diff --git a/GPy/examples/laplace_approximations.py b/GPy/examples/laplace_approximations.py index 4d8e96b8..27f063dc 100644 --- a/GPy/examples/laplace_approximations.py +++ b/GPy/examples/laplace_approximations.py @@ -69,22 +69,21 @@ def debug_student_t_noise_approx(): print "Clean Gaussian" #A GP should completely break down due to the points as they get a lot of weight # create simple GP model - m = GPy.models.GP_regression(X, Y, kernel=kernel1) - # optimize - m.ensure_default_constraints() - m.optimize() - # plot - if plot: - plt.figure(1) - plt.suptitle('Gaussian likelihood') - plt.subplot(131) - m.plot() - plt.plot(X_full, Y_full) - print m + #m = GPy.models.GP_regression(X, Y, kernel=kernel1) + ## optimize + #m.ensure_default_constraints() + #m.optimize() + ## plot + #if plot: + #plt.figure(1) + #plt.suptitle('Gaussian likelihood') + #plt.subplot(131) + #m.plot() + #plt.plot(X_full, Y_full) + #print m edited_real_sd = initial_var_guess #real_sd - import ipdb; ipdb.set_trace() ### XXX BREAKPOINT print "Clean student t, rasm" t_distribution = GPy.likelihoods.likelihood_functions.student_t(deg_free, sigma=edited_real_sd) stu_t_likelihood = GPy.likelihoods.Laplace(Y.copy(), t_distribution, rasm=True) @@ -95,10 +94,10 @@ def debug_student_t_noise_approx(): m.constrain_positive('t_noi') #m.constrain_fixed('t_noise_variance', real_sd) m.update_likelihood_approximation() + m.optimize('scg', messages=True) print(m) return m #m.optimize('lbfgsb', messages=True, callback=m._update_params_callback) - m.optimize('scg', messages=True) if plot: plt.suptitle('Student-t likelihood') plt.subplot(132) diff --git a/GPy/likelihoods/Laplace.py b/GPy/likelihoods/Laplace.py index 69c0876b..f8ba25f1 100644 --- a/GPy/likelihoods/Laplace.py +++ b/GPy/likelihoods/Laplace.py @@ -79,17 +79,40 @@ class Laplace(likelihood): return (self._Kgradients(dL_d_K_Sigma, dK_dthetaK), self._gradients(dL_d_K_Sigma)) def _shared_gradients_components(self): + Ki, _, _, _ = pdinv(self.K) + Ki_W_i = inv(Ki + self.W) #Do it non numerically stable for now + d3lik_d3fhat = self.likelihood_function.d3lik_d3f(self.data, self.f_hat) + dL_dfhat = -0.5*np.dot(np.diag(Ki_W_i), d3lik_d3fhat) + KW = np.dot(self.K, self.W) + I_KW_i = inv(np.eye(KW.shape[0]) + KW) + return dL_dfhat, Ki, I_KW_i def _Kgradients(self, dL_d_K_Sigma, dK_dthetaK): """ Gradients with respect to prior kernel parameters """ + dL_dfhat, Ki, I_KW_i = self._shared_gradients_components() + K_Wi_i = inv(self.K + inv(self.W)) + dlp = self.likelihood_function.dlik_df(self.data, self.f_hat) + + dL_dthetaK = np.zeros(dK_dthetaK.shape) + for thetaK_i, dK_dthetaK_i in enumerate(dK_dthetaK): + #Explicit + dL_dthetaK[thetaK_i] = 0.5*mdot(self.f_hat.T, Ki, dK_dthetaK_i, Ki, self.f_hat) - 0.5*np.trace(np.dot(K_Wi_i, dK_dthetaK_i)) + #Implicit + df_hat_dthetaK = mdot(I_KW_i, dK_dthetaK, dlp) + dL_dthetaK[thetaK_i] += np.dot(dL_dfhat.T, df_hat_dthetaK) + return dL_dthetaK def _gradients(self, partial): """ Gradients with respect to likelihood parameters """ + dL_dfhat, Ki, I_KW_i = self._shared_gradients_components() + dlik_dthetaL, dlik_grad_dthetaL, dlik_hess_dthetaL = self.likelihood_function._gradients(self.data, self.f_hat) + dL_dthetaL = np.zeros(dlik_dthetaL.shape) + return dL_dthetaL #should be array of