fixing EP and merging it with GP_regression

GPy/examples/ep_fix.py

@@ -25,14 +25,15 @@ seed=default_seed
 data = GPy.util.datasets.toy_linear_1d_classification(seed=seed)
 likelihood = GPy.inference.likelihoods.probit(data['Y'][:, 0:1])
 
-m = GPy.models.GP_EP2(data['X'],likelihood)
+m = GPy.models.GP(data['X'],likelihood=likelihood)
 
-#m.constrain_positive('var')
-#m.constrain_positive('len')
-#m.tie_param('lengthscale')
+m.constrain_positive('var')
+m.constrain_positive('len')
+m.tie_param('lengthscale')
 m.approximate_likelihood()
+print m.checkgrad()
 # Optimize and plot
-#m.optimize()
+m.optimize()
 #m.em(plot_all=False) # EM algorithm
 m.plot()
 

GPy/inference/EP.py

@@ -60,7 +60,7 @@ class Full(EP):
     def fit_EP(self):
         """
         The expectation-propagation algorithm.
-        For nomenclature see Rasmussen & Williams 2006 (pag. 52-60)
+        For nomenclature see Rasmussen & Williams 2006.
         """
         #Prior distribution parameters: p(f|X) = N(f|0,K)
         #self.K = self.kernel.K(self.X,self.X)
@@ -84,8 +84,6 @@ class Full(EP):
         phi = np.empty(self.N,dtype=float)
         mu_hat = np.empty(self.N,dtype=float)
         sigma2_hat = np.empty(self.N,dtype=float)
-        self.mu_hat = mu_hat #TODO erase me
-        self.sigma2_hat = sigma2_hat #TODO erase me
 
         #Approximation
         epsilon_np1 = self.epsilon + 1.
@@ -95,21 +93,16 @@ class Full(EP):
         self.np2 = [self.v_tilde.copy()]
         while epsilon_np1 > self.epsilon or epsilon_np2 > self.epsilon:
             update_order = np.arange(self.N)
-            #random.shuffle(update_order) #TODO uncomment
+            random.shuffle(update_order)
             for i in update_order:
                 #Cavity distribution parameters
                 self.tau_[i] = 1./self.Sigma[i,i] - self.eta*self.tau_tilde[i]
                 self.v_[i] = self.mu[i]/self.Sigma[i,i] - self.eta*self.v_tilde[i]
                 #Marginal moments
                 self.Z_hat[i], mu_hat[i], sigma2_hat[i] = self.likelihood.moments_match(i,self.tau_[i],self.v_[i])
-                self.mu_hat[i] = mu_hat[i] #TODO erase me
-                self.sigma2_hat[i] = sigma2_hat[i] #TODO erase me
-                #if i == 3:
-                #    a = b
                 #Site parameters update
                 Delta_tau = self.delta/self.eta*(1./sigma2_hat[i] - 1./self.Sigma[i,i])
                 Delta_v = self.delta/self.eta*(mu_hat[i]/sigma2_hat[i] - self.mu[i]/self.Sigma[i,i])
-                print Delta_tau
                 self.tau_tilde[i] = self.tau_tilde[i] + Delta_tau
                 self.v_tilde[i] = self.v_tilde[i] + Delta_v
                 #Posterior distribution parameters update
@@ -128,6 +121,7 @@ class Full(EP):
             epsilon_np2 = sum((self.v_tilde-self.np2[-1])**2)/self.N
             self.np1.append(self.tau_tilde.copy())
             self.np2.append(self.v_tilde.copy())
+        return self.tau_tilde[:,None], self.v_tilde[:,None], self.Z_hat[:,None], self.tau_[:,None], self.v_[:,None]
 
 class DTC(EP):
     def fit_EP(self):

GPy/inference/likelihoods.py

@@ -19,7 +19,7 @@ class likelihood:
         self.Y = Y
         self.N = self.Y.shape[0]
 
-    def plot1Da(self,X_new,Mean_new,Var_new,X_u,Mean_u,Var_u):
+    def plot1Da(self,X,mean,var,Z=None,mean_Z=None,var_Z=None):
         """
         Plot the predictive distribution of the GP model for 1-dimensional inputs
 
