Can Gaussian Processes be defined on arbitrary sets? Can they model arbitrary functions?
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I am working with Gaussian Processes, but I am afraid I don't understand them very well. I was under the impression that a Gaussian Process could be defined on an arbitrary set. This would mean that you can take two finite sets, say, $A = {a, b, c, d}$, and $B = {e, f, g, h}$ and take the set of all functions between them. YOu should be able to define Gaussian processes that model functions between these two sets. Instead, it seems that the only sets you can use are $mathbb{R}^n$ for some $n$. Can the domain be an arbitrary set if the range is, say $mathbb{R}^m$ for some $m$?
For that matter, how would you define a random variable between $A$ and $B$? How do you compute a mean when there is no notion of the algebra of real numbers on the set $B$?
Is there a standard way to statistically model functions between arbitrary sets for which we can compute an entropy?
probability-theory
asked Nov 26 at 21:39
Ben Sprott
432312
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down vote
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I am working with Gaussian Processes, but I am afraid I don't understand them very well. I was under the impression that a Gaussian Process could be defined on an arbitrary set. This would mean that you can take two finite sets, say, $A = {a, b, c, d}$, and $B = {e, f, g, h}$ and take the set of all functions between them. YOu should be able to define Gaussian processes that model functions between these two sets. Instead, it seems that the only sets you can use are $mathbb{R}^n$ for some $n$. Can the domain be an arbitrary set if the range is, say $mathbb{R}^m$ for some $m$?
For that matter, how would you define a random variable between $A$ and $B$? How do you compute a mean when there is no notion of the algebra of real numbers on the set $B$?
Is there a standard way to statistically model functions between arbitrary sets for which we can compute an entropy?
probability-theory
asked Nov 26 at 21:39
Ben Sprott
432312
add a comment |
up vote
0
down vote
favorite
up vote
0
down vote
favorite
I am working with Gaussian Processes, but I am afraid I don't understand them very well. I was under the impression that a Gaussian Process could be defined on an arbitrary set. This would mean that you can take two finite sets, say, $A = {a, b, c, d}$, and $B = {e, f, g, h}$ and take the set of all functions between them. YOu should be able to define Gaussian processes that model functions between these two sets. Instead, it seems that the only sets you can use are $mathbb{R}^n$ for some $n$. Can the domain be an arbitrary set if the range is, say $mathbb{R}^m$ for some $m$?
For that matter, how would you define a random variable between $A$ and $B$? How do you compute a mean when there is no notion of the algebra of real numbers on the set $B$?
Is there a standard way to statistically model functions between arbitrary sets for which we can compute an entropy?
probability-theory
asked Nov 26 at 21:39
Ben Sprott
432312
I am working with Gaussian Processes, but I am afraid I don't understand them very well. I was under the impression that a Gaussian Process could be defined on an arbitrary set. This would mean that you can take two finite sets, say, $A = {a, b, c, d}$, and $B = {e, f, g, h}$ and take the set of all functions between them. YOu should be able to define Gaussian processes that model functions between these two sets. Instead, it seems that the only sets you can use are $mathbb{R}^n$ for some $n$. Can the domain be an arbitrary set if the range is, say $mathbb{R}^m$ for some $m$?
For that matter, how would you define a random variable between $A$ and $B$? How do you compute a mean when there is no notion of the algebra of real numbers on the set $B$?
Is there a standard way to statistically model functions between arbitrary sets for which we can compute an entropy?
probability-theory
probability-theory
asked Nov 26 at 21:39
Ben Sprott
432312
asked Nov 26 at 21:39
Ben Sprott
432312
asked Nov 26 at 21:39
Ben Sprott
432312
asked Nov 26 at 21:39
Ben Sprott
432312
asked Nov 26 at 21:39
Ben Sprott
432312
432312
add a comment |
add a comment |
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