EED. Motivation The Giry Monad Basing Markov Transition Systems on the Giry Monad Smooth Equiva- lence Relations Congruences Ernst-Erich Doberkat Factoring Chair for Software Technology Some Technische Universität Dortmund Conclu- ding http://ls10-www.cs.uni-dortmund.de Remarks References 1/1/
Overview EED. 1 Motivation Motivation The Giry Monad 2 The Giry Monad Smooth Equiva- lence Relations 3 Smooth Equivalence Relations Congruences Factoring 4 Congruences Some Conclu- ding Remarks 5 Factoring References 6 Some Concluding Remarks 7 References 2/2/
Motivation EED. You are interested in the behavior of coalitions, not in the behavior of Motivation individual objects (Economics, Biology). You model the system through a The Giry Monad stochastic model (e. g., a Markov transition system). Then it is sometimes Smooth preferable not to compare the states of a modelling system but rather Equiva- lence distributions over these states. Relations Some questions come up Congruences How do you model morphisms? Factoring Some How is equivalent behavior modelled? Conclu- ding This is what will be done: Remarks References A brief introduction to the Giry monad as the mechanism underlying Markov transition systems. A discussion of moving equivalence relations around. A look into morphisms and congruences, including factoring. 3/3/Motivation
The Giry Monad The space of all subprobabilities EED. Measurable Space Measurable Map Motivation A measurable space ( X , A ) is a set Let ( Y , B ) be another measurable The Giry X with a σ -algebra A — a Boolean space. A map f : X → Y is Monad A - B -measurable iff f − 1 [ B ] ∈ A for algebra of subsets of X which is Smooth Equiva- closed under countable unions. all B ∈ B . lence Relations Congruences Measurable spaces with measurable maps form a category Meas . Factoring Some Conclu- Subprobability ding Remarks A subprobability µ on A is a map µ : A → [ 0 , 1 ] with µ ( ∅ ) = 0 such that References µ ( S n ∈ N A n ) = P n ∈ N A n , provided ( A n ) n ∈ N ⊆ A is mutually disjoint. S ( X , A ) The set S ( X , A ) of all subprobabilities is made into a measurable space: take as a σ -algebra A • the smallest σ -algebra which makes the evaluations ev A : µ �→ µ ( A ) measurable. 4/4/The Giry Monad
The Giry Monad S as an endofunctor EED. S : Meas → Meas Easy Motivation The Giry What about morphisms? Let S ( f ) : S ( X , A ) → S ( Y , B ) Monad is A • - B • -measurable. f : ( X , A ) → ( Y , B ) be measurable. Then Smooth f − 1 [ B ] Equiva- ` ´ S ( f ) ( µ ) : B ∋ B �→ µ . lence Relations Congruences Proposition Factoring S is an endofunctor on the category of measurable spaces. Some Conclu- ding Remarks Remark References S is also an endofunctor also on the subcategory of Polish spaces or the subcategory of analytic spaces. Well ... But I promised you a monad. 5/5/The Giry Monad
The Giry Monad Kleisli Tripel EED. Recall Motivation A Kleisli triple ( F , η, − ∗ ) on a category C has The Giry Monad 1 a map F : Obj ( C ) → Obj ( C ) , Smooth Equiva- 2 for each object A a map η A : A → F ( A ) , lence Relations 3 for each morphism f : A → F ( B ) a morphism f ∗ : F ( A ) → F ( B ) Congruences that satisfy these laws Factoring Some A = id F ( A ) , f ∗ ◦ η A = f , g ∗ ◦ f ∗ = ( g ∗ ◦ f ) ∗ η ∗ Conclu- ding Remarks References ( F , e , m ) • • Put m A := id ∗ F ( A ) and e A := η A . One shows that m : F 2 → F and e : 1 → F l C satisfy the usual laws for a monad. 6/6/The Giry Monad
The Giry Monad EED. Motivation The Giry Monad S ? Smooth Equiva- Put for measurable K : ( X , A ) → S ( Y , B ) lence Relations Congruences Z K ∗ ( µ )( B ) := K ( x )( B ) µ ( dx ) . Factoring X Some Conclu- and ding ( Remarks 1 , x ∈ D , η A ( x )( D ) := References 0 , otherwise . 7/7/The Giry Monad
The Giry Monad EED. Proposition ( S , η, − ∗ ) is a Kleisli tripel. The corresponding monad is the Giry monad. Motivation The Giry Monad Kleisli Morphisms Smooth Equiva- A stochastic relation K : ( X , A ) � ( Y , B ) is a A - B • -measurable map. lence Relations Stochastic relations are just the morphisms in the Kleisli category associated Congruences with this monad. Kleisli composition L ∗ K for K : ( X , A ) � ( Y , B ) and Factoring L : ( Y , B ) � ( Z , C ) is given through Some Conclu- ding Z ` L ∗ K ´ Remarks ( x )( D ) = L ( y )( D ) K ( x )( dy ) Y References Remark on Language In Probability Theory stochastic relations are referred to as sub Markov kernels; the Kleisli product of stochastic relations is called there the convolution of kernels. 8/8/The Giry Monad
