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Adding weibull likelihood, requires 'extra_data' to be passed to likelihood, i.e. the censoring information
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Alan Saul
parent
65481d7a73
commit
9bbb11b825
2 changed files with 104 additions and 19 deletions
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@ -30,7 +30,7 @@ def pddet(A):
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class Laplace(likelihood):
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"""Laplace approximation to a posterior"""
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def __init__(self, data, likelihood_function, rasm=True):
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def __init__(self, data, likelihood_function, extra_data=None, rasm=True):
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"""
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Laplace Approximation
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@ -44,13 +44,15 @@ class Laplace(likelihood):
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Arguments
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---------
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:data: @todo
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:data: array of data the likelihood function is approximating
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:likelihood_function: likelihood function - subclass of likelihood_function
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:extra_data: additional data used by some likelihood functions, for example survival likelihoods need censoring data
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:rasm: Flag of whether to use rasmussens numerically stable mode finding or simple ncg optimisation
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"""
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self.data = data
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self.likelihood_function = likelihood_function
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self.extra_data = extra_data
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self.rasm = rasm
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#Inital values
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@ -179,7 +181,7 @@ class Laplace(likelihood):
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self.f_hat = self.ncg_mode(K)
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#At this point get the hessian matrix
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self.W = -np.diag(self.likelihood_function.link_hess(self.data, self.f_hat))
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self.W = -np.diag(self.likelihood_function.link_hess(self.data, self.f_hat, extra_data=self.extra_data))
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if not self.likelihood_function.log_concave:
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self.W[self.W < 0] = 1e-6 # FIXME-HACK: This is a hack since GPy can't handle negative variances which can occur
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@ -194,7 +196,7 @@ class Laplace(likelihood):
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Ki_W_i = self.K - mdot(self.K, self.W_12, self.Bi, self.W_12, self.K)
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self.ln_Ki_W_i_det = np.linalg.det(Ki_W_i)
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b = np.dot(self.W, self.f_hat) + self.likelihood_function.link_grad(self.data, self.f_hat)[:, None]
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b = np.dot(self.W, self.f_hat) + self.likelihood_function.link_grad(self.data, self.f_hat, extra_data=self.extra_data)[:, None]
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solve_chol = cho_solve((self.B_chol, True), mdot(self.W_12, (K, b)))
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a = b - mdot(self.W_12, solve_chol)
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self.f_Ki_f = np.dot(self.f_hat.T, a)
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@ -203,7 +205,7 @@ class Laplace(likelihood):
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self.ln_z_hat = (- 0.5*self.f_Ki_f
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- 0.5*self.ln_K_det
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+ 0.5*self.ln_Ki_W_i_det
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+ self.likelihood_function.link_function(self.data, self.f_hat)
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+ self.likelihood_function.link_function(self.data, self.f_hat, extra_data=self.extra_data)
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)
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return self._compute_GP_variables()
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@ -236,16 +238,16 @@ class Laplace(likelihood):
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#FIXME: Can we get rid of this horrible reshaping?
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#ONLY WORKS FOR 1D DATA
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def obj(f):
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res = -1 * (self.likelihood_function.link_function(self.data[:, 0], f) - 0.5 * np.dot(f.T, np.dot(self.Ki, f))
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res = -1 * (self.likelihood_function.link_function(self.data[:, 0], f, extra_data=self.extra_data) - 0.5 * np.dot(f.T, np.dot(self.Ki, f))
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+ self.NORMAL_CONST)
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return float(res)
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def obj_grad(f):
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res = -1 * (self.likelihood_function.link_grad(self.data[:, 0], f) - np.dot(self.Ki, f))
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res = -1 * (self.likelihood_function.link_grad(self.data[:, 0], f, extra_data=self.extra_data) - np.dot(self.Ki, f))
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return np.squeeze(res)
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def obj_hess(f):
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res = -1 * (--np.diag(self.likelihood_function.link_hess(self.data[:, 0], f)) - self.Ki)
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res = -1 * (--np.diag(self.likelihood_function.link_hess(self.data[:, 0], f, extra_data=self.extra_data)) - self.Ki)
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return np.squeeze(res)
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f_hat = sp.optimize.fmin_ncg(obj, f, fprime=obj_grad, fhess=obj_hess)
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@ -267,7 +269,7 @@ class Laplace(likelihood):
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def obj(a, f):
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#Careful of shape of data!
