Lots of name changing and went through all likelihood gradients again

GPy/examples/laplace_approximations.py

@@ -69,22 +69,21 @@ def debug_student_t_noise_approx():
     print "Clean Gaussian"
     #A GP should completely break down due to the points as they get a lot of weight
     # create simple GP model
-    m = GPy.models.GP_regression(X, Y, kernel=kernel1)
+    #m = GPy.models.GP_regression(X, Y, kernel=kernel1)
-    # optimize
+    ## optimize
-    m.ensure_default_constraints()
+    #m.ensure_default_constraints()
-    m.optimize()
+    #m.optimize()
-    # plot
+    ## plot
-    if plot:
+    #if plot:
-        plt.figure(1)
+        #plt.figure(1)
-        plt.suptitle('Gaussian likelihood')
+        #plt.suptitle('Gaussian likelihood')
-        plt.subplot(131)
+        #plt.subplot(131)
-        m.plot()
+        #m.plot()
-        plt.plot(X_full, Y_full)
+        #plt.plot(X_full, Y_full)
-    print m
+    #print m
 
     edited_real_sd = initial_var_guess #real_sd
 
-    import ipdb; ipdb.set_trace() ### XXX BREAKPOINT
     print "Clean student t, rasm"
     t_distribution = GPy.likelihoods.likelihood_functions.student_t(deg_free, sigma=edited_real_sd)
     stu_t_likelihood = GPy.likelihoods.Laplace(Y.copy(), t_distribution, rasm=True)
@@ -95,10 +94,10 @@ def debug_student_t_noise_approx():
     m.constrain_positive('t_noi')
     #m.constrain_fixed('t_noise_variance', real_sd)
     m.update_likelihood_approximation()
+    m.optimize('scg', messages=True)
     print(m)
     return m
     #m.optimize('lbfgsb', messages=True, callback=m._update_params_callback)
-    m.optimize('scg', messages=True)
     if plot:
         plt.suptitle('Student-t likelihood')
         plt.subplot(132)

GPy/likelihoods/Laplace.py

@@ -79,17 +79,40 @@ class Laplace(likelihood):
         return (self._Kgradients(dL_d_K_Sigma, dK_dthetaK), self._gradients(dL_d_K_Sigma))
 
     def _shared_gradients_components(self):
+        Ki, _, _, _ = pdinv(self.K)
+        Ki_W_i = inv(Ki + self.W) #Do it non numerically stable for now
+        d3lik_d3fhat = self.likelihood_function.d3lik_d3f(self.data, self.f_hat)
+        dL_dfhat = -0.5*np.dot(np.diag(Ki_W_i), d3lik_d3fhat)
+        KW = np.dot(self.K, self.W)
+        I_KW_i = inv(np.eye(KW.shape[0]) + KW)
+        return dL_dfhat, Ki, I_KW_i
 
     def _Kgradients(self, dL_d_K_Sigma, dK_dthetaK):
         """
         Gradients with respect to prior kernel parameters
         """
+        dL_dfhat, Ki, I_KW_i = self._shared_gradients_components()
+        K_Wi_i = inv(self.K + inv(self.W))
+        dlp = self.likelihood_function.dlik_df(self.data, self.f_hat)
+
+        dL_dthetaK = np.zeros(dK_dthetaK.shape)
+        for thetaK_i, dK_dthetaK_i in enumerate(dK_dthetaK):
+            #Explicit
+            dL_dthetaK[thetaK_i] = 0.5*mdot(self.f_hat.T, Ki, dK_dthetaK_i, Ki, self.f_hat) - 0.5*np.trace(np.dot(K_Wi_i, dK_dthetaK_i))
+            #Implicit
+            df_hat_dthetaK = mdot(I_KW_i, dK_dthetaK, dlp)
+            dL_dthetaK[thetaK_i] += np.dot(dL_dfhat.T, df_hat_dthetaK)
+
         return dL_dthetaK
 
