first commit in new structure

GPy/inference/ep.py

@@ -0,0 +1,392 @@
+import numpy as np
+from scipy import stats
+from ..util.linalg import pdinv,mdot,jitchol,chol_inv,DSYR,tdot,dtrtrs
+from likelihood import likelihood
+
+class EP(likelihood):
+    def __init__(self,data,noise_model):
+        """
+        Expectation Propagation
+
+        :param data: data to model
+        :type data: numpy array
+        :param noise_model: noise distribution
+        :type noise_model: A GPy noise model
+
+        """
+        self.noise_model = noise_model
+        self.data = data
+        self.num_data, self.output_dim = self.data.shape
+        self.is_heteroscedastic = True
+        self.num_params = 0
+
+        #Initial values - Likelihood approximation parameters:
+        #p(y|f) = t(f|tau_tilde,v_tilde)
+        self.tau_tilde = np.zeros(self.num_data)
+        self.v_tilde = np.zeros(self.num_data)
+
+        #initial values for the GP variables
+        self.Y = np.zeros((self.num_data,1))
+        self.covariance_matrix = np.eye(self.num_data)
+        self.precision = np.ones(self.num_data)[:,None]
+        self.Z = 0
+        self.YYT = None
+        self.V = self.precision * self.Y
+        self.VVT_factor = self.V
+        self.trYYT = 0.
+
+        super(EP, self).__init__()
+
+    def restart(self):
+        self.tau_tilde = np.zeros(self.num_data)
+        self.v_tilde = np.zeros(self.num_data)
+        self.Y = np.zeros((self.num_data,1))
+        self.covariance_matrix = np.eye(self.num_data)
+        self.precision = np.ones(self.num_data)[:,None]
+        self.Z = 0
+        self.YYT = None
+        self.V = self.precision * self.Y
+        self.VVT_factor = self.V
+        self.trYYT = 0.
+
+    def predictive_values(self,mu,var,full_cov,**noise_args):
+        if full_cov:
+            raise NotImplementedError, "Cannot make correlated predictions with an EP likelihood"
+        return self.noise_model.predictive_values(mu,var,**noise_args)
+
+    def log_predictive_density(self, y_test, mu_star, var_star):
+        """
+        Calculation of the log predictive density
+
+        .. math:
+            p(y_{*}|D) = p(y_{*}|f_{*})p(f_{*}|\mu_{*}\\sigma^{2}_{*})
+
+        :param y_test: test observations (y_{*})
+        :type y_test: (Nx1) array
+        :param mu_star: predictive mean of gaussian p(f_{*}|mu_{*}, var_{*})
+        :type mu_star: (Nx1) array
+        :param var_star: predictive variance of gaussian p(f_{*}|mu_{*}, var_{*})
+        :type var_star: (Nx1) array
+        """
+        return self.noise_model.log_predictive_density(y_test, mu_star, var_star)
+
+    def _get_params(self):
+        #return np.zeros(0)
+        return self.noise_model._get_params()
+
+    def _get_param_names(self):
+        #return []
+        return self.noise_model._get_param_names()
+
+    def _set_params(self,p):
+        #pass # TODO: the EP likelihood might want to take some parameters...
+        self.noise_model._set_params(p)
+
+    def _gradients(self,partial):
+        #return np.zeros(0) # TODO: the EP likelihood might want to take some parameters...
+        return self.noise_model._gradients(partial)
+
+    def _compute_GP_variables(self):
+        #Variables to be called from GP
+        mu_tilde = self.v_tilde/self.tau_tilde #When calling EP, this variable is used instead of Y in the GP model
+        sigma_sum = 1./self.tau_ + 1./self.tau_tilde
+        mu_diff_2 = (self.v_/self.tau_ - mu_tilde)**2
+        self.Z = np.sum(np.log(self.Z_hat)) + 0.5*np.sum(np.log(sigma_sum)) + 0.5*np.sum(mu_diff_2/sigma_sum) #Normalization constant, aka Z_ep
+        self.Z += 0.5*self.num_data*np.log(2*np.pi)
+
+        self.Y =  mu_tilde[:,None]
+        self.YYT = np.dot(self.Y,self.Y.T)
+        self.covariance_matrix = np.diag(1./self.tau_tilde)
+        self.precision = self.tau_tilde[:,None]
+        self.V = self.precision * self.Y
+        self.VVT_factor = self.V
+        self.trYYT = np.trace(self.YYT)
+
+    def fit_full(self, K, epsilon=1e-3,power_ep=[1.,1.]):
+        """
+        The expectation-propagation algorithm.
+        For nomenclature see Rasmussen & Williams 2006.