@@ -30,10 +30,18 @@ class likelihood:
         :param Mean_u: mean values at X_u
         :param Var_new: variance values at X_u
         """
-        assert X_new.shape[1] == 1, 'Number of dimensions must be 1'
-        gpplot(X_new,Mean_new,Var_new)
-        pb.errorbar(X_u.flatten(),Mean_u.flatten(),2*np.sqrt(Var_u.flatten()),fmt='r+')
-        pb.plot(X_u,Mean_u,'ro')
+        assert X.shape[1] == 1, 'Number of dimensions must be 1'
+        gpplot(X,mean,var.flatten())
+        pb.errorbar(Z.flatten(),mean_Z.flatten(),2*np.sqrt(var_Z.flatten()),fmt='r+')
+        pb.plot(Z,mean_Z,'ro')
+
+    def plot1Db(self,X_obs,X,phi,Z=None):
+        assert X_obs.shape[1] == 1, 'Number of dimensions must be 1'
+        gpplot(X,phi,np.zeros(X.shape[0]))
+        pb.plot(X_obs,(self.Y+1)/2,'kx',mew=1.5)
+        pb.ylim(-0.2,1.2)
+        if Z is not None:
+            pb.plot(Z,Z*0+.5,'r|',mew=1.5,markersize=12)
 
     def plot2D(self,X,X_new,F_new,U=None):
         """
@@ -88,16 +96,11 @@ class probit(likelihood):
         sigma2_hat = 1./tau_i - (phi/((tau_i**2+tau_i)*Z_hat))*(z+phi/Z_hat)
         return Z_hat, mu_hat, sigma2_hat
 
-    def plot1Db(self,X,X_new,F_new,U=None):
-        assert X.shape[1] == 1, 'Number of dimensions must be 1'
-        gpplot(X_new,F_new,np.zeros(X_new.shape[0]))
-        pb.plot(X,(self.Y+1)/2,'kx',mew=1.5)
-        pb.ylim(-0.2,1.2)
-        if U is not None:
-            pb.plot(U,U*0+.5,'r|',mew=1.5,markersize=12)
 
-    def predictive_mean(self,mu,variance):
-        return stats.norm.cdf(mu/np.sqrt(1+variance))
+    def predictive_mean(self,mu,var):
+        mu = mu.flatten()
+        var = var.flatten()
+        return stats.norm.cdf(mu/np.sqrt(1+var))
 