     def _gradients(self, partial):
         """
         Gradients with respect to likelihood parameters
         """
+        dL_dfhat, Ki, I_KW_i = self._shared_gradients_components()
+        dlik_dthetaL, dlik_grad_dthetaL, dlik_hess_dthetaL = self.likelihood_function._gradients(self.data, self.f_hat)
+        dL_dthetaL = np.zeros(dlik_dthetaL.shape)
+
         return dL_dthetaL #should be array of length *params-being optimized*, for student t just optimising 1 parameter, this is (1,)
 
     def _compute_GP_variables(self):
@@ -197,7 +220,7 @@ class Laplace(likelihood):
         #At this point get the hessian matrix
         #print "Data: ", self.data
         #print "fhat: ", self.f_hat
-        self.W = -np.diag(self.likelihood_function.link_hess(self.data, self.f_hat, extra_data=self.extra_data))
+        self.W = -np.diag(self.likelihood_function.d2lik_d2f(self.data, self.f_hat, extra_data=self.extra_data))
 
         if not self.likelihood_function.log_concave:
             self.W[self.W < 0] = 1e-6  # FIXME-HACK: This is a hack since GPy can't handle negative variances which can occur
@@ -212,7 +235,7 @@ class Laplace(likelihood):
         Ki_W_i = self.K - mdot(self.K, self.W_12, self.Bi, self.W_12, self.K)
         self.ln_Ki_W_i_det = np.linalg.det(Ki_W_i)
 
-        b = np.dot(self.W, self.f_hat) + self.likelihood_function.link_grad(self.data, self.f_hat, extra_data=self.extra_data)[:, None]
+        b = np.dot(self.W, self.f_hat) + self.likelihood_function.dlik_df(self.data, self.f_hat, extra_data=self.extra_data)[:, None]
         solve_chol = cho_solve((self.B_chol, True), mdot(self.W_12, (self.K, b)))
         a = b - mdot(self.W_12, solve_chol)
         self.Ki_f = a
@@ -259,11 +282,11 @@ class Laplace(likelihood):
             return float(res)
 
         def obj_grad(f):
-            res = -1 * (self.likelihood_function.link_grad(self.data[:, 0], f, extra_data=self.extra_data) - np.dot(self.Ki, f))
+            res = -1 * (self.likelihood_function.dlik_df(self.data[:, 0], f, extra_data=self.extra_data) - np.dot(self.Ki, f))
             return np.squeeze(res)
 
         def obj_hess(f):
-            res = -1 * (--np.diag(self.likelihood_function.link_hess(self.data[:, 0], f, extra_data=self.extra_data)) - self.Ki)
+            res = -1 * (--np.diag(self.likelihood_function.d2lik_d2f(self.data[:, 0], f, extra_data=self.extra_data)) - self.Ki)
             return np.squeeze(res)
 
         f_hat = sp.optimize.fmin_ncg(obj, f, fprime=obj_grad, fhess=obj_hess)
@@ -294,7 +317,7 @@ class Laplace(likelihood):
         i = 0
         while difference > epsilon and i < MAX_ITER and rs < MAX_RESTART:
             #f_old = f.copy()
-            W = -np.diag(self.likelihood_function.link_hess(self.data, f, extra_data=self.extra_data))
+            W = -np.diag(self.likelihood_function.d2lik_d2f(self.data, f, extra_data=self.extra_data))
             if not self.likelihood_function.log_concave:
                 W[W < 0] = 1e-6     # FIXME-HACK: This is a hack since GPy can't handle negative variances which can occur
                                     # If the likelihood is non-log-concave. We wan't to say that there is a negative variance
@@ -303,7 +326,7 @@ class Laplace(likelihood):
             B, L, W_12 = self._compute_B_statistics(K, W)
 
             W_f = np.dot(W, f)
-            grad = self.likelihood_function.link_grad(self.data, f, extra_data=self.extra_data)[:, None]
+            grad = self.likelihood_function.dlik_df(self.data, f, extra_data=self.extra_data)[:, None]
             #Find K_i_f
             b = W_f + grad
 