+
+        :param epsilon: Convergence criterion, maximum squared difference allowed between mean updates to stop iterations (float)
+        :type epsilon: float
+        :param power_ep: Power EP parameters
+        :type power_ep: list of floats
+
+        """
+        self.epsilon = epsilon
+        self.eta, self.delta = power_ep
+
+        #Initial values - Posterior distribution parameters: q(f|X,Y) = N(f|mu,Sigma)
+        mu = np.zeros(self.num_data)
+        Sigma = K.copy()
+
+        """
+        Initial values - Cavity distribution parameters:
+        q_(f|mu_,sigma2_) = Product{q_i(f|mu_i,sigma2_i)}
+        sigma_ = 1./tau_
+        mu_ = v_/tau_
+        """
+        self.tau_ = np.empty(self.num_data,dtype=float)
+        self.v_ = np.empty(self.num_data,dtype=float)
+
+        #Initial values - Marginal moments
+        z = np.empty(self.num_data,dtype=float)
+        self.Z_hat = np.empty(self.num_data,dtype=float)
+        phi = np.empty(self.num_data,dtype=float)
+        mu_hat = np.empty(self.num_data,dtype=float)
+        sigma2_hat = np.empty(self.num_data,dtype=float)
+
+        #Approximation
+        epsilon_np1 = self.epsilon + 1.
+        epsilon_np2 = self.epsilon + 1.
+       	self.iterations = 0
+        self.np1 = [self.tau_tilde.copy()]
+        self.np2 = [self.v_tilde.copy()]
+        while epsilon_np1 > self.epsilon or epsilon_np2 > self.epsilon:
+            update_order = np.random.permutation(self.num_data)
+            for i in update_order:
+                #Cavity distribution parameters
+                self.tau_[i] = 1./Sigma[i,i] - self.eta*self.tau_tilde[i]
+                self.v_[i] = mu[i]/Sigma[i,i] - self.eta*self.v_tilde[i]
+                #Marginal moments
+                self.Z_hat[i], mu_hat[i], sigma2_hat[i] = self.noise_model.moments_match(self.data[i],self.tau_[i],self.v_[i])
+                #Site parameters update
+                Delta_tau = self.delta/self.eta*(1./sigma2_hat[i] - 1./Sigma[i,i])
+                Delta_v = self.delta/self.eta*(mu_hat[i]/sigma2_hat[i] - mu[i]/Sigma[i,i])
+                self.tau_tilde[i] += Delta_tau
+                self.v_tilde[i] += Delta_v
+                #Posterior distribution parameters update
+                DSYR(Sigma,Sigma[:,i].copy(), -float(Delta_tau/(1.+ Delta_tau*Sigma[i,i])))
+                mu = np.dot(Sigma,self.v_tilde)
+                self.iterations += 1
+            #Sigma recomptutation with Cholesky decompositon
+            Sroot_tilde_K = np.sqrt(self.tau_tilde)[:,None]*K
+            B = np.eye(self.num_data) + np.sqrt(self.tau_tilde)[None,:]*Sroot_tilde_K
+            L = jitchol(B)
+            V,info = dtrtrs(L,Sroot_tilde_K,lower=1)
+            Sigma = K - np.dot(V.T,V)
+            mu = np.dot(Sigma,self.v_tilde)
+            epsilon_np1 = sum((self.tau_tilde-self.np1[-1])**2)/self.num_data
+            epsilon_np2 = sum((self.v_tilde-self.np2[-1])**2)/self.num_data
+            self.np1.append(self.tau_tilde.copy())
+            self.np2.append(self.v_tilde.copy())
+
+        return self._compute_GP_variables()
+
+    def fit_DTC(self, Kmm, Kmn, epsilon=1e-3,power_ep=[1.,1.]):
+        """
+        The expectation-propagation algorithm with sparse pseudo-input.
+        For nomenclature see ... 2013.
+
+        :param epsilon: Convergence criterion, maximum squared difference allowed between mean updates to stop iterations (float)
+        :type epsilon: float
+        :param power_ep: Power EP parameters
+        :type power_ep: list of floats
+
+        """
+        self.epsilon = epsilon
+        self.eta, self.delta = power_ep
+
+        num_inducing = Kmm.shape[0]
+
+        #TODO: this doesn't work with uncertain inputs!