     def _log_likelihood_gradients():
         raise NotImplementedError

GPy/models/GP.py

@@ -0,0 +1,312 @@
+# Copyright (c) 2012, GPy authors (see AUTHORS.txt).
+# Licensed under the BSD 3-clause license (see LICENSE.txt)
+
+
+import numpy as np
+import pylab as pb
+from .. import kern
+from ..core import model
+from ..util.linalg import pdinv,mdot
+from ..util.plot import gpplot, Tango
+from ..inference.EP import Full
+from ..inference.likelihoods import likelihood,probit,poisson,gaussian
+
+class GP(model):
+    """
+    Gaussian Process model for regression
+
+    :param X: input observations
+    :param Y: observed values
+    :param kernel: a GPy kernel, defaults to rbf+white
+    :param normalize_X:  whether to normalize the input data before computing (predictions will be in original scales)
+    :type normalize_X: False|True
+    :param normalize_Y:  whether to normalize the input data before computing (predictions will be in original scales)
+    :type normalize_Y: False|True
+    :param Xslices: how the X,Y data co-vary in the kernel (i.e. which "outputs" they correspond to). See (link:slicing)
+    :rtype: model object
+
+    .. Note:: Multiple independent outputs are allowed using columns of Y
+
+    """
+
+    def __init__(self,X,Y=None,kernel=None,normalize_X=False,normalize_Y=False, Xslices=None,likelihood=None,epsilon_ep=1e-3,epsion_em=.1,powerep=[1.,1.]):
+        #TODO: specify beta parameter explicitely
+
+        # parse arguments
+        self.Xslices = Xslices
+        self.X = X
+        self.N, self.Q = self.X.shape
+        assert len(self.X.shape)==2
+        if kernel is None:
+            kernel = kern.rbf(X.shape[1]) + kern.bias(X.shape[1]) + kern.white(X.shape[1])
+        else:
+            assert isinstance(kernel, kern.kern)
+        self.kern = kernel
+
+        #here's some simple normalisation
+        if normalize_X:
+            self._Xmean = X.mean(0)[None,:]
+            self._Xstd = X.std(0)[None,:]
+            self.X = (X.copy() - self._Xmean) / self._Xstd
+            if hasattr(self,'Z'):
+                self.Z = (self.Z - self._Xmean) / self._Xstd
+        else:
+            self._Xmean = np.zeros((1,self.X.shape[1]))
+            self._Xstd = np.ones((1,self.X.shape[1]))
+
+
+        # Y - likelihood related variables, these might change whether using EP or not
+        if likelihood is None:
+            assert Y is not None, "Either Y or likelihood must be defined"
+            self.likelihood = gaussian(Y)
+        else:
+            self.likelihood = likelihood
+        assert len(self.likelihood.Y.shape)==2
+        assert self.X.shape[0] == self.likelihood.Y.shape[0]
+        self.N, self.D = self.likelihood.Y.shape
+
+        if isinstance(self.likelihood,gaussian):
+            self.EP = False
+            self.Y = Y
+
+            #here's some simple normalisation
+            if normalize_Y:
+                self._Ymean = Y.mean(0)[None,:]
+                self._Ystd = Y.std(0)[None,:]
+                self.Y = (Y.copy()- self._Ymean) / self._Ystd
+            else:
+                self._Ymean = np.zeros((1,self.Y.shape[1]))
+                self._Ystd = np.ones((1,self.Y.shape[1]))
+
+            if self.D > self.N:
+                # then it's more efficient to store YYT
+                self.YYT = np.dot(self.Y, self.Y.T)
+            else:
+                self.YYT = None
+
+        else:
+            # Y is defined after approximating the likelihood
+            self.EP = True
+            self.eta,self.delta = powerep
+            self.epsilon_ep = epsilon_ep
+            self.tau_tilde = np.ones([self.N,self.D])
+            self.v_tilde = np.zeros([self.N,self.D])
+            self.tau_ = np.ones([self.N,self.D])
+            self.v_ = np.zeros([self.N,self.D])
+            self.Z_hat = np.ones([self.N,self.D])
+
+        model.__init__(self)
+
+    def _set_params(self,p):
+        # TODO: remove beta when using EP
+        self.kern._set_params_transformed(p)
+        if not self.EP:
+            self.K = self.kern.K(self.X,slices1=self.Xslices)
+            self.Ki, self.L, self.Li, self.K_logdet = pdinv(self.K)
+        else:
+            self._ep_covariance()
+
+    def _get_params(self):
+        # TODO: remove beta when using EP
+        return self.kern._get_params_transformed()
+
+    def _get_param_names(self):
+        # TODO: remove beta when using EP
