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This paper involves with asymmetric Kernels. They claim that this arises due to local parameters. But this is not really true. They induce a particular asymmetric structure in the Kernel yet still call these "Gaussian processes".

What is an interpretation of this that makes sense? It isn't really a Gaussian process any more: it seems like one interpretation is that they are inducing some causal graph (directed instead of undirected random field) that gives rise to this asymmetric structure. This is just a guess.

Is there any better way to see how this is still somehow a "Gaussian Process" even thought the kernel (covariance function) is not symmetric?

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  • $\begingroup$ See here for related material stats.stackexchange.com/questions/375035/… $\endgroup$
    – safetyduck
    Oct 1, 2021 at 12:22
  • $\begingroup$ This asymmetry is not related to causality. $\endgroup$ Oct 1, 2021 at 12:58
  • $\begingroup$ @MehmetSüzen what do you mean not related? I am looking for a model with Gaussian innovations that gives rise to this structure in the condition probabilities. I thought of a dag causal structure as I think any simple random Gaussian field should have symmetric kernel? $\endgroup$
    – safetyduck
    Oct 1, 2021 at 13:57
  • $\begingroup$ Definitely, you are absolutely right. DAGs are promising in building causal structures. I was only referring to this particular paper that they did not address causality explicitly. $\endgroup$ Oct 1, 2021 at 16:37
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    $\begingroup$ Interesting. True, this is probably not a vanilla GPR anymore. $\endgroup$ Oct 1, 2021 at 16:42

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Not an expert, but to my understanding their approach would break the PSD constraint on the corresponding covariance matrix. They don't address this in their paper at all. Like you mentioned, they call it a GP, but all they are using from the GP is the weighted sum portion that pertains to the mean. They ignore the covariance completely.

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