+
+        """
+        Prior approximation parameters:
+        q(f|X) = int_{df}{N(f|KfuKuu_invu,diag(Kff-Qff)*N(u|0,Kuu)} = N(f|0,Sigma0)
+        Sigma0 = Qnn = Knm*Kmmi*Kmn
+        """
+        KmnKnm = np.dot(Kmn,Kmn.T)
+        Lm = jitchol(Kmm)
+        Lmi = chol_inv(Lm)
+        Kmmi = np.dot(Lmi.T,Lmi)
+        KmmiKmn = np.dot(Kmmi,Kmn)
+        Qnn_diag = np.sum(Kmn*KmmiKmn,-2)
+        LLT0 = Kmm.copy()
+
+        #Kmmi, Lm, Lmi, Kmm_logdet = pdinv(Kmm)
+        #KmnKnm = np.dot(Kmn, Kmn.T)
+        #KmmiKmn = np.dot(Kmmi,Kmn)
+        #Qnn_diag = np.sum(Kmn*KmmiKmn,-2)
+        #LLT0 = Kmm.copy()
+
+        """
+        Posterior approximation: q(f|y) = N(f| mu, Sigma)
+        Sigma = Diag + P*R.T*R*P.T + K
+        mu = w + P*Gamma
+        """
+        mu = np.zeros(self.num_data)
+        LLT = Kmm.copy()
+        Sigma_diag = Qnn_diag.copy()
+
+        """
+        Initial values - Cavity distribution parameters:
+        q_(g|mu_,sigma2_) = Product{q_i(g|mu_i,sigma2_i)}
+        sigma_ = 1./tau_
+        mu_ = v_/tau_
+        """
+        self.tau_ = np.empty(self.num_data,dtype=float)
+        self.v_ = np.empty(self.num_data,dtype=float)
+
+        #Initial values - Marginal moments
+        z = np.empty(self.num_data,dtype=float)
+        self.Z_hat = np.empty(self.num_data,dtype=float)
+        phi = np.empty(self.num_data,dtype=float)
+        mu_hat = np.empty(self.num_data,dtype=float)
+        sigma2_hat = np.empty(self.num_data,dtype=float)
+
+        #Approximation
+        epsilon_np1 = 1
+        epsilon_np2 = 1
+       	self.iterations = 0
+        np1 = [self.tau_tilde.copy()]
+        np2 = [self.v_tilde.copy()]
+        while epsilon_np1 > self.epsilon or epsilon_np2 > self.epsilon:
+            update_order = np.random.permutation(self.num_data)
+            for i in update_order:
+                #Cavity distribution parameters
+                self.tau_[i] = 1./Sigma_diag[i] - self.eta*self.tau_tilde[i]
+                self.v_[i] = mu[i]/Sigma_diag[i] - self.eta*self.v_tilde[i]
+                #Marginal moments
+                self.Z_hat[i], mu_hat[i], sigma2_hat[i] = self.noise_model.moments_match(self.data[i],self.tau_[i],self.v_[i])
+                #Site parameters update
+                Delta_tau = self.delta/self.eta*(1./sigma2_hat[i] - 1./Sigma_diag[i])
+                Delta_v = self.delta/self.eta*(mu_hat[i]/sigma2_hat[i] - mu[i]/Sigma_diag[i])
+                self.tau_tilde[i] += Delta_tau
+                self.v_tilde[i] += Delta_v
+                #Posterior distribution parameters update
+                DSYR(LLT,Kmn[:,i].copy(),Delta_tau) #LLT = LLT + np.outer(Kmn[:,i],Kmn[:,i])*Delta_tau
+                L = jitchol(LLT)
+                #cholUpdate(L,Kmn[:,i]*np.sqrt(Delta_tau))
+                V,info = dtrtrs(L,Kmn,lower=1)
+                Sigma_diag = np.sum(V*V,-2)
+                si = np.sum(V.T*V[:,i],-1)
+                mu += (Delta_v-Delta_tau*mu[i])*si
+                self.iterations += 1
+            #Sigma recomputation with Cholesky decompositon
+            LLT = LLT0 + np.dot(Kmn*self.tau_tilde[None,:],Kmn.T)
+            L = jitchol(LLT)
+            V,info = dtrtrs(L,Kmn,lower=1)
+            V2,info = dtrtrs(L.T,V,lower=0)
+            Sigma_diag = np.sum(V*V,-2)
+            Knmv_tilde = np.dot(Kmn,self.v_tilde)
+            mu = np.dot(V2.T,Knmv_tilde)
+            epsilon_np1 = sum((self.tau_tilde-np1[-1])**2)/self.num_data
+            epsilon_np2 = sum((self.v_tilde-np2[-1])**2)/self.num_data
+            np1.append(self.tau_tilde.copy())
+            np2.append(self.v_tilde.copy())
+
+        self._compute_GP_variables()
+
+    def fit_FITC(self, Kmm, Kmn, Knn_diag, epsilon=1e-3,power_ep=[1.,1.]):
+        """
+        The expectation-propagation algorithm with sparse pseudo-input.
+        For nomenclature see Naish-Guzman and Holden, 2008.
