Merge remote-tracking branch 'Falkor/newGP' into newGP

GPy/examples/classification.py

@@ -76,11 +76,10 @@ def toy_linear_1d_classification(model_type='Full', inducing=4, seed=default_see
 
     # create simple GP model
     if model_type=='Full':
-        m = GPy.models.simple_GP_EP(data['X'],likelihood)
+        m = GPy.models.GP_EP(data['X'],likelihood)
     else:
         # create sparse GP EP model
         m = GPy.models.sparse_GP_EP(data['X'],likelihood=likelihood,inducing=inducing,ep_proxy=model_type)
-            
 
     m.constrain_positive('var')
     m.constrain_positive('len')

GPy/examples/ep_fix.py

@@ -0,0 +1,43 @@
+# Copyright (c) 2012, GPy authors (see AUTHORS.txt).
+# Licensed under the BSD 3-clause license (see LICENSE.txt)
+
+
+"""
+Simple Gaussian Processes classification
+"""
+import pylab as pb
+import numpy as np
+import GPy
+pb.ion()
+
+pb.close('all')
+default_seed=10000
+
+model_type='Full'
+inducing=4
+seed=default_seed
+"""Simple 1D classification example.
+:param model_type: type of model to fit ['Full', 'FITC', 'DTC'].
+:param seed : seed value for data generation (default is 4).
+:type seed: int
+:param inducing : number of inducing variables (only used for 'FITC' or 'DTC').
+:type inducing: int
+"""
+data = GPy.util.datasets.toy_linear_1d_classification(seed=seed)
+likelihood = GPy.inference.likelihoods.probit(data['Y'][:, 0:1])
+
+m = GPy.models.GP(data['X'],likelihood=likelihood)
+#m = GPy.models.GP(data['X'],Y=likelihood.Y)
+
+m.constrain_positive('var')
+m.constrain_positive('len')
+m.tie_param('lengthscale')
+if not isinstance(m.likelihood,GPy.inference.likelihoods.gaussian):
+    m.approximate_likelihood()
+print m.checkgrad()
+# Optimize and plot
+m.optimize()
+#m.em(plot_all=False) # EM algorithm
+m.plot()
+
+print(m)

GPy/examples/poisson.py

@@ -0,0 +1,50 @@
+# Copyright (c) 2012, GPy authors (see AUTHORS.txt).
+# Licensed under the BSD 3-clause license (see LICENSE.txt)
+
+
+"""
+Simple Gaussian Processes classification
+"""
+import pylab as pb
+import numpy as np
+import GPy
+pb.ion()
+
+pb.close('all')
+default_seed=10000
+
+model_type='Full'
+inducing=4
+seed=default_seed
+"""Simple 1D classification example.
+:param model_type: type of model to fit ['Full', 'FITC', 'DTC'].
+:param seed : seed value for data generation (default is 4).
+:type seed: int
+:param inducing : number of inducing variables (only used for 'FITC' or 'DTC').
+:type inducing: int
+"""
+
+X = np.arange(0,100,5)[:,None]
+F = np.round(np.sin(X/18.) + .1*X)
+E = np.random.randint(-3,3,20)[:,None]
+Y = F + E
+pb.plot(X,F,'k-')
+pb.plot(X,Y,'ro')
+pb.figure()
+likelihood = GPy.inference.likelihoods.poisson(Y,scale=4.)
+
+m = GPy.models.GP(X,likelihood=likelihood)
+#m = GPy.models.GP(data['X'],Y=likelihood.Y)
+
+m.constrain_positive('var')
+m.constrain_positive('len')
+m.tie_param('lengthscale')
+if not isinstance(m.likelihood,GPy.inference.likelihoods.gaussian):
+    m.approximate_likelihood()
+print m.checkgrad()
+# Optimize and plot
+m.optimize()
+#m.em(plot_all=False) # EM algorithm
+m.plot()
+
+print(m)

GPy/examples/sparse_ep_fix.py

@@ -0,0 +1,76 @@
+# Copyright (c) 2012, GPy authors (see AUTHORS.txt).
+# Licensed under the BSD 3-clause license (see LICENSE.txt)
+
+
+import numpy as np
+"""
+Sparse Gaussian Processes regression with an RBF kernel
+"""
+import pylab as pb
+import numpy as np
+import GPy
+np.random.seed(2)
+pb.ion()
+N = 500
+M = 5
+
+######################################
+## 1 dimensional example
+
+# sample inputs and outputs
+X = np.random.uniform(-3.,3.,(N,1))
+#Y = np.sin(X)+np.random.randn(N,1)*0.05
+F = np.sin(X)+np.random.randn(N,1)*0.05
+Y = np.ones([F.shape[0],1])
+Y[F<0] = -1
+likelihood = GPy.inference.likelihoods.probit(Y)
+
+# construct kernel
+rbf =  GPy.kern.rbf(1)
+noise = GPy.kern.white(1)
+kernel = rbf + noise
+
+# create simple GP model
+#m1 = GPy.models.sparse_GP_regression(X, Y, kernel, M=M)
+m1 = GPy.models.sparse_GP(X, kernel, M=M,likelihood= likelihood)
+
+# contrain all parameters to be positive
+m1.constrain_positive('(variance|lengthscale|precision)')
+#m1.constrain_positive('(variance|lengthscale)')
+#m1.constrain_fixed('prec',10.)
+
+
+#check gradient FIXME unit test please
+m1.checkgrad()
+# optimize and plot
+m1.optimize('tnc', messages = 1)
+m1.plot()
+# print(m1)
+
+######################################
+## 2 dimensional example
+
+# # sample inputs and outputs
+# X = np.random.uniform(-3.,3.,(N,2))
+# Y = np.sin(X[:,0:1]) * np.sin(X[:,1:2])+np.random.randn(N,1)*0.05
+
+# # construct kernel
+# rbf =  GPy.kern.rbf(2)
+# noise = GPy.kern.white(2)
+# kernel = rbf + noise
+
+# # create simple GP model
+# m2 = GPy.models.sparse_GP_regression(X,Y,kernel, M = 50)
+# create simple GP model
+
+# # contrain all parameters to be positive (but not inducing inputs)
+# m2.constrain_positive('(variance|lengthscale|precision)')
+
+# #check gradient FIXME unit test please
+# m2.checkgrad()
+
+# # optimize and plot
+# pb.figure()
+# m2.optimize('tnc', messages = 1)
+# m2.plot()
+# print(m2)

