diff --git a/GPy/likelihoods/student_t.py b/GPy/likelihoods/student_t.py
index 04ad93e6..79745ff6 100644
--- a/GPy/likelihoods/student_t.py
+++ b/GPy/likelihoods/student_t.py
@@ -36,9 +36,6 @@ class StudentT(Likelihood):
 
         self.log_concave = False
 
-    #def parameters_changed(self):
-        #self.variance = (self.v / float(self.v - 2)) * self.sigma2
-
     def update_gradients(self, grads):
         """
         Pull out the gradients, be careful as the order must match the order
@@ -231,8 +228,8 @@ class StudentT(Likelihood):
         df = float(self.v[:])
         s2 = float(self.sigma2[:])
         dlogpdf_dv =  0.5*digamma(0.5*(df+1)) - 0.5*digamma(0.5*df) - 1.0/(2*df)
-        dlogpdf_dv += (1.0/(2*df))*(df+1)*e/(e2 + s2*df)
-        dlogpdf_dv -= np.log(1 + e2/(s2*df))
+        dlogpdf_dv += 0.5*(df+1)*e2/(df*(e2 + s2*df))
+        dlogpdf_dv -= 0.5*np.log1p(e2/(s2*df))
         return dlogpdf_dv
 
     def dlogpdf_dlink_dv(self, inv_link_f, y, Y_metadata=None):
@@ -248,9 +245,8 @@ class StudentT(Likelihood):
         e2 = np.square(e)
         df = float(self.v[:])
         s2 = float(self.sigma2[:])
-        #derivative of hess = ((self.v + 1)*(e**2 - self.v*self.sigma2)) / ((self.sigma2*self.v + e**2)**2)
         e2_s2v = e**2 + s2*df
-        d2logpdf_df2_dv = (e2 - s2*df - s2*(df + 1))/e2_s2v**2 - 2*s2*(df+1)*(e2 - s2*df)/e2_s2v
+        d2logpdf_df2_dv = (-s2*(df+1) + e2 - s2*df)/e2_s2v**2 - 2*s2*(df+1)*(e2 - s2*df)/e2_s2v**3
         return d2logpdf_df2_dv
 
     def dlogpdf_link_dtheta(self, f, y, Y_metadata=None):
diff --git a/GPy/testing/likelihood_tests.py b/GPy/testing/likelihood_tests.py
index 7fa5886f..bef90835 100644
--- a/GPy/testing/likelihood_tests.py
+++ b/GPy/testing/likelihood_tests.py
@@ -93,6 +93,9 @@ def dparam_checkgrad(func, dfunc, params, params_names, args, constraints=None,
             if not grad.checkgrad(verbose=True):
                 gradchecking = False
 
+            if not grad.checkgrad(verbose=True):
+                gradchecking = False
+
     return gradchecking
 
 
@@ -116,6 +119,7 @@ class TestNoiseModels(object):
         self.integer_Y = np.where(tmp > 0, tmp, 0)
 
         self.var = 0.2
+        self.deg_free = 4.0
 
         #Make a bigger step as lower bound can be quite curved
         self.step = 1e-4
@@ -135,56 +139,56 @@ class TestNoiseModels(object):
                 }
         """
         self.noise_models = {"Student_t_default": {
-            "model": GPy.likelihoods.StudentT(deg_free=5, sigma2=self.var),
+            "model": GPy.likelihoods.StudentT(deg_free=self.deg_free, sigma2=self.var),
             "grad_params": {
-                "names": [".*t_scale2"],
-                "vals": [self.var],
-                "constraints": [(".*t_scale2", self.constrain_positive), (".*deg_free", self.constrain_fixed)]
+                "names": [".*t_scale2", ".*deg_free"],
+                "vals": [self.var, self.deg_free],
+                "constraints": [(".*t_scale2", self.constrain_positive), (".*deg_free", self.constrain_positive)]
             },
             "laplace": True
             },
             "Student_t_1_var": {
-                "model": GPy.likelihoods.StudentT(deg_free=5, sigma2=self.var),
+                "model": GPy.likelihoods.StudentT(deg_free=self.deg_free, sigma2=self.var),
                 "grad_params": {
-                    "names": [".*t_scale2"],
-                    "vals": [1.0],
-                    "constraints": [(".*t_scale2", self.constrain_positive), (".*deg_free", self.constrain_fixed)]
+                    "names": [".*t_scale2", ".*deg_free"],
+                    "vals": [1.0, 8.0],
+                    "constraints": [(".*t_scale2", self.constrain_positive), (".*deg_free", self.constrain_positive)]
                 },
                 "laplace": True
             },
             "Student_t_small_deg_free": {
                 "model": GPy.likelihoods.StudentT(deg_free=1.5, sigma2=self.var),
                 "grad_params": {
-                    "names": [".*t_scale2"],
-                    "vals": [self.var],
-                    "constraints": [(".*t_scale2", self.constrain_positive), (".*deg_free", self.constrain_fixed)]
+                    "names": [".*t_scale2", ".*deg_free"],
+                    "vals": [self.var, 1.5],
+                    "constraints": [(".*t_scale2", self.constrain_positive), (".*deg_free", self.constrain_positive)]
                 },
                 "laplace": True
             },
             "Student_t_small_var": {
-                "model": GPy.likelihoods.StudentT(deg_free=5, sigma2=self.var),
+                "model": GPy.likelihoods.StudentT(deg_free=self.deg_free, sigma2=self.var),
                 "grad_params": {
-                    "names": [".*t_scale2"],
-                    "vals": [0.001],
-                    "constraints": [(".*t_scale2", self.constrain_positive), (".*deg_free", self.constrain_fixed)]
+                    "names": [".*t_scale2", ".*deg_free"],
+                    "vals": [0.001, self.deg_free],
+                    "constraints": [(".*t_scale2", self.constrain_positive), (".*deg_free", self.constrain_positive)]
                 },
                 "laplace": True
             },
             "Student_t_large_var": {
-                "model": GPy.likelihoods.StudentT(deg_free=5, sigma2=self.var),
+                "model": GPy.likelihoods.StudentT(deg_free=self.deg_free, sigma2=self.var),
                 "grad_params": {
-                    "names": [".*t_scale2"],
-                    "vals": [10.0],
-                    "constraints": [(".*t_scale2", self.constrain_positive), (".*deg_free", self.constrain_fixed)]
+                    "names": [".*t_scale2", ".*deg_free"],
+                    "vals": [10.0, self.deg_free],
+                    "constraints": [(".*t_scale2", self.constrain_positive), (".*deg_free", self.constrain_positive)]
                 },
                 "laplace": True
             },
             "Student_t_approx_gauss": {
                 "model": GPy.likelihoods.StudentT(deg_free=1000, sigma2=self.var),
                 "grad_params": {
-                    "names": [".*t_scale2"],
-                    "vals": [self.var],
-                    "constraints": [(".*t_scale2", self.constrain_positive), (".*deg_free", self.constrain_fixed)]
+                    "names": [".*t_scale2", ".*deg_free"],
+                    "vals": [self.var, 1000],
+                    "constraints": [(".*t_scale2", self.constrain_positive), (".*deg_free", self.constrain_positive)]
                 },
                 "laplace": True
             },