+
+        :param epsilon: Convergence criterion, maximum squared difference allowed between mean updates to stop iterations (float)
+        :type epsilon: float
+        :param power_ep: Power EP parameters
+        :type power_ep: list of floats
+        """
+        self.epsilon = epsilon
+        self.eta, self.delta = power_ep
+
+        num_inducing = Kmm.shape[0]
+
+        """
+        Prior approximation parameters:
+        q(f|X) = int_{df}{N(f|KfuKuu_invu,diag(Kff-Qff)*N(u|0,Kuu)} = N(f|0,Sigma0)
+        Sigma0 = diag(Knn-Qnn) + Qnn, Qnn = Knm*Kmmi*Kmn
+        """
+        Lm = jitchol(Kmm)
+        Lmi = chol_inv(Lm)
+        Kmmi = np.dot(Lmi.T,Lmi)
+        P0 = Kmn.T
+        KmnKnm = np.dot(P0.T, P0)
+        KmmiKmn = np.dot(Kmmi,P0.T)
+        Qnn_diag = np.sum(P0.T*KmmiKmn,-2)
+        Diag0 = Knn_diag - Qnn_diag
+        R0 = jitchol(Kmmi).T
+
+        """
+        Posterior approximation: q(f|y) = N(f| mu, Sigma)
+        Sigma = Diag + P*R.T*R*P.T + K
+        mu = w + P*Gamma
+        """
+        self.w = np.zeros(self.num_data)
+        self.Gamma = np.zeros(num_inducing)
+        mu = np.zeros(self.num_data)
+        P = P0.copy()
+        R = R0.copy()
+        Diag = Diag0.copy()
+        Sigma_diag = Knn_diag
+        RPT0 = np.dot(R0,P0.T)
+
+        """
+        Initial values - Cavity distribution parameters:
+        q_(g|mu_,sigma2_) = Product{q_i(g|mu_i,sigma2_i)}
+        sigma_ = 1./tau_
+        mu_ = v_/tau_
+        """
+        self.tau_ = np.empty(self.num_data,dtype=float)
+        self.v_ = np.empty(self.num_data,dtype=float)
+
+        #Initial values - Marginal moments
+        z = np.empty(self.num_data,dtype=float)
+        self.Z_hat = np.empty(self.num_data,dtype=float)
+        phi = np.empty(self.num_data,dtype=float)
+        mu_hat = np.empty(self.num_data,dtype=float)
+        sigma2_hat = np.empty(self.num_data,dtype=float)
+
+        #Approximation
+        epsilon_np1 = 1
+        epsilon_np2 = 1
+       	self.iterations = 0
+        self.np1 = [self.tau_tilde.copy()]
+        self.np2 = [self.v_tilde.copy()]
+        while epsilon_np1 > self.epsilon or epsilon_np2 > self.epsilon:
+            update_order = np.random.permutation(self.num_data)
+            for i in update_order:
+                #Cavity distribution parameters
+                self.tau_[i] = 1./Sigma_diag[i] - self.eta*self.tau_tilde[i]
+                self.v_[i] = mu[i]/Sigma_diag[i] - self.eta*self.v_tilde[i]
+                #Marginal moments
+                self.Z_hat[i], mu_hat[i], sigma2_hat[i] = self.noise_model.moments_match(self.data[i],self.tau_[i],self.v_[i])
+                #Site parameters update
+                Delta_tau = self.delta/self.eta*(1./sigma2_hat[i] - 1./Sigma_diag[i])
+                Delta_v = self.delta/self.eta*(mu_hat[i]/sigma2_hat[i] - mu[i]/Sigma_diag[i])
+                self.tau_tilde[i] += Delta_tau
+                self.v_tilde[i] += Delta_v
+                #Posterior distribution parameters update
+                dtd1 = Delta_tau*Diag[i] + 1.
+                dii = Diag[i]
+                Diag[i] = dii - (Delta_tau * dii**2.)/dtd1
+                pi_ = P[i,:].reshape(1,num_inducing)
+                P[i,:] = pi_ - (Delta_tau*dii)/dtd1 * pi_
+                Rp_i = np.dot(R,pi_.T)
+                RTR = np.dot(R.T,np.dot(np.eye(num_inducing) - Delta_tau/(1.+Delta_tau*Sigma_diag[i]) * np.dot(Rp_i,Rp_i.T),R))
+                R = jitchol(RTR).T
+                self.w[i] += (Delta_v - Delta_tau*self.w[i])*dii/dtd1
+                self.Gamma += (Delta_v - Delta_tau*mu[i])*np.dot(RTR,P[i,:].T)
+                RPT = np.dot(R,P.T)
+                Sigma_diag = Diag + np.sum(RPT.T*RPT.T,-1)
+                mu = self.w + np.dot(P,self.Gamma)
+                self.iterations += 1
+            #Sigma recomptutation with Cholesky decompositon
+            Iplus_Dprod_i = 1./(1.+ Diag0 * self.tau_tilde)
+            Diag = Diag0 * Iplus_Dprod_i
+            P = Iplus_Dprod_i[:,None] * P0
+            safe_diag = np.where(Diag0 < self.tau_tilde, self.tau_tilde/(1.+Diag0*self.tau_tilde), (1. - Iplus_Dprod_i)/Diag0)
+            L = jitchol(np.eye(num_inducing) + np.dot(RPT0,safe_diag[:,None]*RPT0.T))
+            R,info = dtrtrs(L,R0,lower=1)
+            RPT = np.dot(R,P.T)
+            Sigma_diag = Diag + np.sum(RPT.T*RPT.T,-1)
+            self.w = Diag * self.v_tilde
+            self.Gamma = np.dot(R.T, np.dot(RPT,self.v_tilde))
+            mu = self.w + np.dot(P,self.Gamma)
+            epsilon_np1 = sum((self.tau_tilde-self.np1[-1])**2)/self.num_data
+            epsilon_np2 = sum((self.v_tilde-self.np2[-1])**2)/self.num_data
+            self.np1.append(self.tau_tilde.copy())
+            self.np2.append(self.v_tilde.copy())
+
+        return self._compute_GP_variables()

GPy/inference/laplace.py

@@ -0,0 +1,403 @@
+# Copyright (c) 2013, GPy authors (see AUTHORS.txt).