GPy/inference/EP.py

@@ -1,9 +1,6 @@
-# Copyright (c) 2012, GPy authors (see AUTHORS.txt).
-# Licensed under the BSD 3-clause license (see LICENSE.txt)
-
-
 import numpy as np
 import random
+import pylab as pb #TODO erase me
 from scipy import stats, linalg
 from .likelihoods import likelihood
 from ..core import model
@@ -11,21 +8,45 @@ from ..util.linalg import pdinv,mdot,jitchol
 from ..util.plot import gpplot
 from .. import kern
 
-class EP_base:
-    """
-    Expectation Propagation.
+class EP:
+    def __init__(self,covariance,likelihood,Kmn=None,Knn_diag=None,epsilon=1e-3,power_ep=[1.,1.]):
+        """
+        Expectation Propagation
 
-    This is just the base class for expectation propagation. We'll extend it for full and sparse EP.
-    """
-    def __init__(self,likelihood,epsilon=1e-3,powerep=[1.,1.]):
+        Arguments
+        ---------
+        X : input observations
+        likelihood : Output's likelihood (likelihood class)
+        kernel :  a GPy kernel (kern class)
+        inducing : Either an array specifying the inducing points location or a sacalar defining their number. None value for using a non-sparse model is used.
+        power_ep : Power-EP parameters (eta,delta) - 2x1 numpy array (floats)
+        epsilon : Convergence criterion, maximum squared difference allowed between mean updates to stop iterations (float)
+        """
         self.likelihood = likelihood
+        assert covariance.shape[0] == covariance.shape[1]
+        if Kmn is not None:
+            self.Kmm = covariance
+            self.Kmn = Kmn
+            self.M = self.Kmn.shape[0]
+            self.N = self.Kmn.shape[1]
+            assert self.M < self.N, 'The number of inducing inputs must be smaller than the number of observations'
+        else:
+            self.K = covariance
+            self.N = self.K.shape[0]
+        if Knn_diag is not None:
+            self.Knn_diag = Knn_diag
+            assert len(Knn_diag) == self.N, 'Knn_diagonal has size different from N'
+
         self.epsilon = epsilon
-        self.eta, self.delta = powerep
+        self.eta, self.delta = power_ep
         self.jitter = 1e-12
 
-        #Initial values - Likelihood approximation parameters:
-        #p(y|f) = t(f|tau_tilde,v_tilde)
-        self.restart_EP()
+        """
+        Initial values - Likelihood approximation parameters:
+        p(y|f) = t(f|tau_tilde,v_tilde)
+        """
+        self.tau_tilde = np.zeros(self.N)
+        self.v_tilde = np.zeros(self.N)
 
     def restart_EP(self):
         """
@@ -35,26 +56,11 @@ class EP_base:
         self.v_tilde = np.zeros(self.N)
         self.mu = np.zeros(self.N)
 
-class Full(EP_base):
-    """
-    :param likelihood: Output's likelihood (e.g. probit)
-    :type likelihood: GPy.inference.likelihood instance
-    :param K: prior covariance matrix
-    :type K: np.ndarray (N x N)
-    :param likelihood: Output's likelihood (e.g. probit)
-    :type likelihood: GPy.inference.likelihood instance
-    :param epsilon: Convergence criterion, maximum squared difference allowed between mean updates to stop iterations (float)
-    :param powerep: Power-EP parameters (eta,delta) - 2x1 numpy array (floats)
-    """
-    def __init__(self,K,likelihood,*args,**kwargs):
-        assert K.shape[0] == K.shape[1]
-        self.K = K
-        self.N = self.K.shape[0]
-        EP_base.__init__(self,likelihood,*args,**kwargs)
-    def fit_EP(self,messages=False):
+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)
@@ -69,16 +75,15 @@ class Full(EP_base):
         sigma_ = 1./tau_
         mu_ = v_/tau_
         """
-
-        self.tau_ = np.empty(self.N,dtype=np.float64)
-        self.v_ = np.empty(self.N,dtype=np.float64)
+        self.tau_ = np.empty(self.N,dtype=float)
+        self.v_ = np.empty(self.N,dtype=float)
 