length *params-being optimized*, for student t just optimising 1 parameter, this is (1,) def _compute_GP_variables(self): @@ -197,7 +220,7 @@ class Laplace(likelihood): #At this point get the hessian matrix #print "Data: ", self.data #print "fhat: ", self.f_hat - self.W = -np.diag(self.likelihood_function.link_hess(self.data, self.f_hat, extra_data=self.extra_data)) + self.W = -np.diag(self.likelihood_function.d2lik_d2f(self.data, self.f_hat, extra_data=self.extra_data)) if not self.likelihood_function.log_concave: self.W[self.W < 0] = 1e-6 # FIXME-HACK: This is a hack since GPy can't handle negative variances which can occur @@ -212,7 +235,7 @@ class Laplace(likelihood): Ki_W_i = self.K - mdot(self.K, self.W_12, self.Bi, self.W_12, self.K) self.ln_Ki_W_i_det = np.linalg.det(Ki_W_i) - b = np.dot(self.W, self.f_hat) + self.likelihood_function.link_grad(self.data, self.f_hat, extra_data=self.extra_data)[:, None] + b = np.dot(self.W, self.f_hat) + self.likelihood_function.dlik_df(self.data, self.f_hat, extra_data=self.extra_data)[:, None] solve_chol = cho_solve((self.B_chol, True), mdot(self.W_12, (self.K, b))) a = b - mdot(self.W_12, solve_chol) self.Ki_f = a @@ -259,11 +282,11 @@ class Laplace(likelihood): return float(res) def obj_grad(f): - res = -1 * (self.likelihood_function.link_grad(self.data[:, 0], f, extra_data=self.extra_data) - np.dot(self.Ki, f)) + res = -1 * (self.likelihood_function.dlik_df(self.data[:, 0], f, extra_data=self.extra_data) - np.dot(self.Ki, f)) return np.squeeze(res) def obj_hess(f): - res = -1 * (--np.diag(self.likelihood_function.link_hess(self.data[:, 0], f, extra_data=self.extra_data)) - self.Ki) + res = -1 * (--np.diag(self.likelihood_function.d2lik_d2f(self.data[:, 0], f, extra_data=self.extra_data)) - self.Ki) return np.squeeze(res) f_hat = sp.optimize.fmin_ncg(obj, f, fprime=obj_grad, fhess=obj_hess) @@ -294,7 +317,7 @@ class Laplace(likelihood): i = 0 while difference > epsilon and i < MAX_ITER and rs < MAX_RESTART: #f_old = f.copy() - W = -np.diag(self.likelihood_function.link_hess(self.data, f, extra_data=self.extra_data)) + W = -np.diag(self.likelihood_function.d2lik_d2f(self.data, f, extra_data=self.extra_data)) if not self.likelihood_function.log_concave: W[W < 0] = 1e-6 # FIXME-HACK: This is a hack since GPy can't handle negative variances which can occur # If the likelihood is non-log-concave. We wan't to say that there is a negative variance @@ -303,7 +326,7 @@ class Laplace(likelihood): B, L, W_12 = self._compute_B_statistics(K, W) W_f = np.dot(W, f) - grad = self.likelihood_function.link_grad(self.data, f, extra_data=self.extra_data)[:, None] + grad = self.likelihood_function.dlik_df(self.data, f, extra_data=self.extra_data)[:, None] #Find K_i_f b = W_f + grad diff --git a/GPy/likelihoods/likelihood_functions.py b/GPy/likelihoods/likelihood_functions.py index 646293d2..d75e7218 100644 --- a/GPy/likelihoods/likelihood_functions.py +++ b/GPy/likelihoods/likelihood_functions.py @@ -159,10 +159,10 @@ class student_t(likelihood_function): d2ln p(yi|fi)_d2fifj """ def __init__(self, deg_free, sigma=2): + #super(student_t, self).__init__() self.v = deg_free self.sigma = sigma self.log_concave = False - #super(student_t, self).