+# Licensed under the BSD 3-clause license (see LICENSE.txt)
+#
+#Parts of this file were influenced by the Matlab GPML framework written by
+#Carl Edward Rasmussen & Hannes Nickisch, however all bugs are our own.
+#
+#The GPML code is released under the FreeBSD License.
+#Copyright (c) 2005-2013 Carl Edward Rasmussen & Hannes Nickisch. All rights reserved.
+#
+#The code and associated documentation is available from
+#http://gaussianprocess.org/gpml/code.
+
+import numpy as np
+import scipy as sp
+from likelihood import likelihood
+from ..util.linalg import mdot, jitchol, pddet, dpotrs
+from functools import partial as partial_func
+import warnings
+
+class Laplace(likelihood):
+    """Laplace approximation to a posterior"""
+
+    def __init__(self, data, noise_model, extra_data=None):
+        """
+        Laplace Approximation
+
+        Find the moments \hat{f} and the hessian at this point
+        (using Newton-Raphson) of the unnormalised posterior
+
+        Compute the GP variables (i.e. generate some Y^{squiggle} and
+        z^{squiggle} which makes a gaussian the same as the laplace
+        approximation to the posterior, but normalised
+
+        Arguments
+        ---------
+
+        :param data: array of data the likelihood function is approximating
+        :type data: NxD
+        :param noise_model: likelihood function - subclass of noise_model
+        :type noise_model: noise_model
+        :param extra_data: additional data used by some likelihood functions,
+        """
+        self.data = data
+        self.noise_model = noise_model
+        self.extra_data = extra_data
+
+        #Inital values
+        self.N, self.D = self.data.shape
+        self.is_heteroscedastic = True
+        self.Nparams = 0
+        self.NORMAL_CONST = ((0.5 * self.N) * np.log(2 * np.pi))
+
+        self.restart()
+        likelihood.__init__(self)
+
+    def restart(self):
+        """
+        Reset likelihood variables to their defaults
+        """
+        #Initial values for the GP variables
+        self.Y = np.zeros((self.N, 1))
+        self.covariance_matrix = np.eye(self.N)
+        self.precision = np.ones(self.N)[:, None]
+        self.Z = 0
+        self.YYT = None
+
+        self.old_Ki_f = None
+        self.bad_fhat = False
+
+    def predictive_values(self,mu,var,full_cov,**noise_args):
+        if full_cov:
+            raise NotImplementedError, "Cannot make correlated predictions with an EP likelihood"
+        return self.noise_model.predictive_values(mu,var,**noise_args)
+
+    def log_predictive_density(self, y_test, mu_star, var_star):
+        """
+        Calculation of the log predictive density
+
+        .. math:
+            p(y_{*}|D) = p(y_{*}|f_{*})p(f_{*}|\mu_{*}\\sigma^{2}_{*})
+
+        :param y_test: test observations (y_{*})
+        :type y_test: (Nx1) array
+        :param mu_star: predictive mean of gaussian p(f_{*}|mu_{*}, var_{*})
+        :type mu_star: (Nx1) array
+        :param var_star: predictive variance of gaussian p(f_{*}|mu_{*}, var_{*})
+        :type var_star: (Nx1) array
+        """
+        return self.noise_model.log_predictive_density(y_test, mu_star, var_star)
+
+    def _get_params(self):
+        return np.asarray(self.noise_model._get_params())
+
+    def _get_param_names(self):
+        return self.noise_model._get_param_names()
+
+    def _set_params(self, p):
+        return self.noise_model._set_params(p)
+
+    def _shared_gradients_components(self):
+        d3lik_d3fhat = self.noise_model.d3logpdf_df3(self.f_hat, self.data, extra_data=self.extra_data)
+        dL_dfhat = 0.5*(np.diag(self.Ki_W_i)[:, None]*d3lik_d3fhat).T #why isn't this -0.5?