         #Initial values - Marginal moments
-        z = np.empty(self.N,dtype=np.float64)
-        self.Z_hat = np.empty(self.N,dtype=np.float64)
-        phi = np.empty(self.N,dtype=np.float64)
-        mu_hat = np.empty(self.N,dtype=np.float64)
-        sigma2_hat = np.empty(self.N,dtype=np.float64)
+        z = np.empty(self.N,dtype=float)
+        self.Z_hat = np.empty(self.N,dtype=float)
+        phi = np.empty(self.N,dtype=float)
+        mu_hat = np.empty(self.N,dtype=float)
+        sigma2_hat = np.empty(self.N,dtype=float)
 
         #Approximation
         epsilon_np1 = self.epsilon + 1.
@@ -87,7 +92,8 @@ class Full(EP_base):
         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.N)
+            update_order = np.arange(self.N)
+            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]
@@ -104,6 +110,7 @@ class Full(EP_base):
                 self.Sigma = self.Sigma - Delta_tau/(1.+ Delta_tau*self.Sigma[i,i])*np.dot(si,si.T)
                 self.mu = np.dot(self.Sigma,self.v_tilde)
                 self.iterations += 1
+                print self.tau_tilde[i] #TODO erase me
             #Sigma recomptutation with Cholesky decompositon
             Sroot_tilde_K = np.sqrt(self.tau_tilde)[:,None]*(self.K)
             B = np.eye(self.N) + np.sqrt(self.tau_tilde)[None,:]*Sroot_tilde_K
@@ -111,35 +118,98 @@ class Full(EP_base):
             V,info = linalg.flapack.dtrtrs(L,Sroot_tilde_K,lower=1)
             self.Sigma = self.K - np.dot(V.T,V)
             self.mu = np.dot(self.Sigma,self.v_tilde)
-            epsilon_np1 = np.mean(self.tau_tilde-self.np1[-1]**2)
-            epsilon_np2 = np.mean(self.v_tilde-self.np2[-1]**2)
+            epsilon_np1 = sum((self.tau_tilde-self.np1[-1])**2)/self.N
+            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())
-            if messages:
-                print "EP iteration %i, epsiolon %d"%(self.iterations,epsilon_np1)
+        return self.tau_tilde[:,None], self.v_tilde[:,None], self.Z_hat[:,None], self.tau_[:,None], self.v_[:,None]
 
-class FITC(EP_base):
-    """
-    :param likelihood: Output's likelihood (e.g. probit)
-    :type likelihood: GPy.inference.likelihood instance
-    :param Knn_diag: The diagonal elements of Knn is a 1D vector
-    :param Kmn: The 'cross' variance between inducing inputs and data
-    :param Kmm: the covariance matrix of the inducing inputs
-    :param likelihood: Output's likelihood (e.g. probit)
-    :type likelihood: GPy.inference.likelihood instance
-    :param epsilon: Convergence criterion, maximum squared difference allowed between mean updates to stop iterations (float)
-    :param powerep: Power-EP parameters (eta,delta) - 2x1 numpy array (floats)
-    """
-    def __init__(self,likelihood,Knn_diag,Kmn,Kmm,*args,**kwargs):
-        self.Knn_diag = Knn_diag
-        self.Kmn = Kmn
-        self.Kmm = Kmm
-        self.M = self.Kmn.shape[0]
-        self.N = self.Kmn.shape[1]
-        assert self.M <= self.N, 'The number of inducing inputs must be smaller than the number of observations'
-        assert len(Knn_diag) == self.N, 'Knn_diagonal has size different from N'
-        EP_base.__init__(self,likelihood,*args,**kwargs)
+class DTC(EP):
+    def fit_EP(self):
+        """
+        The expectation-propagation algorithm with sparse pseudo-input.
+        For nomenclature see ... 2013.
+        """
 