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return -0.5*np.dot(a.T, f) + self.likelihood_function.link_function(self.data, f)
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return -0.5*np.dot(a.T, f) + self.likelihood_function.link_function(self.data, f, extra_data=self.extra_data)
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difference = np.inf
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epsilon = 1e-6
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@ -276,7 +278,7 @@ class Laplace(likelihood):
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i = 0
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while difference > epsilon and i < MAX_ITER and rs < MAX_RESTART:
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f_old = f.copy()
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W = -np.diag(self.likelihood_function.link_hess(self.data, f))
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W = -np.diag(self.likelihood_function.link_hess(self.data, f, extra_data=self.extra_data))
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if not self.likelihood_function.log_concave:
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W[W < 0] = 1e-6 # FIXME-HACK: This is a hack since GPy can't handle negative variances which can occur
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# If the likelihood is non-log-concave. We wan't to say that there is a negative variance
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@ -285,7 +287,7 @@ class Laplace(likelihood):
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B, L, W_12 = self._compute_B_statistics(K, W)
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W_f = np.dot(W, f)
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grad = self.likelihood_function.link_grad(self.data, f)[:, None]
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grad = self.likelihood_function.link_grad(self.data, f, extra_data=self.extra_data)[:, None]
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#Find K_i_f
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b = W_f + grad
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@ -4,6 +4,7 @@ import numpy as np
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from GPy.likelihoods.likelihood_functions import likelihood_function
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from scipy import stats
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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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@ -24,15 +25,16 @@ class student_t(likelihood_function):
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self.log_concave = False
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@property
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def variance(self):
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def variance(self, extra_data=None):
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return (self.v / float(self.v - 2)) * (self.sigma**2)
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def link_function(self, y, f):
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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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: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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@ -49,7 +51,7 @@ class student_t(likelihood_function):
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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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def link_grad(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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@ -57,6 +59,7 @@ class student_t(likelihood_function):
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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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@ -67,17 +70,18 @@ class student_t(likelihood_function):
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grad = ((self.v + 1) * e) / (self.v * (self.sigma**2) + (e**2))
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return np.squeeze(grad)
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def link_hess(self, y, f):
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def link_hess(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 diaganol of hessian, since every where else it is 0
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Will return diagonal 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: 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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y = np.squeeze(y)
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@ -139,7 +143,7 @@ class student_t(likelihood_function):
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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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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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@ -152,7 +156,7 @@ class student_t(likelihood_function):
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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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* ((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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@ -167,4 +171,83 @@ class student_t(likelihood_function):
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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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import ipdb; ipdb.set_trace() ### XXX BREAKPOINT
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return mu, np.nan*mu, p_025, p_975
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class weibull_survival(likelihood_function):
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"""Weibull t likelihood distribution for survival analysis with censoring
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For nomanclature see Bayesian Survival Analysis
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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, shape, scale):
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self.shape = shape
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self.scale = scale
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#FIXME: This should be in the superclass
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self.log_concave = True
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def link_function(self, y, f, extra_data=None):
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"""
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link_function $\ln p(y|f)$, i.e. log likelihood
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$$\ln p(y|f) = v_{i}(\ln \alpha + (\alpha - 1)\ln y_{i} + f_{i}) - y_{i}^{\alpha}\exp(f_{i})$$
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:y: time of event data
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:f: latent variables f
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:extra_data: the censoring indicator, 1 for censored, 0 for not
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:returns: float(likelihood evaluated for this point)
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"""
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y = np.squeeze(y)
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f = np.squeeze(f)
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assert y.shape == f.shape
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v = extra_data
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objective = v*(np.log(self.shape) + (self.shape - 1)*np.log(y) + f) - (y**self.shape)*np.exp(f) # FIXME: CHECK THIS WITH BOOK, wheres scale?
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return np.sum(objective)
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def link_grad(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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$$\frac{d}{df} \ln p(y_{i}|f_{i}) = v_{i} - y_{i}\exp(f_{i})
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:y: data
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:f: latent variables f
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:extra_data: the censoring indicator, 1 for censored, 0 for not
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:returns: gradient of likelihood evaluated at points
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"""
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y = np.squeeze(y)
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f = np.squeeze(f)
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assert y.shape == f.shape
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v = extra_data
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grad = v - (y**self.shape)*np.exp(f)
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return np.squeeze(grad)
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def link_hess(self, y, f, extra_data=None):
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"""
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Hessian at this point (if we are only looking at the link function not the prior) the hessian will be 0 unless i == j
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i.e. second derivative link_function at y given f f_j w.r.t f and f_j
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Will return diagonal of hessian, since every where else it is 0
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$$\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: data
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:f: latent variables f
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:extra_data: extra_data which is not used hessian
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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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y = np.squeeze(y)
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f = np.squeeze(f)
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assert y.shape == f.shape
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hess = (y**self.shape)*np.exp(f)
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return np.squeeze(hess)
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