GPy/likelihoods/likelihood_functions.py

@@ -159,10 +159,10 @@ class student_t(likelihood_function):
     d2ln p(yi|fi)_d2fifj
     """
     def __init__(self, deg_free, sigma=2):
+        #super(student_t, self).__init__()
         self.v = deg_free
         self.sigma = sigma
         self.log_concave = False
-        #super(student_t, self).__init__()
 
         self._set_params(np.asarray(sigma))
 
@@ -174,8 +174,6 @@ class student_t(likelihood_function):
 
     def _set_params(self, x):
         self.sigma = float(x)
-        #self.covariance_matrix = np.eye(self.N)*self._variance
-        #self.precision = 1./self._variance
 
     @property
     def variance(self, extra_data=None):
@@ -185,6 +183,8 @@ class student_t(likelihood_function):
         """link_function $\ln p(y|f)$
         $$\ln p(y_{i}|f_{i}) = \ln \Gamma(\frac{v+1}{2}) - \ln \Gamma(\frac{v}{2})\sqrt{v \pi}\sigma - \frac{v+1}{2}\ln (1 + \frac{1}{v}\left(\frac{y_{i} - f_{i}}{\sigma}\right)^2$$
 
+        For wolfram alpha import parts for derivative of sigma are -log(sqrt(v*pi)*s) -(1/2)*(v + 1)*log(1 + (1/v)*((y-f)/(s))^2))
+
         :y: data
         :f: latent variables f
         :extra_data: extra_data which is not used in student t distribution
@@ -198,17 +198,16 @@ class student_t(likelihood_function):
         e = y - f
         objective = (gammaln((self.v + 1) * 0.5)
                      - gammaln(self.v * 0.5)
-                     + np.log(self.sigma * np.sqrt(self.v * np.pi))
+                     - np.log(self.sigma * np.sqrt(self.v * np.pi))
-                     - (self.v + 1) * 0.5
+                     - (self.v + 1) * 0.5 * np.log(1 + ((e**2 / self.sigma**2) / self.v))
-                     * np.log(1 + ((e**2 / self.sigma**2) / self.v))
                     )
         return np.sum(objective)
 
-    def link_grad(self, y, f, extra_data=None):
+    def dlik_df(self, y, f, extra_data=None):
         """
         Gradient of the link function at y, given f w.r.t f
 
-        $$\frac{d}{df}p(y_{i}|f_{i}) = \frac{(v + 1)(y - f)}{v \sigma^{2} + (y_{i} - f_{i})^{2}}$$
+        $$\frac{dp(y_{i}|f_{i})}{df} = \frac{-(v+1)(f_{i}-y_{i})}{(f_{i}-y_{i})^{2} + \sigma^{2}v}$$
 
         :y: data
         :f: latent variables f
@@ -220,17 +219,17 @@ class student_t(likelihood_function):
         f = np.squeeze(f)
         assert y.shape == f.shape
         e = y - f
-        grad = ((self.v + 1) * e) / (self.v * (self.sigma**2) + (e**2))
+        grad = -((self.v + 1) * e) / (self.v * (self.sigma**2) + (e**2))
         return np.squeeze(grad)
 
-    def link_hess(self, y, f, extra_data=None):
+    def d2lik_d2f(self, y, f, extra_data=None):
         """
         Hessian at this point (if we are only looking at the link function not the prior) the hessian will be 0 unless i == j
         i.e. second derivative link_function at y given f f_j  w.r.t f and f_j
 
         Will return diagonal of hessian, since every where else it is 0
 
-        $$\frac{d^{2}p(y_{i}|f_{i})}{df^{2}} = \frac{(v + 1)(y - f)}{v \sigma^{2} + (y_{i} - f_{i})^{2}}$$
+        $$\frac{d^{2}p(y_{i}|f_{i})}{d^{3}f} = \frac{(v+1)((f_{i}-y_{i})^{2} - \sigma^{2}v)}{((f_{i}-y_{i})^{2} + \sigma^{2}v)^{2}}$$
 