+        """
+        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
+        """
+        self.Kmmi, self.Kmm_hld = pdinv(self.Kmm)
+        self.KmnKnm = np.dot(self.Kmn, self.Kmn.T)
+        self.KmmiKmn = np.dot(self.Kmmi,self.Kmn)
+        self.Qnn_diag = np.sum(self.Kmn*self.KmmiKmn,-2)
+        self.LLT0 = self.Kmm.copy()
+
+        """
+        Posterior approximation: q(f|y) = N(f| mu, Sigma)
+        Sigma = Diag + P*R.T*R*P.T + K
+        mu = w + P*gamma
+        """
+        self.mu = np.zeros(self.N)
+        self.LLT = self.Kmm.copy()
+        self.Sigma_diag = self.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.N,dtype=float)
+        self.v_ = np.empty(self.N,dtype=float)
+
+        #Initial values - Marginal moments
+        z = np.empty(self.N,dtype=float)
+        self.Z_hat = np.empty(self.N,dtype=float)
+        phi = np.empty(self.N,dtype=float)
+        mu_hat = np.empty(self.N,dtype=float)
+        sigma2_hat = np.empty(self.N,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.arange(self.N)
+            random.shuffle(update_order)
+            for i in update_order:
+                #Cavity distribution parameters
+                self.tau_[i] = 1./self.Sigma_diag[i] - self.eta*self.tau_tilde[i]
+                self.v_[i] = self.mu[i]/self.Sigma_diag[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])
+                #Site parameters update
+                Delta_tau = self.delta/self.eta*(1./sigma2_hat[i] - 1./self.Sigma_diag[i])
+                Delta_v = self.delta/self.eta*(mu_hat[i]/sigma2_hat[i] - self.mu[i]/self.Sigma_diag[i])
+                self.tau_tilde[i] = self.tau_tilde[i] + Delta_tau
+                self.v_tilde[i] = self.v_tilde[i] + Delta_v
+                #Posterior distribution parameters update
+                self.LLT = self.LLT + np.outer(self.Kmn[:,i],self.Kmn[:,i])*Delta_tau
+                L = jitchol(self.LLT)
+                V,info = linalg.flapack.dtrtrs(L,self.Kmn,lower=1)
+                self.Sigma_diag = np.sum(V*V,-2)
+                si = np.sum(V.T*V[:,i],-1)
+                self.mu = self.mu + (Delta_v-Delta_tau*self.mu[i])*si
+                self.iterations += 1
+            #Sigma recomputation with Cholesky decompositon
+            self.LLT0 = self.LLT0 + np.dot(self.Kmn*self.tau_tilde[None,:],self.Kmn.T)
+            self.L = jitchol(self.LLT)
+            V,info = linalg.flapack.dtrtrs(L,self.Kmn,lower=1)
+            V2,info = linalg.flapack.dtrtrs(L.T,V,lower=0)
+            self.Sigma_diag = np.sum(V*V,-2)
+            Knmv_tilde = np.dot(self.Kmn,self.v_tilde)
+            self.mu = np.dot(V2.T,Knmv_tilde)
+            epsilon_np1 = sum((self.tau_tilde-self.np1[-1])**2)/self.N
+            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 FITC(EP):
     def fit_EP(self):
         """
         The expectation-propagation algorithm with sparse pseudo-input.
@@ -178,15 +248,15 @@ class FITC(EP_base):
         sigma_ = 1./tau_
         mu_ = v_/tau_
         """
-        self.tau_ = np.empty(self.N,dtype=np.float64)
-        self.v_ = np.empty(self.N,dtype=np.float64)
+        self.tau_ = np.empty(self.N,dtype=float)
+        self.v_ = np.empty(self.N,dtype=float)
 
         #Initial values - Marginal moments
-        z = np.empty(self.N,dtype=np.float64)
-        self.Z_hat = np.empty(self.N,dtype=np.float64)
-        phi = np.empty(self.N,dtype=np.float64)
-        mu_hat = np.empty(self.N,dtype=np.float64)
-        sigma2_hat = np.empty(self.N,dtype=np.float64)
+        z = np.empty(self.N,dtype=float)
+        self.Z_hat = np.empty(self.N,dtype=float)
+        phi = np.empty(self.N,dtype=float)
+        mu_hat = np.empty(self.N,dtype=float)
+        sigma2_hat = np.empty(self.N,dtype=float)
 
         #Approximation
         epsilon_np1 = 1
@@ -238,3 +308,4 @@ class FITC(EP_base):
             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]

GPy/inference/likelihoods.py

@@ -21,7 +21,7 @@ class likelihood:
         self.location = location
         self.scale = scale
 
-    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
 
@@ -32,10 +32,18 @@ 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,Mean_u,2*np.sqrt(Var_u),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):
         """
@@ -90,16 +98,11 @@ 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,power_ep=[1.,1.]):
+        #TODO: make beta parameter explicit
+
+        # 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 = power_ep
+            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,power_ep=[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

