What is a probability monad? Paolo Perrone Massachusetts Institute - PowerPoint PPT Presentation

What is a probability monad? Paolo Perrone Massachusetts Institute of Technology (MIT) Categorical Probability 2020 Tutorial video

Monads as extensions Definition: Let C be a category. A monad on C consists of: • A functor T : C → C; • A natural transformation η : id C ⇒ T called unit ; • A natural transformation µ : TT ⇒ T called composition ; such that the following diagrams commute: T η η T T µ T TT T TT TTT TT µ T µ µ µ id id µ T T TT T 2 of 27

Monads as extensions Idea: A monad is like a consistent way of extending spaces to include generalized elements and generalized functions of a specific kind . 3 of 27

Monads as extensions Idea: A monad is like a consistent way of extending spaces to include generalized elements and generalized functions of a specific kind . A functor T : C → C consists of: 3 of 27

Monads as extensions Idea: A monad is like a consistent way of extending spaces to include generalized elements and generalized functions of a specific kind . A functor T : C → C consists of: 1. To each space X , an “extended” space TX . 3 of 27

Monads as extensions Idea: A monad is like a consistent way of extending spaces to include generalized elements and generalized functions of a specific kind . A functor T : C → C consists of: 1. To each space X , an “extended” space TX . 2. Given f : X → Y , an “extension” Tf : TX → TY . 3 of 27

Monads as extensions Idea: A monad is like a consistent way of extending spaces to include generalized elements and generalized functions of a specific kind . A functor T : C → C consists of: 1. To each space X , an “extended” space TX . 2. Given f : X → Y , an “extension” Tf : TX → TY . x 1 x 1 x 1 x 2 X TX x 2 x 2 x 2 x 3 x 1 x 2 x 3 x 3 x 3 x 3 x 1 3 of 27

Monads as extensions Idea: A monad is like a consistent way of extending spaces to include generalized elements and generalized functions of a specific kind . A functor T : C → C consists of: 1. To each space X , an “extended” space TX . 2. Given f : X → Y , an “extension” Tf : TX → TY . X Y 3 of 27

Monads as extensions Idea: A monad is like a consistent way of extending spaces to include generalized elements and generalized functions of a specific kind . A functor T : C → C consists of: 1. To each space X , an “extended” space TX . 2. Given f : X → Y , an “extension” Tf : TX → TY . X Y 3 of 27

Monads as extensions Idea: A monad is like a consistent way of extending spaces to include generalized elements and generalized functions of a specific kind . A functor T : C → C consists of: 1. To each space X , an “extended” space TX . 2. Given f : X → Y , an “extension” Tf : TX → TY . X Y 3 of 27

Monads as extensions x 1 x 1 x 1 x 2 X TX x 2 x 2 x 2 x 3 x 1 x 2 x 3 x 3 x 3 x 3 x 1 4 of 27

Monads as extensions x 1 x 1 x 1 x 2 X TX x 2 x 2 x 2 x 3 x 1 x 2 x 3 x 3 x 3 x 3 x 1 4 of 27

Monads as extensions x 1 x 1 x 1 x 2 X TX x 2 x 2 x 2 x 3 x 1 x 2 x 3 x 3 x 3 x 3 x 1 A natural transformation η : id C ⇒ T consists of: 1. To each X a map η X : X → TX , usually monic. 4 of 27

Monads as extensions x 1 x 1 x 1 x 2 X TX x 2 x 2 x 2 x 3 x 1 x 2 x 3 x 3 x 3 x 3 x 1 A natural transformation η : id C ⇒ T consists of: 1. To each X a map η X : X → TX , usually monic. 2. This diagram must commute: f X Y η X η Y Tf TX TY 4 of 27

Monads as extensions A natural transformation µ : TT ⇒ T , is: 1. For each X a map µ X : TTX → TX ; 2. Again a naturality diagram as before. 5 of 27

Monads as extensions A natural transformation µ : TT ⇒ T , is: 1. For each X a map µ X : TTX → TX ; 2. Again a naturality diagram as before. 5 of 27

Monads as extensions A natural transformation µ : TT ⇒ T , is: 1. For each X a map µ X : TTX → TX ; 2. Again a naturality diagram as before. 5 of 27

Monads as extensions Definition: Let T be a monad on C. A Kleisli morphism from X to Y is a morphism X → TY . 6 of 27

Monads as extensions Definition: Let T be a monad on C. A Kleisli morphism from X to Y is a morphism X → TY . X Y 6 of 27

Monads as extensions Definition: Given Kleisli morphisms k : X → TY and h : Y → TZ , their Kleisli composition is the morphism h ◦ kl k given by: k Th µ X TY TTZ TZ 7 of 27

Monads as extensions Definition: Given Kleisli morphisms k : X → TY and h : Y → TZ , their Kleisli composition is the morphism h ◦ kl k given by: k Th µ X TY TTZ TZ k X Y Z 7 of 27