+        return self.kern._get_param_names_transformed()
+
+    def approximate_likelihood(self):
+        assert not isinstance(self.likelihood, gaussian), "EP is only available for non-gaussian likelihoods"
+        self.ep_approx = Full(self.K,self.likelihood,epsilon=self.epsilon_ep,powerep=[self.eta,self.delta])
+        self.tau_tilde, self.v_tilde, self.Z_hat, self.tau_, self.v_=self.ep_approx.fit_EP()
+        # Y: EP likelihood is defined as a regression model for mu_tilde
+        self.Y = self.v_tilde/self.tau_tilde
+        self._Ymean = np.zeros((1,self.Y.shape[1]))
+        self._Ystd = np.ones((1,self.Y.shape[1]))
+        if self.D > self.N:
+            # then it's more efficient to store YYT
+            self.YYT = np.dot(self.Y, self.Y.T)
+        else:
+            self.YYT = None
+        self.mu_ = self.v_/self.tau_
+        self._ep_covariance()
+
+    def _ep_covariance(self):
+        # Kernel plus noise variance term
+        self.K = self.kern.K(self.X,slices1=self.Xslices) + np.diag(1./self.tau_tilde.flatten())
+        self.Ki, self.L, self.Li, self.K_logdet = pdinv(self.K)
+
+    def _model_fit_term(self):
+        """
+        Computes the model fit using YYT if it's available
+        """
+        if self.YYT is None:
+            return -0.5*np.sum(np.square(np.dot(self.Li,self.Y)))
+        else:
+            return -0.5*np.sum(np.multiply(self.Ki, self.YYT))
+
+    def _normalization_term(self):
+        """
+        Computes the marginal likelihood normalization constants
+        """
+        sigma_sum = 1./self.tau_ + 1./self.tau_tilde
+        mu_diff_2 = (self.mu_ - self.Y)**2
+        penalty_term = np.sum(np.log(self.Z_hat))
+        return penalty_term + 0.5*np.sum(np.log(sigma_sum)) + 0.5*np.sum(mu_diff_2/sigma_sum)
+
+    def log_likelihood(self):
+        """
+        The log marginal likelihood for an EP model can be written as the log likelihood of
+        a regression model for a new variable Y* = v_tilde/tau_tilde, with a covariance
+        matrix K* = K + diag(1./tau_tilde) plus a normalization term.
+        """
+        complexity_term = -0.5*self.D*self.Kplus_logdet
+        normalization_term = 0 if self.EP == False else self.normalization_term()
+        return complexity_term + normalization_term + self._model_fit_term()
+
+
+    def log_likelihood(self):
+        complexity_term = -0.5*self.N*self.D*np.log(2.*np.pi) - 0.5*self.D*self.K_logdet
+        return complexity_term + self._model_fit_term()
+
+    def dL_dK(self):
+        if self.YYT is None:
+            alpha = np.dot(self.Ki,self.Y)
+            dL_dK = 0.5*(np.dot(alpha,alpha.T)-self.D*self.Ki)
+        else:
+            dL_dK = 0.5*(mdot(self.Ki, self.YYT, self.Ki) - self.D*self.Ki)
+
+        return dL_dK
+
+    def _log_likelihood_gradients(self):
+        return self.kern.dK_dtheta(partial=self.dL_dK(),X=self.X)
+
+    def predict(self,Xnew, slices=None, full_cov=False):
+        """
+
+        Predict the function(s) at the new point(s) Xnew.
+
+        Arguments
+        ---------
+        :param Xnew: The points at which to make a prediction
+        :type Xnew: np.ndarray, Nnew x self.Q
+        :param slices:  specifies which outputs kernel(s) the Xnew correspond to (see below)
+        :type slices: (None, list of slice objects, list of ints)
+        :param full_cov: whether to return the folll covariance matrix, or just the diagonal
+        :type full_cov: bool
+        :rtype: posterior mean,  a Numpy array, Nnew x self.D
+        :rtype: posterior variance, a Numpy array, Nnew x Nnew x (self.D)
+
+        .. Note:: "slices" specifies how the the points X_new co-vary wich the training points.
+
+             - If None, the new points covary throigh every kernel part (default)
+             - If a list of slices, the i^th slice specifies which data are affected by the i^th kernel part
+             - If a list of booleans, specifying which kernel parts are active
+
+           If full_cov and self.D > 1, the return shape of var is Nnew x Nnew x self.D. If self.D == 1, the return shape is Nnew x Nnew.
+           This is to allow for different normalisations of the output dimensions.
+
+
+        """
+
+        #normalise X values
+        Xnew = (Xnew.copy() - self._Xmean) / self._Xstd
+        mu, var, phi = self._raw_predict(Xnew, slices, full_cov)
+
+        #un-normalise