@@ -1,160 +0,0 @@
-# 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 scipy import stats, linalg
-from .. import kern
-from ..inference.Expectation_Propagation import Full
-from ..inference.likelihoods import likelihood,probit#,poisson,gaussian
-from ..core import model
-from ..util.linalg import pdinv,jitchol
-from ..util.plot import gpplot
-
-class GP_EP(model):
-    def __init__(self,X,likelihood,kernel=None,epsilon_ep=1e-3,epsion_em=.1,powerep=[1.,1.]):
-        """
-        Simple Gaussian Process with Non-Gaussian likelihood
-
-        Arguments
-        ---------
-        :param X: input observations (NxD numpy.darray)
-        :param likelihood: a GPy likelihood (likelihood class)
-        :param kernel: a GPy kernel (kern class)
-        :param epsilon_ep: convergence criterion for the Expectation Propagation algorithm, defaults to 0.1 (float)
-        :param powerep: power-EP parameters [$\eta$,$\delta$], defaults to [1.,1.] (list)
-        :rtype: GPy model class.
-        """
-        if kernel is None:
-            kernel = kern.rbf(X.shape[1]) + kern.bias(X.shape[1]) + kern.white(X.shape[1])
-
-        assert isinstance(kernel,kern.kern), 'kernel is not a kern instance'
-        self.likelihood = likelihood
-        self.Y = self.likelihood.Y
-        self.kernel = kernel
-        self.X = X
-        self.N, self.D = self.X.shape
-        self.eta,self.delta = powerep
-        self.epsilon_ep = epsilon_ep
-        self.jitter = 1e-12
-        self.K = self.kernel.K(self.X)
-        model.__init__(self)
-
-    def _set_params(self,p):
-        self.kernel._set_params_transformed(p)
-
-    def _get_params(self):
-        return self.kernel._get_params_transformed()
-
-    def _get_param_names(self):
-        return self.kernel._get_param_names_transformed()
-
-    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()
-
-    def posterior_param(self):
-        self.K = self.kernel.K(self.X)
-        self.Sroot_tilde_K =  np.sqrt(self.ep_approx.tau_tilde)[:,None]*self.K
-        B = np.eye(self.N) + np.sqrt(self.ep_approx.tau_tilde)[None,:]*self.Sroot_tilde_K
-        #self.L = np.linalg.cholesky(B)
-        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)
-
-    def log_likelihood(self):
-        """
-        Returns
-        -------
-        The EP approximation to the log-marginal likelihood
-        """
-        self.posterior_param()
-        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
-
-    def _log_likelihood_gradients(self):
-        dK_dp = self.kernel.dK_dtheta(self.X)
-        self.dK_dp = dK_dp
-        aux1,info_1 = linalg.flapack.dtrtrs(self.L,np.dot(self.Sroot_tilde_K,self.ep_approx.v_tilde),lower=1)
-        b = self.ep_approx.v_tilde - np.sqrt(self.ep_approx.tau_tilde)*linalg.flapack.dtrtrs(self.L.T,aux1)[0]
-        U,info_u = linalg.flapack.dtrtrs(self.L,np.diag(np.sqrt(self.ep_approx.tau_tilde)),lower=1)
-        dL_dK = 0.5*(np.outer(b,b)-np.dot(U.T,U))
-        self.dL_dK = dL_dK
-        return np.array([np.sum(dK_dpi*dL_dK) for dK_dpi in dK_dp.T])
-
-    def predict(self,X):
-        #TODO: check output dimensions
-        self.posterior_param()
-        K_x = self.kernel.K(self.X,X)
-        Kxx = self.kernel.K(X)
-        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)
-        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)
-        variance = Kxx - np.dot(v.T,v)
-       	vdiag = np.diag(variance)
-        y=self.likelihood.predictive_mean(f,vdiag)
-        return f,vdiag,y
-
-    def plot(self):
-        """
-        Plot the fitted model: training function values, inducing points used, mean estimate and confidence intervals.
-        """
-        if self.X.shape[1]==1:
-            pb.figure()
-            xmin,xmax = self.X.min(),self.X.max()
-            xmin, xmax = xmin-0.2*(xmax-xmin), xmax+0.2*(xmax-xmin)
-            Xnew = np.linspace(xmin,xmax,100)[:,None]
-            mu_f, var_f, mu_phi = self.predict(Xnew)
-            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))
-            pb.subplot(212)
-            self.likelihood.plot1Db(self.X,Xnew,mu_phi)
-        elif self.X.shape[1]==2:
-            pb.figure()
-            x1min,x1max = self.X[:,0].min(0),self.X[:,0].max(0)
-            x2min,x2max = self.X[:,1].min(0),self.X[:,1].max(0)
-            x1min, x1max = x1min-0.2*(x1max-x1min), x1max+0.2*(x1max-x1min)
-            x2min, x2max = x2min-0.2*(x2max-x2min), x2max+0.2*(x1max-x1min)
-            axis1 = np.linspace(x1min,x1max,50)
-            axis2 = np.linspace(x2min,x2max,50)
-            XX1, XX2 = [e.flatten() for e in np.meshgrid(axis1,axis2)]
-            Xnew = np.c_[XX1.flatten(),XX2.flatten()]
-            f,v,p = self.predict(Xnew)
-            self.likelihood.plot2D(self.X,Xnew,p)
-        else:
-            raise NotImplementedError, "Cannot plot GPs with more than two input dimensions"
-
-    def em(self,max_f_eval=1e4,epsilon=.1,plot_all=False): #TODO check this makes sense
-        """
-        Fits sparse_EP and optimizes the hyperparametes iteratively until convergence is achieved.
-        """
-        self.epsilon_em = epsilon
-        log_likelihood_change = self.epsilon_em + 1.
-        self.parameters_path = [self.kernel._get_params()]
-        self.approximate_likelihood()
-        self.site_approximations_path = [[self.ep_approx.tau_tilde,self.ep_approx.v_tilde]]
-        self.log_likelihood_path = [self.log_likelihood()]
-        iteration = 0
-        while log_likelihood_change > self.epsilon_em:
-            print 'EM iteration', iteration
-            self.optimize(max_f_eval = max_f_eval)
-            log_likelihood_new = self.log_likelihood()
-            log_likelihood_change = log_likelihood_new - self.log_likelihood_path[-1]
-            if log_likelihood_change < 0:
-                print 'log_likelihood decrement'
-                self.kernel._set_params_transformed(self.parameters_path[-1])
-                self.kernM._set_params_transformed(self.parameters_path[-1])
-            else:
-                self.approximate_likelihood()
-                self.log_likelihood_path.append(self.log_likelihood())
-                self.parameters_path.append(self.kernel._get_params())
-                self.site_approximations_path.append([self.ep_approx.tau_tilde,self.ep_approx.v_tilde])
-            iteration += 1