Monads as extensions Definition: Given Kleisli morphisms k : X → TY and h : Y → TZ , their Kleisli composition is the morphism h ◦ kl k given by: k Th µ X TY TTZ TZ k h X Y Z 7 of 27

Monads as extensions Definition: Given Kleisli morphisms k : X → TY and h : Y → TZ , their Kleisli composition is the morphism h ◦ kl k given by: k Th µ X TY TTZ TZ k Th X Y Z 7 of 27

Monads as extensions Definition: Given Kleisli morphisms k : X → TY and h : Y → TZ , their Kleisli composition is the morphism h ◦ kl k given by: k Th µ X TY TTZ TZ k µ ◦ Th X Y Z 7 of 27

Monads as extensions Definition: Given Kleisli morphisms k : X → TY and h : Y → TZ , their Kleisli composition is the morphism h ◦ kl k given by: k Th µ X TY TTZ TZ X Y Z 7 of 27

Monads as extensions Exercise: Prove that Kleisli morphisms form a category thanks to the commutativity of these diagrams: T η η T T µ TX TTX TX TTX TTTX TTX µ T µ µ µ µ TX TX TTX TX where the identity morphisms of the Kleisli category are given by the units η : X → TX . 8 of 27

Probability monads Idea [Giry, 1982]: Spaces of “random elements” generalizing usual elements. 9 of 27

Probability monads Idea [Giry, 1982]: Spaces of “random elements” generalizing usual elements. • Base category C 9 of 27

Probability monads Idea [Giry, 1982]: Spaces of “random elements” generalizing usual elements. • Base category C X 9 of 27

Probability monads Idea [Giry, 1982]: Spaces of “random elements” generalizing usual elements. • Base category C • Functor X �→ PX PX X 9 of 27

Probability monads Idea [Giry, 1982]: Spaces of “random elements” generalizing usual elements. • Base category C • Functor X �→ PX • Unit δ : X → PX PX X 9 of 27

Probability monads Idea [Giry, 1982]: Spaces of “random elements” generalizing usual elements. • Base category C • Functor X �→ PX • Unit δ : X → PX 1/2 1/2 9 of 27

Probability monads Idea [Giry, 1982]: Spaces of “random elements” generalizing usual elements. • Base category C • Functor X �→ PX • Unit δ : X → PX 1/2 1/2 1/2 1/2 9 of 27

Probability monads Idea [Giry, 1982]: Spaces of “random elements” generalizing usual elements. ? 1/2 1/2 • Base category C • Functor X �→ PX • Unit δ : X → PX 1/2 1/2 1/2 1/2 9 of 27

Probability monads Idea [Giry, 1982]: Spaces of “random elements” generalizing usual elements. ? ? 1/2 1/2 • Base category C 3/4 1/4 • Functor X �→ PX • Unit δ : X → PX 1/2 1/2 1/2 1/2 9 of 27

Probability monads Idea [Giry, 1982]: Spaces of “random elements” generalizing usual elements. ? ? 1/2 1/2 • Base category C 3/4 1/4 • Functor X �→ PX • Unit δ : X → PX 1/2 1/2 1/2 1/2 • Composition E : PPX → PX PX PPX 9 of 27

Probability monads A Kleisli morphism from X to Y is a morphism X → PY . We can interpret this as a “random function” or “random transition”. X Y 10 of 27

Probability monads Given Kleisli morphisms k : X → PY and h : Y → PZ , their Kleisli composition is the morphism h ◦ kl k given by: k Ph E X PY PPZ PZ X Y Z 11 of 27

Probability monads Given Kleisli morphisms k : X → PY and h : Y → PZ , their Kleisli composition is the morphism h ◦ kl k given by: k Ph E X PY PPZ PZ X Y Z 11 of 27

Probability monads Given Kleisli morphisms k : X → PY and h : Y → PZ , their Kleisli composition is the morphism h ◦ kl k given by: k Ph E X PY PPZ PZ X Y Z 11 of 27

Probability monads Given Kleisli morphisms k : X → PY and h : Y → PZ , their Kleisli composition is the morphism h ◦ kl k given by: k Ph E X PY PPZ PZ X Y Z 11 of 27

Probability monads Given Kleisli morphisms k : X → PY and h : Y → PZ , their Kleisli composition is the morphism h ◦ kl k given by: k Ph E X PY PPZ PZ X Y Z 11 of 27

Probability monads Given Kleisli morphisms k : X → PY and h : Y → PZ , their Kleisli composition is the morphism h ◦ kl k given by: k Ph E X PY PPZ PZ X Y Z 11 of 27

The distribution monad on Set Definition: Let X be a set. A f.s. distribution on X is a function p : X → [0 , 1] such that • It is nonzero for finitely many x ∈ X ; • � x ∈ X p ( x ) = 1. We denote by DX the set of f.s. distributions on X . 12 of 27

The distribution monad on Set Definition: Let X be a set. A f.s. distribution on X is a function p : X → [0 , 1] such that • It is nonzero for finitely many x ∈ X ; • � x ∈ X p ( x ) = 1. We denote by DX the set of f.s. distributions on X . X 12 of 27