Added some comments

python/likelihoods/likelihood_function.py

@@ -5,6 +5,9 @@ class student_t(GPy.likelihoods.likelihood_function):
     """Student t likelihood distribution
     For nomanclature see Bayesian Data Analysis 2003 p576
 
+    $$\ln(\frac{\Gamma(\frac{(v+1)}{2})}{\Gamma(\sqrt(v \pi \Gamma(\frac{v}{2}))})+ \ln(1+\frac{(y_i-f_i)^2}{\sigma v})^{-\frac{(v+1)}{2}}$$
+    TODO:Double check this
+
     Laplace:
     Needs functions to calculate
     ln p(yi|fi)
@@ -17,6 +20,8 @@ class student_t(GPy.likelihoods.likelihood_function):
 
     def link_function(self, y_i, f_i):
         """link_function $\ln p(y_i|f_i)$
+        $$\ln \Gamma(\frac{v+1}{2}) - \ln \Gamma(\frac{v}{2}) - \ln \frac{v \pi \sigma}{2} - \frac{v+1}{2}\ln (1 + \frac{(y_{i} - f_{i})^{2}}{v\sigma})$$
+        TODO: Double check this
 
         :y_i: datum number i
         :f_i: latent variable f_i
@@ -24,11 +29,15 @@ class student_t(GPy.likelihoods.likelihood_function):
 
         """
         e = y_i - f_i
-        return gammaln((v+1)*0.5) - gammaln(v*0.5) - np.ln(v*np.pi*sigma)*0.5 - (v+1)*0.5*np.ln(1 + ((e/sigma)**2)/v)
+        return gammaln((v+1)*0.5) - gammaln(v*0.5) - np.ln(v*np.pi*sigma)*0.5 - (v+1)*0.5*np.ln(1 + ((e/sigma)**2)/v) #Check the /v!
 
     def link_grad(self, y_i, f_i):
         """gradient of the link function at y_i, given f_i w.r.t f_i
 
+        derivative of log((gamma((v+1)/2)/gamma(sqrt(v*pi*gamma(v/2))))*(1+(t^2)/(a*v))^((-(v+1))/2)) with respect to t
+        $$\frac{(y_i - f_i)(v + 1)}{\sigma v (y_{i} - f_{i})^{2}}$$
+        TODO: Double check this
+
         :y_i: datum number i
         :f_i: latent variable f_i
         :returns: float(gradient of likelihood evaluated at this point)
@@ -40,6 +49,8 @@ class student_t(GPy.likelihoods.likelihood_function):
         """hessian at this point (the hessian will be 0 unless i == j)
         i.e. second derivative w.r.t f_i and f_j
 
+        second derivative of
+
         :y_i: @todo
         :f_i: @todo
         :f_j: @todo