GPy/models/__init__.py

@@ -6,7 +6,8 @@ from GP_regression import GP_regression
 from sparse_GP_regression import sparse_GP_regression
 from GPLVM import GPLVM
 from warped_GP import warpedGP
-from GP_EP import GP_EP
 from generalized_FITC import generalized_FITC
 from sparse_GPLVM import sparse_GPLVM
 from uncollapsed_sparse_GP import uncollapsed_sparse_GP
+from GP import GP
+from sparse_GP import sparse_GP

GPy/models/generalized_FITC.py

@@ -9,7 +9,8 @@ from .. import kern
 from ..core import model
 from ..util.linalg import pdinv,mdot
 from ..util.plot import gpplot
-from ..inference.Expectation_Propagation import FITC
+#from ..inference.Expectation_Propagation import FITC
+from ..inference.EP import FITC
 from ..inference.likelihoods import likelihood,probit
 
 class generalized_FITC(model):

GPy/models/sparse_GP.py

@@ -0,0 +1,258 @@
+# 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 ..util.linalg import mdot, jitchol, chol_inv, pdinv
+from ..util.plot import gpplot
+from .. import kern
+from GP import GP
+from ..inference.EP import Full
+from ..inference.likelihoods import likelihood,probit,poisson,gaussian
+
+#Still TODO:
+# make use of slices properly (kernel can now do this)
+# enable heteroscedatic noise (kernel will need to compute psi2 as a (NxMxM) array)
+
+class sparse_GP(GP):
+    """
+    Variational sparse GP model (Regression)
+
+    :param X: inputs
+    :type X: np.ndarray (N x Q)
+    :param Y: observed data
+    :type Y: np.ndarray of observations (N x D)
+    :param kernel : the kernel/covariance function. See link kernels
+    :type kernel: a GPy kernel
+    :param Z: inducing inputs (optional, see note)
+    :type Z: np.ndarray (M x Q) | None
+    :param X_uncertainty: The uncertainty in the measurements of X (Gaussian variance)
+    :type X_uncertainty: np.ndarray (N x Q) | None
+    :param Zslices: slices for the inducing inputs (see slicing TODO: link)
+    :param M : Number of inducing points (optional, default 10. Ignored if Z is not None)
+    :type M: int
+    :param beta: noise precision. TODO> ignore beta if doing EP
+    :type beta: float
+    :param normalize_(X|Y) : whether to normalize the data before computing (predictions will be in original scales)
+    :type normalize_(X|Y): bool
+    """
+
+    def __init__(self,X,Y,kernel=None, X_uncertainty=None, beta=100., Z=None,Zslices=None,M=10,normalize_X=False,normalize_Y=False,likelihood=None,method_ep='DTC',epsilon_ep=1e-3,epsilon_em=.1,power_ep=[1.,1.]):
+
+        if Z is None:
+            self.Z = np.random.permutation(X.copy())[:M]
+            self.M = M
+        else:
+            assert Z.shape[1]==X.shape[1]
+            self.Z = Z
+            self.M = Z.shape[0]
+        if X_uncertainty is None:
+            self.has_uncertain_inputs=False
+        else:
+            assert X_uncertainty.shape==X.shape
+            self.has_uncertain_inputs=True
+            self.X_uncertainty = X_uncertainty
+
+
+        self.beta = beta #FIXME
+        GP.__init__(self, X, Y, kernel=kernel, normalize_X=normalize_X, normalize_Y=normalize_Y,likelihood=likelihood,epsilon_ep=epsilon_ep,epsion_em=epsilon_em,power_ep=power_ep)
+        self.beta = beta if isinstance(likelihood,gaussian) else self.tau_tilde #TODO this should be defined in GP.__init__
+
+
+        #normalise X uncertainty also
+        if self.has_uncertain_inputs:
+            self.X_uncertainty /= np.square(self._Xstd)
+
+    def _set_params(self, p):
+        if not self.EP:
+            self.Z = p[:self.M*self.Q].reshape(self.M, self.Q)
+            self.beta = p[self.M*self.Q]
+            self.kern._set_params(p[self.Z.size + 1:])
+            self.beta2 = self.beta**2
+            self._compute_kernel_matrices()
+            self._computations()
+        else:
+            self.Z = p[:self.M*self.Q].reshape(self.M, self.Q)
+            self.kern._set_params(p[self.Z.size:])
+            #self._compute_kernel_matrices() this is replaced by _ep_covariance
+            self._ep_covariance()
+            self._ep_computations()
+
+    def approximate_likelihood(self):
+        assert not isinstance(self.likelihood, gaussian), "EP is only available for non-gaussian likelihoods"
+        if self.ep_proxy == 'DTC':
+            self.ep_approx = DTC(self.Kmm,self.likelihood,self.psi1,epsilon=self.epsilon_ep,power_ep=[self.eta,self.delta])