+        I_KW_i = np.eye(self.N) - np.dot(self.K, self.Wi_K_i)
+        return dL_dfhat, I_KW_i
+
+    def _Kgradients(self):
+        """
+        Gradients with respect to prior kernel parameters dL_dK to be chained
+        with dK_dthetaK to give dL_dthetaK
+        :returns: dL_dK matrix
+        :rtype: Matrix (1 x num_kernel_params)
+        """
+        dL_dfhat, I_KW_i = self._shared_gradients_components()
+        dlp = self.noise_model.dlogpdf_df(self.f_hat, self.data, extra_data=self.extra_data)
+
+        #Explicit
+        #expl_a = np.dot(self.Ki_f, self.Ki_f.T)
+        #expl_b = self.Wi_K_i
+        #expl = 0.5*expl_a - 0.5*expl_b
+        #dL_dthetaK_exp = dK_dthetaK(expl, X)
+
+        #Implicit
+        impl = mdot(dlp, dL_dfhat, I_KW_i)
+
+        #No longer required as we are computing these in the gp already
+        #otherwise we would take them away and add them back
+        #dL_dthetaK_imp = dK_dthetaK(impl, X)
+        #dL_dthetaK = dL_dthetaK_exp + dL_dthetaK_imp
+        #dL_dK = expl + impl
+
+        #No need to compute explicit as we are computing dZ_dK to account
+        #for the difference between the K gradients of a normal GP,
+        #and the K gradients including the implicit part
+        dL_dK = impl
+        return dL_dK
+
+    def _gradients(self, partial):
+        """
+        Gradients with respect to likelihood parameters (dL_dthetaL)
+
+        :param partial: Not needed by this likelihood
+        :type partial: lambda function
+        :rtype: array of derivatives (1 x num_likelihood_params)
+        """
+        dL_dfhat, I_KW_i = self._shared_gradients_components()
+        dlik_dthetaL, dlik_grad_dthetaL, dlik_hess_dthetaL = self.noise_model._laplace_gradients(self.f_hat, self.data, extra_data=self.extra_data)
+
+        #len(dlik_dthetaL)
+        num_params = len(self._get_param_names())
+        # make space for one derivative for each likelihood parameter
+        dL_dthetaL = np.zeros(num_params)
+        for thetaL_i in range(num_params):
+            #Explicit
+            dL_dthetaL_exp = ( np.sum(dlik_dthetaL[:, thetaL_i])
+                             #- 0.5*np.trace(mdot(self.Ki_W_i, (self.K, np.diagflat(dlik_hess_dthetaL[thetaL_i]))))
+                             + np.dot(0.5*np.diag(self.Ki_W_i)[:,None].T, dlik_hess_dthetaL[:, thetaL_i])
+                             )
+
+            #Implicit
+            dfhat_dthetaL = mdot(I_KW_i, self.K, dlik_grad_dthetaL[:, thetaL_i])
+            dL_dthetaL_imp = np.dot(dL_dfhat, dfhat_dthetaL)
+            dL_dthetaL[thetaL_i] = dL_dthetaL_exp + dL_dthetaL_imp
+
+        return dL_dthetaL
+
+    def _compute_GP_variables(self):
+        """
+        Generate data Y which would give the normal distribution identical
+        to the laplace approximation to the posterior, but normalised
+
+        GPy expects a likelihood to be gaussian, so need to caluclate
+        the data Y^{\tilde} that makes the posterior match that found
+        by a laplace approximation to a non-gaussian likelihood but with
+        a gaussian likelihood
+
+        Firstly,
+        The hessian of the unormalised posterior distribution is (K^{-1} + W)^{-1},
+        i.e. z*N(f|f^{\hat}, (K^{-1} + W)^{-1}) but this assumes a non-gaussian likelihood,
+        we wish to find the hessian \Sigma^{\tilde}
+        that has the same curvature but using our new simulated data Y^{\tilde}
+        i.e. we do N(Y^{\tilde}|f^{\hat}, \Sigma^{\tilde})N(f|0, K) = z*N(f|f^{\hat}, (K^{-1} + W)^{-1})
+        and we wish to find what Y^{\tilde} and \Sigma^{\tilde}
+        We find that Y^{\tilde} = W^{-1}(K^{-1} + W)f^{\hat} and \Sigma^{tilde} = W^{-1}
+
+        Secondly,
+        GPy optimizes the log marginal log p(y) = -0.5*ln|K+\Sigma^{\tilde}| - 0.5*Y^{\tilde}^{T}(K^{-1} + \Sigma^{tilde})^{-1}Y + lik.Z
+        So we can suck up any differences between that and our log marginal likelihood approximation
+        p^{\squiggle}(y) = -0.5*f^{\hat}K^{-1}f^{\hat} + log p(y|f^{\hat}) - 0.5*log |K||K^{-1} + W|
+        which we want to optimize instead, by equating them and rearranging, the difference is added onto
+        the log p(y) that GPy optimizes by default
+
+        Thirdly,
+        Since we have gradients that depend on how we move f^{\hat}, we have implicit components
+        aswell as the explicit dL_dK, we hold these differences in dZ_dK and add them to dL_dK in the
