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Dragged likelihood_function changes in
This commit is contained in:
Alan Saul
parent
f641ab54a8
commit
b9a7a40795
1 changed files with 383 additions and 1 deletions
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@ -3,12 +3,13 @@
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import numpy as np
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from scipy import stats
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from scipy import stats, integrate
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import scipy as sp
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import pylab as pb
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from ..util.plot import gpplot
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from ..util.univariate_Gaussian import std_norm_pdf,std_norm_cdf
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import link_functions
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from scipy.special import gammaln, gamma
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class LikelihoodFunction(object):
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"""
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@ -24,6 +25,7 @@ class LikelihoodFunction(object):
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assert isinstance(link,link_functions.LinkFunction)
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self.link = link
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self.moments_match = self._moments_match_numerical
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self.log_concave = True
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def _preprocess_values(self,Y):
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return Y
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@ -164,3 +166,383 @@ class Poisson(LikelihoodFunction):
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p_025 = tmp[:,0]
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p_975 = tmp[:,1]
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return mean,np.nan*mean,p_025,p_975 # better variance here TODO
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class Student_t(LikelihoodFunction):
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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=5, sigma2=2, link=None):
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super(Student_t, self).__init__(link)
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self.v = deg_free
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self.sigma2 = sigma2
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self._set_params(np.asarray(sigma2))
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self.log_concave = False
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def _get_params(self):
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return np.asarray(self.sigma2)
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def _get_param_names(self):
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return ["t_noise_std2"]
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def _set_params(self, x):
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self.sigma2 = float(x)
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@property
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def variance(self, extra_data=None):
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return (self.v / float(self.v - 2)) * self.sigma2
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def link_function(self, y, f, extra_data=None):
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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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For wolfram alpha import parts for derivative of sigma are -log(sqrt(v*pi)*s) -(1/2)*(v + 1)*log(1 + (1/v)*((y-f)/(s))^2))
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:y: data
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:f: latent variables f
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:extra_data: extra_data which is not used in student t distribution
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:returns: float(likelihood evaluated for this point)
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"""
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assert y.shape == f.shape
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e = y - f
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#A = gammaln((self.v + 1) * 0.5)
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#B = - gammaln(self.v * 0.5)
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#C = - 0.5*np.log(self.sigma2 * self.v * np.pi)
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#D = + (-(self.v + 1)*0.5)*np.log(1 + ((e**2)/self.sigma2)/np.float(self.v))
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objective = (+ gammaln((self.v + 1) * 0.5)
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- gammaln(self.v * 0.5)
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- 0.5*np.log(self.sigma2 * self.v * np.pi)
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+ (-(self.v + 1)*0.5)*np.log(1 + ((e**2)/self.sigma2)/np.float(self.v))
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)
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#print "C: {} D: {} obj: {}".format(C, np.sum(D), objective.sum())
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return np.sum(objective)
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def dlik_df(self, y, f, extra_data=None):
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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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$$\frac{dp(y_{i}|f_{i})}{df} = \frac{(v+1)(y_{i}-f_{i})}{(y_{i}-f_{i})^{2} + \sigma^{2}v}$$
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:y: data
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:f: latent variables f
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:extra_data: extra_data which is not used in student t distribution
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:returns: gradient of likelihood evaluated at points
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"""
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assert y.shape == f.shape
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e = y - f
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grad = ((self.v + 1) * e) / (self.v * self.sigma2 + (e**2))
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return grad
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def d2lik_d2f(self, y, f, extra_data=None):
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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 diagonal of hessian, since every where else it is 0, as the likelihood factorizes over cases
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(the distribution for y_{i} depends only on f_{i} not on f_{j!=i}
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$$\frac{d^{2}p(y_{i}|f_{i})}{d^{3}f} = \frac{(v+1)((y_{i}-f_{i})^{2} - \sigma^{2}v)}{((y_{i}-f_{i})^{2} + \sigma^{2}v)^{2}}$$
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:y: data
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:f: latent variables f
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:extra_data: extra_data which is not used in student t distribution