+        elif self.ep_proxy == 'FITC':
+            self.Knn_diag = self.kern.psi0(self.Z,self.X, self.X_uncertainty) #TODO psi0 already calculates this
+            self.ep_approx = FITC(self.Kmm,self.likelihood,self.psi1,self.Knn_diag,epsilon=self.epsilon_ep,power_ep=[self.eta,self.delta])
+        else:
+            self.ep_approx = Full(self.X,self.likelihood,self.kernel,inducing=None,epsilon=self.epsilon_ep,power_ep=[self.eta,self.delta])
+        self.beta, 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.beta
+        self._Ymean = np.zeros((1,self.Y.shape[1]))
+        self._Ystd = np.ones((1,self.Y.shape[1]))
+        self.trbetaYYT = np.sum(self.beta*np.square(self.Y))
+        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()
+        self._computations()
+
+    def _ep_covariance(self):
+        self.Kmm = self.kern.K(self.Z)
+        if self.has_uncertain_inputs:
+            self.psi0 = self.kern.psi0(self.Z,self.X, self.X_uncertainty).sum()
+            self.psi1 = self.kern.psi1(self.Z,self.X, self.X_uncertainty).T
+            self.psi2 = self.kern.psi2(self.Z,self.X, self.X_uncertainty) #FIXME include beta
+        else:
+            #self.psi0 = self.kern.Kdiag(self.X,slices=self.Xslices).sum()
+            self.Knn_diag = self.kern.Kdiag(self.X,slices=self.Xslices)
+            self.psi0 = (self.beta*self.Knn_diag).sum() #TODO check dimensions
+            self.psi1 = self.kern.K(self.Z,self.X)
+            #self.psi2 = np.dot(self.psi1,self.psi1.T)
+            self.psi2 = np.dot(self.psi1,self.beta*self.psi1.T)
+
+    def _compute_kernel_matrices(self):
+        # kernel computations, using BGPLVM notation
+        #TODO: slices for psi statistics (easy enough)
+
+        self.Kmm = self.kern.K(self.Z)
+        if self.has_uncertain_inputs:
+            self.psi0 = self.kern.psi0(self.Z,self.X, self.X_uncertainty).sum()
+            self.psi1 = self.kern.psi1(self.Z,self.X, self.X_uncertainty).T
+            self.psi2 = self.kern.psi2(self.Z,self.X, self.X_uncertainty)
+        else:
+            self.psi0 = self.kern.Kdiag(self.X,slices=self.Xslices).sum()
+            self.psi1 = self.kern.K(self.Z,self.X)
+            self.psi2 = np.dot(self.psi1,self.psi1.T)
+
+    def _ep_computations(self):
+        # TODO find routine to multiply triangular matrices
+        self.V = self.beta*self.Y
+        self.psi1V = np.dot(self.psi1, self.V)
+        self.psi1VVpsi1 = np.dot(self.psi1V, self.psi1V.T)
+        self.Kmmi, self.Lm, self.Lmi, self.Kmm_logdet = pdinv(self.Kmm)
+        #self.A = mdot(self.Lmi, self.beta*self.psi2, self.Lmi.T)
+        self.A = mdot(self.Lmi, self.psi2, self.Lmi.T)
+        self.B = np.eye(self.M) + self.A
+        self.Bi, self.LB, self.LBi, self.B_logdet = pdinv(self.B)
+        self.LLambdai = np.dot(self.LBi, self.Lmi)
+        #self.trace_K = self.psi0 - np.sum(np.dot(self.Lmi,self.psi1)**2,-1) #TODO check
+        self.trace_K = self.psi0 - np.trace(self.A)
+        self.LBL_inv = mdot(self.Lmi.T, self.Bi, self.Lmi)
+        self.C = mdot(self.LLambdai, self.psi1V)
+        self.G =  mdot(self.LBL_inv, self.psi1VVpsi1, self.LBL_inv.T)
+
+        # Compute dL_dpsi
+        #self.dL_dpsi0 = - 0.5 * self.D * self.beta * np.ones(self.N)
+        self.dL_dpsi0 = - 0.5 * self.D * self.beta.flatten() * np.ones(self.N) #TODO check
+        self.dL_dpsi1 = mdot(self.LLambdai.T,self.C,self.V.T)
+        #self.dL_dpsi2 = - 0.5 * self.beta * (self.D*(self.LBL_inv - self.Kmmi) + self.G)
+        self.dL_dpsi2 = - 0.5 * self.beta * (self.D*(self.LBL_inv - self.Kmmi) + self.G)
+
+        # Compute dL_dKmm
+        self.dL_dKmm = -0.5 * self.D * mdot(self.Lmi.T, self.A, self.Lmi) # dB
+        self.dL_dKmm += -0.5 * self.D * (- self.LBL_inv - 2.*self.beta*mdot(self.LBL_inv, self.psi2, self.Kmmi) + self.Kmmi) # dC
+        self.dL_dKmm +=  np.dot(np.dot(self.G,self.beta*self.psi2) - np.dot(self.LBL_inv, self.psi1VVpsi1), self.Kmmi) + 0.5*self.G # dE
+
+    def _get_params(self):
+        if not self.EP:
+            return np.hstack([self.Z.flatten(),self.beta,self.kern._get_params_transformed()])
+        else:
+            return np.hstack([self.Z.flatten(),self.kern._get_params_transformed()])
+
+    def _get_param_names(self):
+        if not self.EP:
