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Let's imagine I have the following equation $y_t=f(x_t)+e_t$, where $f(x)$ follows a gaussian process, and $e_t\sim N(0,\Sigma)$.

How does one go about to learn the hyperparameters, i.e., $\Sigma$ and those of of the gaussian process, in a Bayesian way?

More specifically, how would we write the $p(Y|X, \theta)$, where $\theta$ are the hyperparameters of the model, and $Y=(y_1,...,y_N)$ and $X=(x_1,...,x_N)$?

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You could define a prior distribution on the hyperparameters and add the log-prior to the log-likelihood to obtain the log-posterior up to a constant.

You could then maximise the log-posterior just as you'd maximise the log-likelihood to obtain a MAP solution, or you could use MCMC methods to obtain samples from this log-posterior - you'd then have samples from the joint sampling distributions of the hyperparameters.

You'd then perhaps use the evidence (marginal likelihood) to compare different models/kernels, or if you're trying to fit more complex models, you could obtain lower bound approximations for the evidence and fine-tune parameters or model structures so that these lower bounds are maximised.

Is this roughly what you were looking for?

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  • $\begingroup$ Could you elaborate a bit more on the MCMC method. How would you write the likelihood? I would like to draw from $p(\theta|X,Y)\propto p(X,Y|\theta)p(\theta)$. My problem is the $p(X,Y|\theta)$, where we've marginalized the function $f$ out... if we had $p(X,f(X), Y|\theta)$ instead, I would know how to find it's expression. $\endgroup$ Oct 12 '18 at 20:40
  • $\begingroup$ I'm not sure I understand. $Y$ is completely determined by $X$ and $e_t$ in your example. In that example, it's extremely easy to write the likelihood, as X is also an MVN, you'd just be adding $\Sigma$ to the GP's covariance matrix. So, if $e_t \sim N(0, \sigma_n^2 \mathcal I)$, you could simply add the $\sigma_n^2 1(x = x')$ to your kernel $\endgroup$
    – adityar
    Oct 12 '18 at 21:36
  • $\begingroup$ Thanks for the comment, but 2 things I don't understand. 1) When we fix X, we're not fixing Y, since f(x) follows a GP. Could you write the likelihood of the model in my question in your answer? 2) Why do you say that X is MVN (matrix-valued Normal?) ? $\endgroup$ Oct 12 '18 at 21:43
  • $\begingroup$ Sorry, I meant $f(X)$ is a multivariate normal, as it's a GP. X is a constant variable right? So, both $f(x_t)$ and $e_t$ are MVNs, hence their sum is as well. Their sum is a GP as well. $\endgroup$
    – adityar
    Oct 12 '18 at 21:48

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