+        mu = mu*self._Ystd + self._Ymean
+        if full_cov:
+            if self.D==1:
+                var *= np.square(self._Ystd)
+            else:
+                var = var[:,:,None] * np.square(self._Ystd)
+        else:
+            if self.D==1:
+                var *= np.square(np.squeeze(self._Ystd))
+            else:
+                var = var[:,None] * np.square(self._Ystd)
+
+        return mu,var,phi
+
+    def _raw_predict(self,_Xnew,slices, full_cov=False):
+        """Internal helper function for making predictions, does not account for normalisation"""
+        Kx = self.kern.K(self.X,_Xnew, slices1=self.Xslices,slices2=slices)
+        mu = np.dot(np.dot(Kx.T,self.Ki),self.Y)
+        KiKx = np.dot(self.Ki,Kx)
+        if full_cov:
+            Kxx = self.kern.K(_Xnew, slices1=slices,slices2=slices)
+            var = Kxx - np.dot(KiKx.T,Kx)
+        else:
+            Kxx = self.kern.Kdiag(_Xnew, slices=slices)
+            var = Kxx - np.sum(np.multiply(KiKx,Kx),0)
+        phi = None if not self.EP else self.likelihood.predictive_mean(mu,var)
+        return mu, var, phi
+
+    def plot(self,samples=0,plot_limits=None,which_data='all',which_functions='all',resolution=None):
+        """
+        :param samples: the number of a posteriori samples to plot
+        :param which_data: which if the training data to plot (default all)
+        :type which_data: 'all' or a slice object to slice self.X, self.Y
+        :param plot_limits: The limits of the plot. If 1D [xmin,xmax], if 2D [[xmin,ymin],[xmax,ymax]]. Defaluts to data limits
+        :param which_functions: which of the kernel functions to plot (additively)
+        :type which_functions: list of bools
+        :param resolution: the number of intervals to sample the GP on. Defaults to 200 in 1D and 50 (a 50x50 grid) in 2D
+
+        Plot the posterior of the GP.
+          - In one dimension, the function is plotted with a shaded region identifying two standard deviations.
+          - In two dimsensions, a contour-plot shows the mean predicted function
+          - In higher dimensions, we've no implemented this yet !TODO!
+
+        Can plot only part of the data and part of the posterior functions using which_data and which_functions
+        """
+        if which_functions=='all':
+            which_functions = [True]*self.kern.Nparts
+        if which_data=='all':
+            which_data = slice(None)
+
+        X = self.X[which_data,:]
+        Y = self.Y[which_data,:]
+
+        Xorig = X*self._Xstd + self._Xmean
+        Yorig = Y*self._Ystd + self._Ymean if not self.EP else self.likelihood.Y
+
+        if plot_limits is None:
+            xmin,xmax = Xorig.min(0),Xorig.max(0)
+            xmin, xmax = xmin-0.2*(xmax-xmin), xmax+0.2*(xmax-xmin)
+        elif len(plot_limits)==2:
+            xmin, xmax = plot_limits
+        else:
+            raise ValueError, "Bad limits for plotting"
+
+        if self.X.shape[1]==1:
+            Xnew = np.linspace(xmin,xmax,resolution or 200)[:,None]
+            m,v,phi = self.predict(Xnew,slices=which_functions)
+            if self.EP:
+                pb.subplot(211)
+
+            gpplot(Xnew,m,v)
+            if samples:
+                s = np.random.multivariate_normal(m.flatten(),v,samples)
+                pb.plot(Xnew.flatten(),s.T, alpha = 0.4, c='#3465a4', linewidth = 0.8)
+
+            if not self.EP:
+                pb.plot(Xorig,Yorig,'kx',mew=1.5)
+                pb.xlim(xmin,xmax)
+            else:
+                pb.xlim(xmin,xmax)
+                pb.subplot(212)
+                self.likelihood.plot1Db(self.X,Xnew,phi)
+                pb.xlim(xmin,xmax)
+
+        elif self.X.shape[1]==2:
+            resolution = 50 or resolution
+            xx,yy = np.mgrid[xmin[0]:xmax[0]:1j*resolution,xmin[1]:xmax[1]:1j*resolution]
+            Xtest = np.vstack((xx.flatten(),yy.flatten())).T
+            zz,vv,phi = self.predict(Xtest,slices=which_functions)
+            zz = zz.reshape(resolution,resolution)
+            pb.contour(xx,yy,zz,vmin=zz.min(),vmax=zz.max(),cmap=pb.cm.jet)
+            pb.scatter(Xorig[:,0],Xorig[:,1],40,Yorig,linewidth=0,cmap=pb.cm.jet,vmin=zz.min(),vmax=zz.max())
+            pb.xlim(xmin[0],xmax[0])
+            pb.ylim(xmin[1],xmax[1])
+
+        else:
+            raise NotImplementedError, "Cannot plot GPs with more than two input dimensions"