         :y: data
         :f: latent variables f
@@ -245,54 +244,79 @@ class student_t(likelihood_function):
         hess = ((self.v + 1)*(e**2 - self.v*(self.sigma**2))) / ((((self.sigma**2)*self.v) + e**2)**2)
         return np.squeeze(hess)
 
-    def d3link(self, y, f, extra_data=None):
+    def d3lik_d3f(self, y, f, extra_data=None):
         """
         Third order derivative link_function (log-likelihood ) at y given f f_j w.r.t f and f_j
 
-        $$\frac{2(v+1)((y-f)^{3} - 3\sigma^{2}v(y-f))}{((y-f)^{2} + \sigma^{2}v)^{3}}$$
+        $$\frac{d^{3}p(y_{i}|f_{i})}{d^{3}f} = \frac{-2(v+1)((f_{i} - y_{i})^3 - 3(f_{i} - y_{i}) \sigma^{2} v))}{((f_{i} - y_{i}) + \sigma^{2} v)^3}$$
         """
         y = np.squeeze(y)
         f = np.squeeze(f)
         assert y.shape == f.shape
         e = y - f
-        d3link_d3f = (  (2*(self.v + 1)*(-1*e)*(e**2 - 3*(self.sigma**2)*self.v))
+        d3lik_d3f = ( -(2*(self.v + 1)*(e**3 - e*3*self.v*(self.sigma**2))) /
-                      / ((e**2 + (self.sigma**2)*self.v)**3)
+                       ((e**2 + (self.sigma**2)*self.v)**3)
                     )
-        return np.squeeze(d3link_d3f)
+        return np.squeeze(d3lik_d3f)
 
-    def link_hess_grad_std(self, y, f, extra_data=None):
+    def link_dstd(self, y, f, extra_data=None):
         """
-        Gradient of the hessian w.r.t sigma parameter (standard deviation)
+        Gradient of the likelihood (lik) w.r.t sigma parameter (standard deviation)
 
-        $$\frac{2\sigma v(v+1)(\sigma^{2}v - 3(f-y)^2)}{((f-y)^{2} + \sigma^{2}v)^{3}}
+        Terms relavent to derivatives wrt sigma are:
+        -log(sqrt(v*pi)*s) -(1/2)*(v + 1)*log(1 + (1/v)*((y-f)/(s))^2))
+
+        $$\frac{dp(y_{i}|f_{i})}{d\sigma} = -\frac{1}{\sigma} + \frac{(1+v)(y_{i}-f_{i})^2}{\sigma^3 v(1 + \frac{1}{v}(\frac{(y_{i} - f_{i})}{\sigma^2})^2)}$$
         """
         y = np.squeeze(y)
         f = np.squeeze(f)
         assert y.shape == f.shape
         e = y - f
-        hess_grad_sigma = (  (2*self.sigma*self.v*(self.v + 1)*((self.sigma**2)*self.v - 3*(e**2)))
+        dlik_dsigma = ( (1/self.sigma) -
-                           / ((e**2 + (self.sigma**2)*self.v)**3)
+                        ((1+self.v)*(e**2))/((self.sigma**3)*self.v*(1 + (e**2) / ((self.sigma**2)*self.v) ) )
                       )
-        return np.squeeze(hess_grad_sigma)
+        return np.squeeze(dlik_dsigma)
 
-    def link_grad_std(self, y, f, extra_data=None):
+    def dlik_df_dstd(self, y, f, extra_data=None):
         """
-        Gradient of the likelihood w.r.t sigma parameter (standard deviation)
+        Gradient of the dlik_df w.r.t sigma parameter (standard deviation)
 
