What is the meaning of the explanations and tables in the Gaussian process regression model in Python GPy?

Question

To build the model, I input x,y,kernel and got the model

m = GPy.models.GPRegression(x, y, kernel)

then I did

print(m)

And got the description of model and following table

Model: GP regression  
Objective: 2077.2427462362075  
Number of Parameters: 4  
Number of Optimization Parameters: 4  
Updates: True

|      GP_regression      |         value | constraints | priors |
|     rbf.variance        | 333.688148476 |     +ve     |        |
|     rbf.lengthscale     |          (2,) |     +ve     |        |
| Gausian_noise.variance  |           1.0 |     +ve     |        |

What does the table above mean?

And what is Model, Objective, etc.?

I did a lot of research and didn't find any details on it.

Answer 1

1. The table happens to be an organized view of the hyperparameters optimized. In your example, model m has 4 hyperparameters, i.e. one rbf variance, two lengthscales, (this means your x is 2 dimensional), and a noise variance.

m.parameter_names() # will give you the names of the hyperparameters, and this will be of size 4.

m.param_array # will give you the same values in a numpy array.

2. The objective here is the marginal log likelihood (MLL) or log marginal likelihood equation 2.30. The negative of this can be seen with, m.log_likelihood() # For GPy 1.10.0 .