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:returns: array which is diagonal of covariance matrix (second derivative of likelihood evaluated at points)
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"""
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assert y.shape == f.shape
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e = y - f
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hess = ((self.v + 1)*(e**2 - self.v*self.sigma2)) / ((self.sigma2*self.v + e**2)**2)
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return hess
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def d3lik_d3f(self, y, f, extra_data=None):
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"""
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Third order derivative link_function (log-likelihood ) at y given f f_j w.r.t f and f_j
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$$\frac{d^{3}p(y_{i}|f_{i})}{d^{3}f} = \frac{-2(v+1)((y_{i} - f_{i})^3 - 3(y_{i} - f_{i}) \sigma^{2} v))}{((y_{i} - f_{i}) + \sigma^{2} v)^3}$$
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"""
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assert y.shape == f.shape
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e = y - f
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d3lik_d3f = ( -(2*(self.v + 1)*(-e)*(e**2 - 3*self.v*self.sigma2)) /
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((e**2 + self.sigma2*self.v)**3)
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)
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return d3lik_d3f
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def lik_dstd(self, y, f, extra_data=None):
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"""
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Gradient of the likelihood (lik) w.r.t sigma parameter (standard deviation)
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Terms relavent to derivatives wrt sigma are:
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-log(sqrt(v*pi)*s) -(1/2)*(v + 1)*log(1 + (1/v)*((y-f)/(s))^2))
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$$\frac{dp(y_{i}|f_{i})}{d\sigma} = -\frac{1}{\sigma} + \frac{(1+v)(y_{i}-f_{i})^2}{\sigma^3 v(1 + \frac{1}{v}(\frac{(y_{i} - f_{i})}{\sigma^2})^2)}$$
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"""
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assert y.shape == f.shape
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e = y - f
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dlik_dsigma = self.v*(e**2 - self.sigma2)/(2*self.sigma2*(self.sigma2*self.v + e**2))
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return dlik_dsigma
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def dlik_df_dstd(self, y, f, extra_data=None):
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"""
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Gradient of the dlik_df w.r.t sigma parameter (standard deviation)
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$$\frac{d}{d\sigma}(\frac{dp(y_{i}|f_{i})}{df}) = \frac{-2\sigma v(v + 1)(y_{i}-f_{i})}{(y_{i}-f_{i})^2 + \sigma^2 v)^2}$$
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"""
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assert y.shape == f.shape
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e = y - f
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dlik_grad_dsigma = (self.v*(self.v+1)*(-e))/((self.sigma2*self.v + e**2)**2)
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return dlik_grad_dsigma
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def d2lik_d2f_dstd(self, y, f, extra_data=None):
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"""
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Gradient of the hessian (d2lik_d2f) w.r.t sigma parameter (standard deviation)
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$$\frac{d}{d\sigma}(\frac{d^{2}p(y_{i}|f_{i})}{d^{2}f}) = \frac{2\sigma v(v + 1)(\sigma^2 v - 3(y-f)^2)}{((y-f)^2 + \sigma^2 v)^3}$$
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"""
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assert y.shape == f.shape
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e = y - f
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dlik_hess_dsigma = ( (self.v*(self.v+1)*(self.sigma2*self.v - 3*(e**2)))
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/ ((self.sigma2*self.v + (e**2))**3)
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)
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return dlik_hess_dsigma
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def _gradients(self, y, f, extra_data=None):
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#must be listed in same order as 'get_param_names'
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derivs = ([self.lik_dstd(y, f, extra_data=extra_data)],
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[self.dlik_df_dstd(y, f, extra_data=extra_data)],
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[self.d2lik_d2f_dstd(y, f, extra_data=extra_data)]
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) # lists as we might learn many parameters
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# ensure we have gradients for every parameter we want to optimize
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assert len(derivs[0]) == len(self._get_param_names())
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assert len(derivs[1]) == len(self._get_param_names())
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assert len(derivs[2]) == len(self._get_param_names())
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return derivs
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def predictive_values(self, mu, var):
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"""
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Compute mean, and conficence interval (percentiles 5 and 95) of the prediction
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Need to find what the variance is at the latent points for a student t*normal p(y*|f*)p(f*)
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(((g((v+1)/2))/(g(v/2)*s*sqrt(v*pi)))*(1+(1/v)*((y-f)/s)^2)^(-(v+1)/2))
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*((1/(s*sqrt(2*pi)))*exp(-(1/(2*(s^2)))*((y-f)^2)))
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"""
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#We want the variance around test points y which comes from int p(y*|f*)p(f*) df*
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#Var(y*) = Var(E[y*|f*]) + E[Var(y*|f*)]
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#Since we are given f* (mu) which is our mean (expected) value of y*|f* then the variance is the variance around this
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#Which was also given to us as (var)
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#We also need to know the expected variance of y* around samples f*, this is the variance of the student t distribution
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#However the variance of the student t distribution is not dependent on f, only on sigma and the degrees of freedom
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true_var = var + self.variance
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#Now we have an analytical solution for the variances of the distribution p(y*|f*)p(f*) around our test points but we now
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#need the 95 and 5 percentiles.