-        $$\frac{-2\sigma(v+1)(y-f)}{(v\sigma^{2} + (y-f)^{2})^{2}}$$
+        $$\frac{d}{d\sigma}(\frac{dp(y_{i}|f_{i})}{df}) = \frac{2\sigma v(v + 1)(f-y)}{(f-y)^2 + \sigma^2 v)^2}$$
         """
         y = np.squeeze(y)
         f = np.squeeze(f)
         assert y.shape == f.shape
         e = y - f
-        grad_sigma = (  (-2*self.sigma*self.v*(self.v + 1)*e)
+        dlik_grad_dsigma = ((2*self.sigma*self.v*(self.v + 1)*e)
                             / ((self.v*(self.sigma**2) + e**2)**2)
                            )
-        return np.squeeze(grad_sigma)
+        return np.squeeze(dlik_grad_dsigma)
+
+    def d2lik_d2f_dstd(self, y, f, extra_data=None):
+        """
+        Gradient of the hessian (d2lik_d2f) w.r.t sigma parameter (standard deviation)
+
+        $$\frac{d}{d\sigma}(\frac{d^{2}p(y_{i}|f_{i})}{d^{2}f}) = \frac{(v + 1)((f-y)^2 - \sigma^2 v)}{((f-y)^2 + \sigma^2 v)}$$
+        """
+        y = np.squeeze(y)
+        f = np.squeeze(f)
+        assert y.shape == f.shape
+        e = y - f
+        dlik_hess_dsigma = ( ((v + 1)*(e**2 - (self.sigma**2)*self.v)) /
+                             ((e**2 + (self.sigma**2)*self.v)**2)
+                           )
+        return np.squeeze(dlik_hess_dsigma)
 
     def _gradients(self, y, f, extra_data=None):
-        return [self.link_grad_std(y, f, extra_data=extra_data),
+        derivs = ([self.link_dstd(y, f, extra_data=extra_data)],
-                self.link_hess_grad_std(y, f, extra_data=extra_data)] # list as we might learn many parameters
+                  [self.dlik_df_dstd(y, f, extra_data=extra_data)],
+                  [self.d2lik_d2f_dstd(y, f, extra_data=extra_data)]
+                 ) # lists as we might learn many parameters
+        # ensure we have gradients for every parameter we want to optimize
+        assert len(derivs[0]) == len(self._get_param_names())
+        assert len(derivs[1]) == len(self._get_param_names())
+        assert len(derivs[2]) == len(self._get_param_names())
+        return derivs
 
     def predictive_values(self, mu, var):
         """
@@ -412,7 +436,7 @@ class weibull_survival(likelihood_function):
         objective = v*(np.log(self.shape) + (self.shape - 1)*np.log(y) + f) - (y**self.shape)*np.exp(f)  # FIXME: CHECK THIS WITH BOOK, wheres scale?
         return np.sum(objective)
 
-    def link_grad(self, y, f, extra_data=None):
+    def dlik_df(self, y, f, extra_data=None):
         """
         Gradient of the link function at y, given f w.r.t f
 
@@ -432,7 +456,7 @@ class weibull_survival(likelihood_function):
         grad = v - (y**self.shape)*np.exp(f)
         return np.squeeze(grad)
 
-    def link_hess(self, y, f, extra_data=None):
+    def d2lik_d2f(self, y, f, extra_data=None):
         """
         Hessian at this point (if we are only looking at the link function not the prior) the hessian will be 0 unless i == j
         i.e. second derivative link_function at y given f f_j  w.r.t f and f_j

GPy/models/GP.py

@@ -147,7 +147,7 @@ class GP(model):
         if isinstance(self.likelihood, Laplace):
             dL_dthetaK_explicit = dL_dthetaK
             #Need to pass in a matrix of ones to get access to raw dK_dthetaK values without being chained
-            fake_dL_dKs = np.eye(self.dL_dK.shape[0]) #FIXME: Check this is right...
+            fake_dL_dKs = np.ones(self.dL_dK.shape) #FIXME: Check this is right...
             dK_dthetaK = self.kern.dK_dtheta(dL_dK=fake_dL_dKs, X=self.X)
 
             dL_dthetaK = self.likelihood._Kgradients(dL_d_K_Sigma=self.dL_dK, dK_dthetaK=dK_dthetaK)