GPy/models/GP_EP.py

@@ -62,7 +62,7 @@ class GP_EP(model):
         self.L = jitchol(B)
         V,info = linalg.flapack.dtrtrs(self.L,self.Sroot_tilde_K,lower=1)
         self.Sigma = self.K - np.dot(V.T,V)
-        self.mu = np.dot(self.Sigma,self.ep_approx.v_tilde)
+        self.mu = np.dot(self.Sigma,self.ep_approx.v_tilde) * self.Z_hat
 
     def log_likelihood(self):
         """

GPy/models/GP_EP2.py

@@ -36,14 +36,11 @@ class GP_EP2(model):
         self.Xslices = Xslices
         assert isinstance(kernel, kern.kern)
         self.likelihood = likelihood
-        #self.Y = self.likelihood.Y #we might not need this
         self.kern = kernel
         self.X = X
         assert len(self.X.shape)==2
-        #assert len(self.Y.shape)==2
-        #assert self.X.shape[0] == self.Y.shape[0]
-        #self.N, self.D = self.Y.shape
-        self.D = 1
+        assert self.X.shape[0] == self.likelihood.Y.shape[0]
+        self.D = self.likelihood.Y.shape[1]
         self.N, self.Q = self.X.shape
 
         #here's some simple normalisation
@@ -75,14 +72,17 @@ class GP_EP2(model):
         """
         self.eta,self.delta = powerep
         self.epsilon_ep = epsilon_ep
-        self.tau_tilde = np.zeros([self.N,self.D])
+        self.tau_tilde = np.ones([self.N,self.D])
         self.v_tilde = np.zeros([self.N,self.D])
+        self.tau_ = np.ones([self.N,self.D])
+        self.v_ = np.zeros([self.N,self.D])
+        self.Z_hat = np.ones([self.N,self.D])
         model.__init__(self)
 
     def _set_params(self,p):
         self.kern._set_params_transformed(p)
         self.K = self.kern.K(self.X,slices1=self.Xslices)
-        self.posterior_params()
+        self._ep_params()
 
     def _get_params(self):
         return self.kern._get_params_transformed()
@@ -92,52 +92,63 @@ class GP_EP2(model):
 
     def approximate_likelihood(self):
         self.ep_approx = Full(self.K,self.likelihood,epsilon=self.epsilon_ep,powerep=[self.eta,self.delta])
-        self.ep_approx.fit_EP()
-        self.tau_tilde = self.ep_approx.tau_tilde[:,None]
-        self.v_tilde = self.ep_approx.tau_tilde[:,None]
-        self.posterior_params()
-        self.Y = self.v_tilde/self.tau_tilde
-        self._Ymean = np.zeros((1,self.Y.shape[1]))
-        self._Ystd = np.ones((1,self.Y.shape[1]))
-        #self.YYT = np.dot(self.Y, self.Y.T)
+        self.tau_tilde, self.v_tilde, self.Z_hat, self.tau_, self.v_=self.ep_approx.fit_EP()
+        self._ep_params()
 
-    def posterior_params(self):
-        self.Sroot_tilde_K =  np.sqrt(self.tau_tilde.flatten())[:,None]*self.K
+    def _ep_params(self):
+        # Posterior mean and Variance computation
+        self.Sroot_tilde_K =  np.sqrt(self.tau_tilde)*self.K
         B = np.eye(self.N) + np.sqrt(self.tau_tilde.flatten())[None,:]*self.Sroot_tilde_K
         self.Bi,self.L,self.Li,B_logdet = pdinv(B)
         V = np.dot(self.Li,self.Sroot_tilde_K)
-        #V,info = linalg.flapack.dtrtrs(self.L,self.Sroot_tilde_K,lower=1)
-        self.Sigma = self.K - np.dot(V.T,V)
-        self.mu = np.dot(self.Sigma,self.v_tilde.flatten())
+        self.Sigma = self.K - np.dot(V.T,V) #posterior variance
+        self.mu = np.dot(self.Sigma,self.v_tilde) #posterior mean
+        # Kernel plus noise variance term
+        self.Kplus = self.K + np.diag(1./self.tau_tilde.flatten())
+        self.Kplusi,self.Lplus,self.Lplusi,self.Kplus_logdet = pdinv(self.Kplus)
+        # Y: EP likelihood is defined as a regression model for mu_tilde
+        self.Y = self.v_tilde/self.tau_tilde
+        self._Ymean = np.zeros((1,self.Y.shape[1]))
+        self._Ystd = np.ones((1,self.Y.shape[1]))
+        self.YYT = None #np.dot(self.Y, self.Y.T)
+        self.mu_ = self.v_/self.tau_
 
