y_i: datum number i
:f_i: latent variable f_i
:y: datum number i
:f: latent variable f
:returns: float(likelihood evaluated for this point)
"""
e = y_i - f_i
return gammaln((v+1)*0.5) - gammaln(v*0.5) - np.ln(v*np.pi*sigma)*0.5 - (v+1)*0.5*np.ln(1 + ((e/sigma)**2)/v) #Check the /v!
e = y - f
#print "Link ", y.shape, f.shape, e.shape
objective = (gammaln((self.v + 1) * 0.5)
- gammaln(self.v * 0.5)
+ np.log(self.sigma * np.sqrt(self.v * np.pi))
- (self.v + 1) * 0.5
* np.log(1 + ((e**2 / self.sigma**2) / self.v))
)
return np.sum(objective)
def link_grad(self, y_i, f_i):
"""gradient of the link function at y_i, given f_i w.r.t f_i
def link_grad(self, y, f):
"""
Gradient of the link function at y, given f w.r.t f
derivative of log((gamma((v+1)/2)/gamma(sqrt(v*pi*gamma(v/2))))*(1+(t^2)/(a*v))^((-(v+1))/2)) with respect to t
$$\frac{(y_i - f_i)(v + 1)}{\sigma v (y_{i} - f_{i})^{2}}$$
TODO: Double check this
$$\frac{d}{df}p(y_{i}|f_{i}) = \frac{(v + 1)(y - f)}{v \sigma^{2} + (y_{i} - f_{i})^{2}}$$
:y_i: datum number i
:f_i: latent variable f_i
:y: datum number i
:f: latent variable f
:returns: float(gradient of likelihood evaluated at this point)
"""
pass
def link_hess(self, y_i, f_i, f_j):
"""hessian at this point (the hessian will be 0 unless i == j)
i.e. second derivative w.r.t f_i and f_j
second derivative of
:y_i: @todo
:f_i: @todo
:f_j: @todo
:returns: @todo
e = y - f
#print "Grad ", y.shape, f.shape, e.shape
grad = ((self.v + 1) * e) / (self.v * (self.sigma**2) + (e**2))
return grad
def link_hess(self, y, f):
"""
if f_i =
pass
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 link_function at y given f f_j w.r.t f and f_j
Will return diaganol of hessian, since every where else it is 0
$$\frac{d^{2}p(y_{i}|f_{i})}{df^{2}} = \frac{(v + 1)(y - f)}{v \sigma^{2} + (y_{i} - f_{i})^{2}}$$
:y: datum number i
:f: latent variable f
:returns: float(second derivative of likelihood evaluated at this point)
"""
e = y - f
hess = ((self.v + 1) * e) / ((((self.sigma**2)*self.v) + e**2)**2)
return hess