+        gp.py code
+        """
+        Wi = 1.0/self.W
+        self.Sigma_tilde = np.diagflat(Wi)
+
+        Y_tilde = Wi*self.Ki_f + self.f_hat
+
+        self.Wi_K_i = self.W12BiW12
+        ln_det_Wi_K = pddet(self.Sigma_tilde + self.K)
+        lik = self.noise_model.logpdf(self.f_hat, self.data, extra_data=self.extra_data)
+        y_Wi_K_i_y = mdot(Y_tilde.T, self.Wi_K_i, Y_tilde)
+
+        Z_tilde = (+ lik
+                   - 0.5*self.ln_B_det
+                   + 0.5*ln_det_Wi_K
+                   - 0.5*self.f_Ki_f
+                   + 0.5*y_Wi_K_i_y
+                   + self.NORMAL_CONST
+                  )
+
+        #Convert to float as its (1, 1) and Z must be a scalar
+        self.Z = np.float64(Z_tilde)
+        self.Y = Y_tilde
+        self.YYT = np.dot(self.Y, self.Y.T)
+        self.covariance_matrix = self.Sigma_tilde
+        self.precision = 1.0 / np.diag(self.covariance_matrix)[:, None]
+
+        #Compute dZ_dK which is how the approximated distributions gradients differ from the dL_dK computed for other likelihoods
+        self.dZ_dK = self._Kgradients()
+        #+ 0.5*self.Wi_K_i - 0.5*np.dot(self.Ki_f, self.Ki_f.T) #since we are not adding the K gradients explicit part theres no need to compute this again
+
+    def fit_full(self, K):
+        """
+        The laplace approximation algorithm, find K and expand hessian
+        For nomenclature see Rasmussen & Williams 2006 - modified for numerical stability
+
+        :param K: Prior covariance matrix evaluated at locations X
+        :type K: NxN matrix
+        """
+        self.K = K.copy()
+
+        #Find mode
+        self.f_hat = self.rasm_mode(self.K)
+
+        #Compute hessian and other variables at mode
+        self._compute_likelihood_variables()
+
+        #Compute fake variables replicating laplace approximation to posterior
+        self._compute_GP_variables()
+
+    def _compute_likelihood_variables(self):
+        """
+        Compute the variables required to compute gaussian Y variables
+        """
+        #At this point get the hessian matrix (or vector as W is diagonal)
+        self.W = -self.noise_model.d2logpdf_df2(self.f_hat, self.data, extra_data=self.extra_data)
+
+        if not self.noise_model.log_concave:
+            #print "Under 1e-10: {}".format(np.sum(self.W < 1e-6))
+            self.W[self.W < 1e-6] = 1e-6  # FIXME-HACK: This is a hack since GPy can't handle negative variances which can occur
+
+        self.W12BiW12, self.ln_B_det = self._compute_B_statistics(self.K, self.W, np.eye(self.N))
+
+        self.Ki_f = self.Ki_f
+        self.f_Ki_f = np.dot(self.f_hat.T, self.Ki_f)
+        self.Ki_W_i = self.K - mdot(self.K, self.W12BiW12, self.K)
+
+    def _compute_B_statistics(self, K, W, a):
+        """
+        Rasmussen suggests the use of a numerically stable positive definite matrix B
+        Which has a positive diagonal element and can be easyily inverted
+
+        :param K: Prior Covariance matrix evaluated at locations X
+        :type K: NxN matrix
+        :param W: Negative hessian at a point (diagonal matrix)
+        :type W: Vector of diagonal values of hessian (1xN)
+        :param a: Matrix to calculate W12BiW12a
+        :type a: Matrix NxN
+        :returns: (W12BiW12, ln_B_det)
+        """
+        if not self.noise_model.log_concave:
+            #print "Under 1e-10: {}".format(np.sum(W < 1e-6))
+            W[W < 1e-6] = 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
+                                # To cause the posterior to become less certain than the prior and likelihood,
+                                # This is a property only held by non-log-concave likelihoods
+
+
+        #W is diagonal so its sqrt is just the sqrt of the diagonal elements
+        W_12 = np.sqrt(W)
+        B = np.eye(self.N) + W_12*K*W_12.T
+        L = jitchol(B)
+
+        W12BiW12a = W_12*dpotrs(L, np.asfortranarray(W_12*a), lower=1)[0]
+        ln_B_det = 2*np.sum(np.log(np.diag(L)))
+        return W12BiW12a, ln_B_det
+
+    def rasm_mode(self, K, MAX_ITER=40):
+        """
+        Rasmussen's numerically stable mode finding
+        For nomenclature see Rasmussen & Williams 2006
+        Influenced by GPML (BSD) code, all errors are our own
+
+        :param K: Covariance matrix evaluated at locations X
+        :type K: NxD matrix
