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In MCMC sampling methods, a transition kernel, as found in Metropolis(/Hastings) algorithm, is the comparison of the likelihood of the current position and the likelihood of the proposed position. However, in support vector machines and gaussian processes, the kernel is an operation defining the similarity of any two points.

In both cases, these definitions might be too simplistic. However, they very loosely are related in that they compare points with one another in support of some decision. Is this the most basic, valid definition of a kernel? Is there a better definition that encompasses any kernel in mathematics?

jbuddy_13
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    Short answer: no. "Kernel" is used in myriad ways in completely different mathematical settings. There is no overarching mathematical concept. You can discover that by researching some of the definitions offered by Wikipedia at https://en.wikipedia.org/wiki/Kernel. Perhaps one could argue that "kernel" is a [radial category](https://en.wikipedia.org/wiki/Cognitive_semantics#Categorization_and_cognition) whose central concept is the kernel of a nut. – whuber Mar 24 '21 at 18:24
  • @whuber, cynically, it sounds that this is the "John" or "Mike" of mathematics. When you make something up and lack creativity, call it a kernel. – jbuddy_13 Mar 24 '21 at 18:26
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    Almost, but not quite: check out the link I added to an intro to cognitive prototype theory. – whuber Mar 24 '21 at 18:27

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