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#FIXME: Hack, just pretend p(y*|f*)p(f*) is a gaussian and use the gaussian's percentiles
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p_025 = mu - 2.*np.sqrt(true_var)
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p_975 = mu + 2.*np.sqrt(true_var)
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return mu, np.nan*mu, p_025, p_975
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def sample_predicted_values(self, mu, var):
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""" Experimental sample approches and numerical integration """
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#p_025 = stats.t.ppf(.025, mu)
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#p_975 = stats.t.ppf(.975, mu)
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num_test_points = mu.shape[0]
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#Each mu is the latent point f* at the test point x*,
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#and the var is the gaussian variance at this point
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#Take lots of samples from this, so we have lots of possible values
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#for latent point f* for each test point x* weighted by how likely we were to pick it
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print "Taking %d samples of f*".format(num_test_points)
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num_f_samples = 10
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num_y_samples = 10
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student_t_means = np.random.normal(loc=mu, scale=np.sqrt(var), size=(num_test_points, num_f_samples))
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print "Student t means shape: ", student_t_means.shape
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#Now we have lots of f*, lets work out the likelihood of getting this by sampling
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#from a student t centred on this point, sample many points from this distribution
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#centred on f*
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#for test_point, f in enumerate(student_t_means):
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#print test_point
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#print f.shape
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#student_t_samples = stats.t.rvs(self.v, loc=f[:,None],
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#scale=self.sigma,
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#size=(num_f_samples, num_y_samples))
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#print student_t_samples.shape
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student_t_samples = stats.t.rvs(self.v, loc=student_t_means[:, None],
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scale=self.sigma,
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size=(num_test_points, num_y_samples, num_f_samples))
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student_t_samples = np.reshape(student_t_samples,
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(num_test_points, num_y_samples*num_f_samples))
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#Now take the 97.5 and 0.25 percentile of these points
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p_025 = stats.scoreatpercentile(student_t_samples, .025, axis=1)[:, None]
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p_975 = stats.scoreatpercentile(student_t_samples, .975, axis=1)[:, None]
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##Alernenately we could sample from int p(y|f*)p(f*|x*) df*
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def t_gaussian(f, mu, var):
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return (((gamma((self.v+1)*0.5)) / (gamma(self.v*0.5)*self.sigma*np.sqrt(self.v*np.pi))) * ((1+(1/self.v)*(((mu-f)/self.sigma)**2))**(-(self.v+1)*0.5))
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* ((1/(np.sqrt(2*np.pi*var)))*np.exp(-(1/(2*var)) *((mu-f)**2)))
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)
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def t_gauss_int(mu, var):
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print "Mu: ", mu
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print "var: ", var
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result = integrate.quad(t_gaussian, 0.025, 0.975, args=(mu, var))
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print "Result: ", result
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return result[0]
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vec_t_gauss_int = np.vectorize(t_gauss_int)
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p = vec_t_gauss_int(mu, var)
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p_025 = mu - p
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p_975 = mu + p
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return mu, np.nan*mu, p_025, p_975
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class Gaussian(LikelihoodFunction):
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"""
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Gaussian likelihood - this is a test class for approximation schemes
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"""
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def __init__(self, variance, D, N, link=None):
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super(Gaussian, self).__init__(link)
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self.D = D
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self.N = N
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self._variance = float(variance)
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self._set_params(np.asarray(variance))
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#Don't support normalizing yet
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self._bias = np.zeros((1, self.D))
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self._scale = np.ones((1, self.D))
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def _get_params(self):
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return np.asarray(self._variance)
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def _get_param_names(self):
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return ["noise_variance"]
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def _set_params(self, x):
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self._variance = float(x)
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self.I = np.eye(self.N)
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self.covariance_matrix = self.I * self._variance
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self.Ki = self.I*(1.0 / self._variance)
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self.ln_K = np.trace(self.covariance_matrix)
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def link_function(self, y, f, extra_data=None):
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"""link_function $\ln p(y|f)$
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$$\ln p(y_{i}|f_{i}) = \ln $$
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:y: data
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:f: latent variables f
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:extra_data: extra_data which is not used in student t distribution
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:returns: float(likelihood evaluated for this point)
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"""
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assert y.shape == f.shape
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e = y - f
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eeT = np.dot(e, e.T)
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objective = (- 0.5*self.D*np.log(2*np.pi)
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- 0.5*self.ln_K
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#- 0.5*np.sum(np.multiply(self.Ki, eeT))
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- 0.5*np.dot(np.dot(e.T, self.Ki), e)
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)
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return np.sum(objective)
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def dlik_df(self, y, f, extra_data=None):
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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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:y: data
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:f: latent variables f
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:extra_data: extra_data which is not used in student t distribution
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:returns: gradient of likelihood evaluated at points
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"""
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assert y.shape == f.shape
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s2_i = (1.0/self._variance)*self.I
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grad = np.dot(s2_i, y) - 0.5*np.dot(s2_i, f)
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return grad
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def d2lik_d2f(self, y, f, extra_data=None):
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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 diagonal of hessian, since every where else it is 0, as the likelihood factorizes over cases
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(the distribution for y_{i} depends only on f_{i} not on f_{j!=i}
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:y: data
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:f: latent variables f
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:extra_data: extra_data which is not used in student t distribution
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:returns: array which is diagonal of covariance matrix (second derivative of likelihood evaluated at points)
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"""
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assert y.shape == f.shape
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s2_i = (1.0/self._variance)*self.I
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hess = 0.5*np.diag(-s2_i)[:, None] # FIXME: CAREFUL THIS MAY NOT WORK WITH MULTIDIMENSIONS?