__init__() self._set_params(np.asarray(sigma)) @@ -174,8 +174,6 @@ class student_t(likelihood_function): def _set_params(self, x): self.sigma = float(x) - #self.covariance_matrix = np.eye(self.N)*self._variance - #self.precision = 1./self._variance @property def variance(self, extra_data=None): @@ -185,6 +183,8 @@ class student_t(likelihood_function): """link_function $\ln p(y|f)$ $$\ln p(y_{i}|f_{i}) = \ln \Gamma(\frac{v+1}{2}) - \ln \Gamma(\frac{v}{2})\sqrt{v \pi}\sigma - \frac{v+1}{2}\ln (1 + \frac{1}{v}\left(\frac{y_{i} - f_{i}}{\sigma}\right)^2$$ + 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)) + :y: data :f: latent variables f :extra_data: extra_data which is not used in student t distribution @@ -198,17 +198,16 @@ class student_t(likelihood_function): e = y - f 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)) - ) + - 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, f, extra_data=None): + def dlik_df(self, y, f, extra_data=None): """ Gradient of the link function at y, given f w.r.t f - $$\frac{d}{df}p(y_{i}|f_{i}) = \frac{(v + 1)(y - f)}{v \sigma^{2} + (y_{i} - f_{i})^{2}}$$ + $$\frac{dp(y_{i}|f_{i})}{df} = \frac{-(v+1)(f_{i}-y_{i})}{(f_{i}-y_{i})^{2} + \sigma^{2}v}$$ :y: data :f: latent variables f @@ -220,17 +219,17 @@ class student_t(likelihood_function): f = np.squeeze(f) assert y.shape == f.shape e = y - f - grad = ((self.v + 1) * e) / (self.v * (self.sigma**2) + (e**2)) + grad = -((self.v + 1) * e) / (self.v * (self.sigma**2) + (e**2)) return np.squeeze(grad) - def link_hess(self, y, f, extra_data=None): + def d2lik_d2f(self, y, f, extra_data=None): """ 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 diagonal of hessian, since every where else it is 0 - $$\frac{d^{2}p(y_{i}|f_{i})}{df^{2}} = \frac{(v + 1)(y - f)}{v \sigma^{2} + (y_{i} - f_{i})^{2}}$$ + $$\frac{d^{2}p(y_{i}|f_{i})}{d^{3}f} = \frac{(v+1)((f_{i}-y_{i})^{2} - \sigma^{2}v)}{((f_{i}-y_{i})^{2} + \sigma^{2}v)^{2}}$$ :y: data :f: latent variables f @@ -245,54 +244,79 @@ class student_t(likelihood_function): hess = ((self.v + 1)*(e**2 - self.v*(self.sigma**2))) / ((((self.sigma**2)*self.v) + e**2)**2) return np.squeeze(hess) - def d3link(self, y, f, extra_data=None): + def d3lik_d3f(self, y, f, extra_data=None): """ Third order derivative link_function (log-likelihood ) at y given f f_j w.r.t f and f_j - $$\frac{2(v+1)((y-f)^{3} - 3\sigma^{2}v(y-f))}{((y-f)^{2} + \sigma^{2}v)^{3}}$$ + $$\frac{d^{3}p(y_{i}|f_{i})}{d^{3}f} = \frac{-2(v+1)((f_{i} - y_{i})^3 - 3(f_{i} - y_{i}) \sigma^{2} v))}{((f_{i} - y_{i}) + \sigma^{2} v)^3}$$ """ y = np.squeeze(y) f = np.squeeze(f) assert y.shape == f.shape e = y - f - d3link_d3f = ( (2*(self.v + 1)*(-1*e)*(e**2 - 3*(self.sigma**2)*self.v)) - / ((e**2 + (self.sigma**2)*self.v)**3) - ) - return np.squeeze(d3link_d3f) + d3lik_d3f = ( -(2*(self.v + 1)*(e**3 - e*3*self.v*(self.sigma**2))) / + ((e**2 + (self.sigma**2)*self.v)**3) + ) + return np.squeeze(d3lik_d3f) - def link_hess_grad_std(self, y, f, extra_data=None): + def link_dstd(self, y, f, extra_data=None): """ - Gradient of the hessian w.r.t sigma parameter (standard deviation) + Gradient of the likelihood (lik) w.r.t sigma parameter (standard deviation) - $$\frac{2\sigma v(v+1)(\sigma^{2}v - 3(f-y)^2)}{((f-y)^{2} + \sigma^{2}v)^{3}} + Terms relavent to derivatives wrt sigma are: + -log(sqrt(v*pi)*s) -(1/2)*(v + 1)*log(1 + (1/v)*((y-f)/(s))^2)) + + $$\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)}$$ """ y = np.squeeze(y) f = np.squeeze(f) assert y.shape == f.shape e = y - f - hess_grad_sigma = ( (2*self.sigma*self.v*(self.v + 1)*((self.sigma**2)*self.v - 3*(e**2))) - / ((e**2 + (self.sigma**2)*self.v)**3) - ) - return np.squeeze(hess_grad_sigma) + dlik_dsigma = ( (1/self.sigma) - + ((1+self.v)*(e**2))/((self.sigma**3)*self.v*(1 + (e**2) / ((self.sigma**2)*self.v) ) ) + ) + return np.squeeze(dlik_dsigma) - def link_grad_std(self, y, f, extra_data=None): + def dlik_df_dstd(self, y, f, extra_data=None): """ - Gradient of the likelihood w.r.t sigma parameter (standard deviation) + Gradient of the dlik_df w.r.t sigma parameter (standard deviation) - $$\frac{-2\sigma(v+1)(y-f)}{(v\sigma^{2} + (y-f)^{2})^{2}}$$ + $$\frac{d}{d\sigma}(\frac{dp(y_{i}|f_{i})}{df}) = \frac{2\sigma v(v + 1)(f-y)}{(f-y)^2 + \sigma^2 v)^2}$$ """ y = np.squeeze(y) f = np.squeeze(f) assert y.shape == f.shape e = y - f - grad_sigma = ( (-2*self.sigma*self.v*(self.v + 1)*e) - / ((self.v*(self.sigma**2) + e**2)**2) - ) - return np.squeeze(grad_sigma) + dlik_grad_dsigma = ((2*self.sigma*self.v*(self.v + 1)*e) + / ((self.v*(self.sigma**2) + e**2)**2) + ) + return np.squeeze(dlik_grad_dsigma) + + def d2lik_d2f_dstd(self, y, f, extra_data=None): + """ + Gradient of the hessian (d2lik_d2f) w.r.t sigma parameter (standard deviation) + + $$\frac{d}{d\sigma}(\frac{d^{2}p(y_{i}|f_{i})}{d^{2}f}) = \frac{(v + 1)((f-y)^2 - \sigma^2 v)}{((f-y)^2 + \sigma^2 v)}$$ + """ + y = np.squeeze(y) + f = np.squeeze(f) + assert y.shape == f.shape + e = y - f + dlik_hess_dsigma = ( ((v + 1)*(e**2 - (self.sigma**2)*self.v)) / + ((e**2 + (self.sigma**2)*self.v)**2) + ) + return np.squeeze(dlik_hess_dsigma) def _gradients(self, y, f, extra_data=None): - return [self.link_grad_std(y, f, extra_data=extra_data), - self.link_hess_grad_std(y, f, extra_data=extra_data)] # list as we might learn many parameters + derivs = ([self.link_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): """ @@ -412,7 +436,7 @@ class weibull_survival(likelihood_function): 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? return np.sum(objective) - def link_grad(self, y, f, extra_data=None): + def dlik_df(self, y, f, extra_data=None): """ Gradient of the link function at y, given f w.r.t f @@ -432,7 +456,7 @@ class weibull_survival(likelihood_function): grad = v - (y**self.shape)*np.exp(f) return np.squeeze(grad) - def link_hess(self, y, f, extra_data=None): + def d2lik_d2f(self, y, f, extra_data=None): """ 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 diff --git a/GPy/models/GP.py b/GPy/models/GP.py index da379eb1..0b5a8db6 100644 --- a/GPy/models/GP.py +++ b/GPy/models/GP.py @@ -147,7 +147,7 @@ class GP(model): if isinstance(self.likelihood, Laplace): dL_dthetaK_explicit = dL_dthetaK #Need to pass in a matrix of ones to get access to raw dK_dthetaK values without being chained - fake_dL_dKs = np.eye(self.dL_dK.shape[0]) #FIXME: Check this is right... + fake_dL_dKs = np.ones(self.dL_dK.shape) #FIXME: Check this is right... dK_dthetaK = self.kern.dK_dtheta(dL_dK=fake_dL_dKs, X=self.X) dL_dthetaK = self.likelihood._Kgradients(dL_d_K_Sigma=self.dL_dK, dK_dthetaK=dK_dthetaK)