Probabilistic functions, and the category Prob
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I have recently been guided towards a model of probabilistic functions. A probabilistic function which models functions of the form $f : X rightarrow Y$, is a map from $X$ into probability distributions over $Y$. It has been suggested that these maps are actually morphisms in a category $Prob$. This means that there is a standard way of composing them to mimic function composition. Furthermore, there should be functors from any concrete category $C$ into $Prob$, and there is a faithful functor that maps morphisms in $C$ into morphisms where the codomain distributions are zero everywhere except at one set element. I understand that this is related to the Giry monad, where we send a measurable set to the set of probability measures on that set. I am told this is related to Lawvere theories.
Can someone verify this, perhaps expound on it, and give me a reference? In this discipline, when/why do we talk about the category $Prob$ vs the probability monad?
probability category-theory
asked Dec 17 '18 at 19:22
Ben SprottBen Sprott
426312
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add a comment |
$begingroup$
I have recently been guided towards a model of probabilistic functions. A probabilistic function which models functions of the form $f : X rightarrow Y$, is a map from $X$ into probability distributions over $Y$. It has been suggested that these maps are actually morphisms in a category $Prob$. This means that there is a standard way of composing them to mimic function composition. Furthermore, there should be functors from any concrete category $C$ into $Prob$, and there is a faithful functor that maps morphisms in $C$ into morphisms where the codomain distributions are zero everywhere except at one set element. I understand that this is related to the Giry monad, where we send a measurable set to the set of probability measures on that set. I am told this is related to Lawvere theories.
Can someone verify this, perhaps expound on it, and give me a reference? In this discipline, when/why do we talk about the category $Prob$ vs the probability monad?
probability category-theory
asked Dec 17 '18 at 19:22
Ben SprottBen Sprott
426312
$endgroup$
3
$begingroup$
Maybe what you want is the Kleisli category of the Giry monad?
$endgroup$
– Daniel Schepler
Dec 17 '18 at 19:48
add a comment |
$begingroup$
I have recently been guided towards a model of probabilistic functions. A probabilistic function which models functions of the form $f : X rightarrow Y$, is a map from $X$ into probability distributions over $Y$. It has been suggested that these maps are actually morphisms in a category $Prob$. This means that there is a standard way of composing them to mimic function composition. Furthermore, there should be functors from any concrete category $C$ into $Prob$, and there is a faithful functor that maps morphisms in $C$ into morphisms where the codomain distributions are zero everywhere except at one set element. I understand that this is related to the Giry monad, where we send a measurable set to the set of probability measures on that set. I am told this is related to Lawvere theories.
Can someone verify this, perhaps expound on it, and give me a reference? In this discipline, when/why do we talk about the category $Prob$ vs the probability monad?
probability category-theory
asked Dec 17 '18 at 19:22
Ben SprottBen Sprott
426312
$endgroup$
I have recently been guided towards a model of probabilistic functions. A probabilistic function which models functions of the form $f : X rightarrow Y$, is a map from $X$ into probability distributions over $Y$. It has been suggested that these maps are actually morphisms in a category $Prob$. This means that there is a standard way of composing them to mimic function composition. Furthermore, there should be functors from any concrete category $C$ into $Prob$, and there is a faithful functor that maps morphisms in $C$ into morphisms where the codomain distributions are zero everywhere except at one set element. I understand that this is related to the Giry monad, where we send a measurable set to the set of probability measures on that set. I am told this is related to Lawvere theories.
Can someone verify this, perhaps expound on it, and give me a reference? In this discipline, when/why do we talk about the category $Prob$ vs the probability monad?
probability category-theory
probability category-theory
asked Dec 17 '18 at 19:22
Ben SprottBen Sprott
426312
asked Dec 17 '18 at 19:22
Ben SprottBen Sprott
426312
edited Dec 17 '18 at 19:37
Ben Sprott
asked Dec 17 '18 at 19:22
Ben SprottBen Sprott
426312
asked Dec 17 '18 at 19:22
Ben SprottBen Sprott
426312
asked Dec 17 '18 at 19:22
Ben SprottBen Sprott
426312
426312
3
$begingroup$
Maybe what you want is the Kleisli category of the Giry monad?
$endgroup$
– Daniel Schepler
Dec 17 '18 at 19:48
add a comment |
3
$begingroup$
Maybe what you want is the Kleisli category of the Giry monad?
$endgroup$
– Daniel Schepler
Dec 17 '18 at 19:48
3
3
$begingroup$
Maybe what you want is the Kleisli category of the Giry monad?
$endgroup$
– Daniel Schepler
Dec 17 '18 at 19:48
$begingroup$
Maybe what you want is the Kleisli category of the Giry monad?
$endgroup$
– Daniel Schepler
Dec 17 '18 at 19:48
add a comment |
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3
$begingroup$
Maybe what you want is the Kleisli category of the Giry monad?
$endgroup$
– Daniel Schepler
Dec 17 '18 at 19:48