+            return sum([['iip_%i_%i'%(i,j) for i in range(self.Z.shape[0])] for j in range(self.Z.shape[1])],[]) + ['noise_precision']+self.kern._get_param_names_transformed()
+        else:
+            return sum([['iip_%i_%i'%(i,j) for i in range(self.Z.shape[0])] for j in range(self.Z.shape[1])],[]) + self.kern._get_param_names_transformed()
+
+    def log_likelihood(self):
+        """
+        Compute the (lower bound on the) log marginal likelihood
+        """
+        beta_logdet = self.N*self.D*np.log(self.beta) if not self.EP else self.D*np.sum(np.log(self.beta))
+        A = -0.5*self.N*self.D*(np.log(2.*np.pi)) - 0.5*beta_logdet
+        B = -0.5*self.beta*self.D*self.trace_K if not self.EP else -0.5*self.D*self.trace_K
+        C = -0.5*self.D * self.B_logdet
+        D = -0.5*self.beta*self.trYYT if not self.EP else -0.5*self.trbetaYYT
+        E = +0.5*np.sum(self.psi1VVpsi1 * self.LBL_inv)
+        return A+B+C+D+E
+
+    def dL_dbeta(self):
+        """
+        Compute the gradient of the log likelihood wrt beta.
+        """
+        #TODO: suport heteroscedatic noise
+        dA_dbeta =   0.5 * self.N*self.D/self.beta
+        dB_dbeta = - 0.5 * self.D * self.trace_K
+        dC_dbeta = - 0.5 * self.D * np.sum(self.Bi*self.A)/self.beta
+        dD_dbeta = - 0.5 * self.trYYT
+        tmp = mdot(self.LBi.T, self.LLambdai, self.psi1V)
+        dE_dbeta = (np.sum(np.square(self.C)) - 0.5 * np.sum(self.A * np.dot(tmp, tmp.T)))/self.beta
+
+        return np.squeeze(dA_dbeta + dB_dbeta + dC_dbeta + dD_dbeta + dE_dbeta)
+
+    def dL_dtheta(self):
+        """
+        Compute and return the derivative of the log marginal likelihood wrt the parameters of the kernel
+        """
+        dL_dtheta = self.kern.dK_dtheta(self.dL_dKmm,self.Z)
+        if self.has_uncertain_inputs:
+            dL_dtheta += self.kern.dpsi0_dtheta(self.dL_dpsi0, self.Z,self.X,self.X_uncertainty)
+            dL_dtheta += self.kern.dpsi1_dtheta(self.dL_dpsi1.T,self.Z,self.X, self.X_uncertainty)
+            dL_dtheta += self.kern.dpsi2_dtheta(self.dL_dpsi2,self.Z,self.X, self.X_uncertainty) # for multiple_beta, dL_dpsi2 will be a different shape
+        else:
+            #re-cast computations in psi2 back to psi1:
+            dL_dpsi1 = self.dL_dpsi1 + 2.*np.dot(self.dL_dpsi2,self.psi1)
+            dL_dtheta += self.kern.dK_dtheta(dL_dpsi1,self.Z,self.X)
+            dL_dtheta += self.kern.dKdiag_dtheta(self.dL_dpsi0, self.X)
+
+        return dL_dtheta
+
+    def dL_dZ(self):
+        """
+        The derivative of the bound wrt the inducing inputs Z
+        """
+        dL_dZ = 2.*self.kern.dK_dX(self.dL_dKmm,self.Z,)#factor of two becase of vertical and horizontal 'stripes' in dKmm_dZ
+        if self.has_uncertain_inputs:
+            dL_dZ += self.kern.dpsi1_dZ(self.dL_dpsi1.T,self.Z,self.X, self.X_uncertainty)
+            dL_dZ += self.kern.dpsi2_dZ(self.dL_dpsi2,self.Z,self.X, self.X_uncertainty)
+        else:
+            #re-cast computations in psi2 back to psi1:
+            dL_dpsi1 = self.dL_dpsi1 + 2.*np.dot(self.dL_dpsi2,self.psi1)
+            dL_dZ += self.kern.dK_dX(dL_dpsi1,self.Z,self.X)
+        return dL_dZ
+
+    def _log_likelihood_gradients(self):
+        return np.hstack([self.dL_dZ().flatten(), self.dL_dbeta(), self.dL_dtheta()])
+
+    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.Z, Xnew)
+        mu = mdot(Kx.T, self.LBL_inv, self.psi1V)
+        if full_cov:
+            noise_term = np.eye(Xnew.shape[0])/self.beta if not self.EP else 0
+            Kxx = self.kern.K(Xnew)
+            var = Kxx - mdot(Kx.T, (self.Kmmi - self.LBL_inv), Kx) + noise_term
+        else:
+            noise_term = 1./self.beta if not self.EP else 0
+            Kxx = self.kern.Kdiag(Xnew)
+            var = Kxx - np.sum(Kx*np.dot(self.Kmmi - self.LBL_inv, Kx),0) + noise_term
+        return mu,var
+
+    def plot(self, *args, **kwargs):
+        """
+        Plot the fitted model: just call the GP_regression plot function and then add inducing inputs
+        """
+        GP_regression.plot(self,*args,**kwargs)
+        if self.Q==1:
+            pb.plot(self.Z,self.Z*0+pb.ylim()[0],'k|',mew=1.5,markersize=12)
+            if self.has_uncertain_inputs:
+                pb.errorbar(self.X[:,0], pb.ylim()[0]+np.zeros(self.N), xerr=2*np.sqrt(self.X_uncertainty.flatten()))
+        if self.Q==2:
+            pb.plot(self.Z[:,0],self.Z[:,1],'wo')