# Choice of Gaussian process in non-parametric regression

I have been trying to understand non-parametric regression using Gaussian processes (GP), which are used to represent prior distributions over the space of functions. The linear model considered is $$\mathbf{y}_i = f(\mathbf{x}_i) + \mathbf{\epsilon}\\=f_i +\mathbf{\epsilon}$$ where $$\mathbf{y}_i$$ is the observations, $$f_i$$ is the output of the function at input location $$\mathbf{x}_i$$ and $$\mathbf{\epsilon}$$ is the additive Gaussian noise. The folowing GP is chosen in this case $$p(f|\mathbf{X},\mathbf{\theta}) = \mathcal{N}(\mathbf{0},k(\mathbf{X},\mathbf{X})).$$ where, $$k(\cdot;\cdot)$$ is the covariance function and $$\mathbf{\theta}$$ is the hyper-parameters.

I would appreciate if someone can shed light on the choice of this prior and what it actually does? Also, what is the approach to formulating the joint likelihood $$p(\mathbf{Y},\mathbf{X},f,\mathbf{\theta})$$ of the entire model.

There are various ways to choose the covariance kernel $$k(\cdot, \cdot; \theta)$$ and the hyperparameters $$\theta$$. Hence, the question per se is probably a bit too broad to be answered reasonably. However, I figured, I could give you an example of a class of covariance kernels that appears to be popular in Gaussian process regression and explain the influence of the hyper parameters for those.
Matérn class covariance kernels are given by $$k(x_1,x_2;\ell, \nu, \sigma) = \sigma^2\frac{2^{1-\nu}}{\Gamma(\nu)}\Bigg(\sqrt{2\nu}\frac{|x_1-x_2|}{\ell}\Bigg)^\nu K_\nu\Bigg(\sqrt{2\nu}\frac{|x_1-x_2|}{\ell}\Bigg),$$ where $$K_\nu$$ is the modified Bessel function of the second kind. Hence, the hyper parameter $$\theta := (\ell, \nu, \sigma)$$.
$$\ell$$ is the correlation length. If $$\ell$$ is small, the process at two points far apart is barely correlated. If $$\ell$$ is large, the process at two points far apart will still be (highly) correlated. Hence, if you assume that the function has a lot of variation in a small area of the domain, the correlation length should be rather short; otherwise long.
$$\nu$$ is the smoothness parameter. Samples from the prior will be $$ceil(\nu)-1$$ times differentiable. Hence, if you know that the function you try to determine is only continuous, but not differentiable, you should choose $$\nu < 1$$ and larger, if you assume more regularity. $$\nu = 1/2$$ corresponds to the exponential covariance, $$\nu = \infty$$ is the Gaussian covariance and produces samples that are infinitely often differentiable.
Each point of the Gaussian process is a normally distributed random variable. Under the prior, it has distribution $$\mathrm{N}(0, \sigma^2)$$. Hence, the standard deviation $$\sigma$$ controls the pointwise variance.
Concerning the joint likelihood. I am not really sure, what you mean. In my understanding the likelihood would be something like $$p(Y|f)$$. This is a function that relates the likelihood of the data to the parameter in the model that is supposed to be estimated. In this case, the parameter is the function $$f$$. If $$\epsilon \sim \mathrm{N}(0, \gamma^2)$$, the likelihood is $$p(Y|f) = \exp\left(-\frac{1}{2 \epsilon^2} \sum_{i=1}^I (y_i - f(x_i))^2 \right).$$ The other parameters are hyperparameters of the prior and do not need to appear in the likelihood in this particular case.