+    def _model_fit_term(self):
+        """
+        Computes the model fit using YYT if it's available
+        """
+        if self.YYT is None:
+            return -0.5*np.sum(np.square(np.dot(self.Lplusi,self.Y)))
+        else:
+            return -0.5*np.sum(np.multiply(self.Kplusi, self.YYT))
 
-    #def _model_fit_term(self):
-    #    """
-    #    Computes the model fit using YYT if it's available
-    #    """
-    #    if self.YYT is None:
-    #        return -0.5*np.sum(np.square(np.dot(self.Li,self.Y)))
-    #    else:
-    #        return -0.5*np.sum(np.multiply(self.Ki, self.YYT))
+    def _normalization_term(self):
+        """
+        Computes the marginal likelihood normalization constants
+        """
+        sigma_sum = 1./self.tau_ + 1./self.tau_tilde
+        mu_diff_2 = (self.mu_ - self.Y)**2
+        penalty_term = np.sum(np.log(self.Z_hat))
+        return penalty_term + 0.5*np.sum(np.log(sigma_sum)) + 0.5*np.sum(mu_diff_2/sigma_sum)
 
     def log_likelihood(self):
-        mu_ = self.ep_approx.v_/self.ep_approx.tau_
-        L1 =.5*sum(np.log(1+self.ep_approx.tau_tilde*1./self.ep_approx.tau_))-sum(np.log(np.diag(self.L)))
-        L2A =.5*np.sum((self.Sigma-np.diag(1./(self.ep_approx.tau_+self.ep_approx.tau_tilde))) * np.dot(self.ep_approx.v_tilde[:,None],self.ep_approx.v_tilde[None,:]))
-        L2B = .5*np.dot(mu_*(self.ep_approx.tau_/(self.ep_approx.tau_tilde+self.ep_approx.tau_)),self.ep_approx.tau_tilde*mu_ - 2*self.ep_approx.v_tilde)
-        L3 = sum(np.log(self.ep_approx.Z_hat))
-        return L1 + L2A + L2B + L3
+        """
+        The log marginal likelihood for an EP model can be written as the log likelihood of
+        a regression model for a new variable Y* = v_tilde/tau_tilde, with a covariance
+        matrix K* = K + diag(1./tau_tilde) plus a normalization term.
+        """
+        complexity_term = -0.5*self.D*self.Kplus_logdet
+        return complexity_term + self._model_fit_term() + self._normalization_term()
 
-    def dL_dK(self): #FIXME
+    def dL_dK(self):
         if self.YYT is None:
-            alpha = np.dot(self.Ki,self.Y)
-            dL_dK = 0.5*(np.dot(alpha,alpha.T)-self.D*self.Ki)
+            alpha = np.dot(self.Kplusi,self.Y)
+            dL_dK = 0.5*(np.dot(alpha,alpha.T)-self.D*self.Kplusi)
         else:
-            dL_dK = 0.5*(mdot(self.Ki, self.YYT, self.Ki) - self.D*self.Ki)
-
+            dL_dK = 0.5*(mdot(self.Kplusi, self.YYT, self.Kplusi) - self.D*self.Kplusi)
         return dL_dK
 