Smooth Equivalence Relations From now on EED. X is assumed to be a Polish space with the Borel sets B ( X ) as the σ -algebra; Motivation B ( X ) is the smallest σ -algebra which contains the open sets of X . The Giry Monad Smooth Countably Generated Equivalence Equiva- lence Relations An equivalence relation ρ on X is called smooth iff there is a sequence Congruences ( A n ) n ∈ N ⊆ B ( X ) of Borel sets with Factoring x ρ x ′ ⇔ ∀ n ∈ N : x ∈ A n ⇔ x ′ ∈ A n ˆ ˜ Some Conclu- ding iff ρ = ker ( f ) for some measurable f : X → R . Remarks References Example Assume X is the state space for a Markov transition system K that interprets a modal logic with a countable number of formulas. Put x ρ K x ′ iff ∀ ϕ : K , x | = ϕ ⇔ K , x ′ | = ϕ. Then ρ K is smooth. This is used when looking at behavioral equivalence of Markov transition systems. 9/9/Smooth Equivalence Relations
Smooth Equivalence Relations EED. Invariant Sets Motivation A Borel subset B ⊆ X is called ρ -invariant iff B = S { [ x ] ρ | x ∈ B } , thus iff The Giry x ∈ B and x ρ x ′ implies x ′ ∈ B . Monad Smooth The invariant Borel sets form a σ -algebra I NV ( B ( X ) , ρ ) on X . Equiva- lence Relations Congruences Lift a Relation X ֒ → S ( X ) Factoring Define the lifting ρ of the smooth equivalence relation ρ from X to S ( X ) Some through Conclu- µ ρ µ ′ iff ∀ B ∈ I NV ( B ( X ) , ρ ) : µ ( B ) = µ ′ ( B ) . ding Remarks References Example Let ρ K be defined through a modal logic. Then µ ρ K µ ′ iff ∀ ϕ : µ = µ ′ ` ` ´ ´ [ [ ϕ ] ] K [ [ ϕ ] ] K . This is used for looking at distributional equivalence of Markov transition systems. 10/10/Smooth Equivalence Relations
Smooth Equivalence Relations To S ( X ) and back EED. → X S ( X ) ֒ Motivation The Giry If ξ is a smooth equivalence relation on S ( X ) , then Monad x ⌊ ξ ⌋ x ′ iff η X ( x ) ξ η X ( x ′ ) Smooth Equiva- lence Relations defines a smooth equivalence relation on X ( η X comes from the monad). Congruences Factoring Some Conclu- Of course Grounded Near Grounded ding Remarks If ρ is smooth on X , ρ = ⌊ ρ ⌋ . ξ = ⌊ ξ ⌋ ξ ⊆ ⌊ ξ ⌋ References Characterization The equivalence relation ξ in S ( X ) is grounded iff ξ = ker ( H ) for a point-affine, surjective and continuous map H : S ( X ) → S ( R ) . 11/11/Smooth Equivalence Relations
Smooth Equivalence Relations Factoring Congruence EED. Let K : X � X be a stochastic relation. The smooth equivalence relation ρ on Motivation X is a congruence for K iff K ( x )( D ) = K ( x ′ )( D ) , whenever x ρ x ′ and The Giry D ⊆ X is a ρ -invariant Borel set. Monad If ρ cannot distinguish x and x ′ , and if it cannot distinguish the elements of Smooth Equiva- D , then the probabilities must be equal. lence Relations Congruences Proposition: Factoring Factoring Some ρ is a congruence for K iff there exists a stochastic relation Conclu- K ρ : X /ρ � S ( X /ρ ) such that this diagram commutes ding Remarks References ε ρ X /ρ X K ρ K S ( X ) S ( X /ρ ) S ( ε ρ ) ( ε ρ is a morphism K → K ρ ) 12/12/Congruences
Smooth Equivalence Relations Factoring EED. Randomized Congruence Motivation The smooth equivalence relation ξ on S ( X ) is a randomized congruence for The Giry Monad the stochastic relation K : X � X iff Smooth ξ is near grounded, Equiva- lence µ ξ µ ′ implies K ∗ ( µ ) ξ K ∗ ( µ ′ ) . Relations Congruences Factoring Factoring for the randomized case? Some Conclu- ding Remarks References Morphism Randomized Morphism If K : X � X and L : Y � Y are If K : X � X and L : Y � Y are stochastic relations, then f : X → Y stochastic relations, then measurable is called a morphism Φ : X � Y is called a randomized K → L iff L ◦ f = S ( f ) ◦ K . morphism K ⇒ L iff L ∗ Φ = Φ ∗ K . 13/13/Factoring
Smooth Equivalence Relations Factoring EED. Morphism Randomized Morphism Motivation If f : K → L is a morphism, then If Φ : K ⇒ L is a randomized morphism, then The Giry ker (Φ ∗ ) is a randomized congruence for K . ker ( f ) is a congruence for K . Monad Smooth Equiva- lence The randomized morphism Φ : K ⇒ L is called near-grounded iff ker (Φ ∗ ) is Relations near grounded. Congruences Factoring Some Conclu- ding For morphism f : K → L there If Φ : K ⇒ L is near-grounded, then Remarks exists a unique morphism there exists a unique randomized References g : K / ker ( f ) → L with morphism Γ : K / ker (Φ) ⇒ L with f Φ K L K L ε ker ( f ) ε ker (Φ) g Γ K / ker ( f ) K / ker (Φ) 14/14/Factoring
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