+        :param MAX_ITER: Maximum number of iterations of newton-raphson before forcing finish of optimisation
+        :type MAX_ITER: scalar
+        :returns: f_hat, mode on which to make laplace approxmiation
+        :rtype: NxD matrix
+        """
+        #old_Ki_f = np.zeros((self.N, 1))
+
+        #Start f's at zero originally of if we have gone off track, try restarting
+        if self.old_Ki_f is None or self.bad_fhat:
+            old_Ki_f = np.random.rand(self.N, 1)/50.0
+            #old_Ki_f = self.Y
+            f = np.dot(K, old_Ki_f)
+        else:
+            #Start at the old best point
+            old_Ki_f = self.old_Ki_f.copy()
+            f = self.f_hat.copy()
+
+        new_obj = -np.inf
+        old_obj = np.inf
+
+        def obj(Ki_f, f):
+            return -0.5*np.dot(Ki_f.T, f) + self.noise_model.logpdf(f, self.data, extra_data=self.extra_data)
+
+        difference = np.inf
+        epsilon = 1e-7
+        #step_size = 1
+        #rs = 0
+        i = 0
+
+        while difference > epsilon and i < MAX_ITER:
+            W = -self.noise_model.d2logpdf_df2(f, self.data, extra_data=self.extra_data)
+
+            W_f = W*f
+            grad = self.noise_model.dlogpdf_df(f, self.data, extra_data=self.extra_data)
+
+            b = W_f + grad
+            W12BiW12Kb, _ = self._compute_B_statistics(K, W.copy(), np.dot(K, b))
+
+            #Work out the DIRECTION that we want to move in, but don't choose the stepsize yet
+            full_step_Ki_f = b - W12BiW12Kb
+            dKi_f = full_step_Ki_f - old_Ki_f
+
+            f_old = f.copy()
+            def inner_obj(step_size, old_Ki_f, dKi_f, K):
+                Ki_f = old_Ki_f + step_size*dKi_f
+                f = np.dot(K, Ki_f)
+                # This is nasty, need to set something within an optimization though
+                self.tmp_Ki_f = Ki_f.copy()
+                self.tmp_f = f.copy()
+                return -obj(Ki_f, f)
+
+            i_o = partial_func(inner_obj, old_Ki_f=old_Ki_f, dKi_f=dKi_f, K=K)
+            #Find the stepsize that minimizes the objective function using a brent line search
+            #The tolerance and maxiter matter for speed! Seems to be best to keep them low and make more full
+            #steps than get this exact then make a step, if B was bigger it might be the other way around though
+            #new_obj = sp.optimize.minimize_scalar(i_o, method='brent', tol=1e-4, options={'maxiter':5}).fun
+            new_obj = sp.optimize.brent(i_o, tol=1e-4, maxiter=10)
+            f = self.tmp_f.copy()
+            Ki_f = self.tmp_Ki_f.copy()
+
+            #Optimize without linesearch
+            #f_old = f.copy()
+            #update_passed = False
+            #while not update_passed:
+                #Ki_f = old_Ki_f + step_size*dKi_f
+                #f = np.dot(K, Ki_f)
+
+                #old_obj = new_obj
+                #new_obj = obj(Ki_f, f)
+                #difference = new_obj - old_obj
+                ##print "difference: ",difference
+                #if difference < 0:
+                    ##print "Objective function rose", np.float(difference)
+                    ##If the objective function isn't rising, restart optimization
+                    #step_size *= 0.8
+                    ##print "Reducing step-size to {ss:.3} and restarting optimization".format(ss=step_size)
+                    ##objective function isn't increasing, try reducing step size
+                    #f = f_old.copy() #it's actually faster not to go back to old location and just zigzag across the mode
+                    #old_obj = new_obj
+                    #rs += 1
+                #else:
+                    #update_passed = True
+
+            #old_Ki_f = self.Ki_f.copy()
+
+            #difference = abs(new_obj - old_obj)
+            #old_obj = new_obj.copy()
+            difference = np.abs(np.sum(f - f_old)) + np.abs(np.sum(Ki_f - old_Ki_f))
+            #difference = np.abs(np.sum(Ki_f - old_Ki_f))/np.float(self.N)
+            old_Ki_f = Ki_f.copy()
+            i += 1
+
+        self.old_Ki_f = old_Ki_f.copy()
+
+        #Warn of bad fits
+        if difference > epsilon:
+            self.bad_fhat = True
+            warnings.warn("Not perfect f_hat fit difference: {}".format(difference))
+        elif self.bad_fhat:
+            self.bad_fhat = False
+            warnings.warn("f_hat now perfect again")
+
+        self.Ki_f = Ki_f
+        return f