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return hess
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def d3lik_d3f(self, y, f, extra_data=None):
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"""
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Third order derivative link_function (log-likelihood ) at y given f f_j w.r.t f and f_j
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$$\frac{d^{3}p(y_{i}|f_{i})}{d^{3}f} = \frac{-2(v+1)((y_{i} - f_{i})^3 - 3(y_{i} - f_{i}) \sigma^{2} v))}{((y_{i} - f_{i}) + \sigma^{2} v)^3}$$
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"""
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assert y.shape == f.shape
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d3lik_d3f = np.diagonal(0*self.I)[:, None] # FIXME: CAREFUL THIS MAY NOT WORK WITH MULTIDIMENSIONS?
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return d3lik_d3f
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def lik_dstd(self, y, f, extra_data=None):
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"""
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Gradient of the likelihood (lik) w.r.t sigma parameter (standard deviation)
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"""
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assert y.shape == f.shape
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e = y - f
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s_4 = 1.0/(self._variance**2)
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dlik_dsigma = -0.5*self.N*1/self._variance + 0.5*s_4*np.trace(np.dot(e.T, np.dot(self.I, e)))
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return dlik_dsigma
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def dlik_df_dstd(self, y, f, extra_data=None):
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"""
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Gradient of the dlik_df w.r.t sigma parameter (standard deviation)
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"""
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assert y.shape == f.shape
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s_4 = 1.0/(self._variance**2)
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dlik_grad_dsigma = -np.dot(s_4, np.dot(self.I, y)) + 0.5*np.dot(s_4, np.dot(self.I, f))
|
||||
return dlik_grad_dsigma
|
||||
|
||||
def d2lik_d2f_dstd(self, y, f, extra_data=None):
|
||||
"""
|
||||
Gradient of the hessian (d2lik_d2f) w.r.t sigma parameter (standard deviation)
|
||||
|
||||
$$\frac{d}{d\sigma}(\frac{d^{2}p(y_{i}|f_{i})}{d^{2}f}) = \frac{2\sigma v(v + 1)(\sigma^2 v - 3(y-f)^2)}{((y-f)^2 + \sigma^2 v)^3}$$
|
||||
"""
|
||||
assert y.shape == f.shape
|
||||
dlik_hess_dsigma = 0.5*np.diag((1.0/(self._variance**2))*self.I)[:, None]
|
||||
return dlik_hess_dsigma
|
||||
|
||||
def _gradients(self, y, f, extra_data=None):
|
||||
#must be listed in same order as 'get_param_names'
|
||||
derivs = ([self.lik_dstd(y, f, extra_data=extra_data)],
|
||||
[self.dlik_df_dstd(y, f, extra_data=extra_data)],
|
||||
[self.d2lik_d2f_dstd(y, f, extra_data=extra_data)]
|
||||
) # lists as we might learn many parameters
|
||||
# ensure we have gradients for every parameter we want to optimize
|
||||
assert len(derivs[0]) == len(self._get_param_names())
|
||||
assert len(derivs[1]) == len(self._get_param_names())
|
||||
assert len(derivs[2]) == len(self._get_param_names())
|
||||
return derivs
|
||||
|
||||
def predictive_values(self, mu, var):
|
||||
mean = mu * self._scale + self._bias
|
||||
true_var = (var + self._variance) * self._scale ** 2
|
||||
_5pc = mean - 2.*np.sqrt(true_var)
|
||||
_95pc = mean + 2.*np.sqrt(true_var)
|
||||
return mean, true_var, _5pc, _95pc
|
||||
|
|
|
|||
Loading…
Add table
Add a link
Reference in a new issue