-    def _log_likelihood_gradients(self): #FIXME
+    def _log_likelihood_gradients(self):
         return self.kern.dK_dtheta(partial=self.dL_dK(),X=self.X)
 
     def predict(self,Xnew, slices=None, full_cov=False):
@@ -189,32 +200,20 @@ class GP_EP2(model):
 
     def _raw_predict(self,_Xnew,slices, full_cov=False):
         """Internal helper function for making predictions, does not account for normalisation"""
-        """
-        Kx = self.kern.K(self.X,_Xnew, slices1=self.Xslices,slices2=slices)
-        mu = np.dot(np.dot(Kx.T,self.Ki),self.Y)
-        KiKx = np.dot(self.Ki,Kx)
+        K_x = self.kern.K(self.X,_Xnew,slices1=self.Xslices,slices2=slices)
+        aux2 = mdot(self.Bi,self.Sroot_tilde_K,self.v_tilde)
+        zeta = np.sqrt(self.tau_tilde)*aux2
+        f = np.dot(K_x.T,self.v_tilde-zeta)
+        v = mdot(self.Li,np.sqrt(self.tau_tilde)*K_x)
         if full_cov:
-            Kxx = self.kern.K(_Xnew, slices1=slices,slices2=slices)
-            var = Kxx - np.dot(KiKx.T,Kx)
+            Kxx = self.kern.K(_Xnew,slices1=slices,slices2=slices)
+            var = Kxx - np.dot(v.T,v)
+            var_diag = np.diag(var)[:,None]
         else:
             Kxx = self.kern.Kdiag(_Xnew, slices=slices)
-            var = Kxx - np.sum(np.multiply(KiKx,Kx),0)
-        return mu, var
-        """
-        K_x = self.kern.K(self.X,_Xnew)
-        Kxx = self.kern.K(_Xnew)
-        #aux1,info = linalg.flapack.dtrtrs(self.L,np.dot(self.Sroot_tilde_K,self.ep_approx.v_tilde),lower=1)
-        #aux2,info = linalg.flapack.dtrtrs(self.L.T, aux1,lower=0)
-        #aux2 = mdot(self.Li.T,self.Li,self.Sroot_tilde_K,self.ep_approx.v_tilde)
-        aux2 = mdot(self.Bi,self.Sroot_tilde_K,self.ep_approx.v_tilde)
-        zeta = np.sqrt(self.ep_approx.tau_tilde)*aux2
-        f = np.dot(K_x.T,self.ep_approx.v_tilde-zeta)
-       	#v,info = linalg.flapack.dtrtrs(self.L,np.sqrt(self.ep_approx.tau_tilde)[:,None]*K_x,lower=1)
-        v = mdot(self.Li,np.sqrt(self.ep_approx.tau_tilde)[:,None]*K_x)
-        variance = Kxx - np.dot(v.T,v)
-       	vdiag = np.diag(variance)
-        y=self.likelihood.predictive_mean(f,vdiag)
-        return f,vdiag,y
+            var_diag = (Kxx - np.sum(v**2,-2))[:,None]
+        phi = self.likelihood.predictive_mean(f,var_diag)
+        return f, var_diag, phi
 
     def plot(self,samples=0,plot_limits=None,which_data='all',which_functions='all',resolution=None):
         """
@@ -257,7 +256,7 @@ class GP_EP2(model):
             #gpplot(Xnew,m,v)
             mu_f, var_f, phi_f = self.predict(Xnew,slices=which_functions)
             pb.subplot(211)
-            self.likelihood.plot1Da(X_new=Xnew,Mean_new=mu_f,Var_new=var_f,X_u=self.X,Mean_u=self.mu,Var_u=np.diag(self.Sigma))
+            self.likelihood.plot1Da(X=Xnew,mean=mu_f,var=var_f,Z=self.X,mean_Z=self.mu,var_Z=np.diag(self.Sigma))
             if samples:
                 s = np.random.multivariate_normal(m.flatten(),v,samples)
                 pb.plot(Xnew.flatten(),s.T, alpha = 0.4, c='#3465a4', linewidth = 0.8)

GPy/models/__init__.py

@@ -11,3 +11,4 @@ from GP_EP2 import GP_EP2
 from generalized_FITC import generalized_FITC
 from sparse_GPLVM import sparse_GPLVM
 from uncollapsed_sparse_GP import uncollapsed_sparse_GP
+from GP import GP