A domain theory for quasi-Borel spaces Ohad Kammar with Matthijs V´ ak´ ar and Sam Staton International Workshop on Domain Theory and its Applications 8 July 2018 Ohad Kammar, Matthijs V´ ak´ ar, and Sam Staton A domain theory for quasi-Borel spaces
Statistical probabilistic programming ⟦ − ⟧ : programs → distributions scale ▶ Continuous types: R , [0 , ∞ ] distribution by r ▶ Probabilistic effects: r : [0 , ∞ ] normally sample ( µ, σ ) : R distributed conditioning/fitting score ( r ) : 1 sample to observed data � � � � let x = sample (0 , 2) � � � � � in score ( normalPd f (1 . 1 | x, 1)); � ⟦ sample (0 , 2) ⟧ � � � � score ( normalPd f (1 . 9 | 2 x, 1)); � � � � score ( normalPd f (2 . 7 | 3 x, 1)); x � � 0.25 0.04 0 0 -4 -2 0 2 4 -4 -2 0 2 4 Ohad Kammar, Matthijs V´ ak´ ar, and Sam Staton A domain theory for quasi-Borel spaces
Statistical probabilistic programming ▶ Commutativity/exchangability � � � � Exact Bayesian inference � � � � let x = M in let y = N in � � � � using disintegration � � � � let y = N in = let x = M in � � � � [Shan-Ramsey’17] � � � � � f ( x, y ) � � f ( x, y ) � � � � � Fubini’s: ∫ ∫ ∫ ∫ ⟦ M ⟧ (d x ) ⟦ N ⟧ (d y ) f ( x, y ) = ⟦ N ⟧ (d y ) ⟦ M ⟧ (d x ) f ( x, y ) probabilitity σ -finite arbitrary distributions distributions distributions " " % s-finite distributions full definability not closed under " push-forward [Staton’17] Ohad Kammar, Matthijs V´ ak´ ar, and Sam Staton A domain theory for quasi-Borel spaces
Statistical probabilistic programming Express continuous distributions with: ▶ Higher-order functions [Heunen et al.’17] measurable cones measure theory and stable quasi-Borel spaces % " measurable functions " [Ehrhard-Pagani-Tasson’18] Theorem (Aumann’61) No σ -algebra over Meas ( R , R ) with measurable evaluation: modular implementation of eval : Meas ( R , R ) × R → R Bayesian inference algorithms [´ Scibior et al.’18a+b] ▶ Inductive types and bounded iteration ▶ Term recursion domain theory ▶ Type recursion [this work] Ohad Kammar, Matthijs V´ ak´ ar, and Sam Staton A domain theory for quasi-Borel spaces
Iso-recursive types: FPC type variable contexts ∆ = { α 1 , . . . , α n } [Fiore-Plotkin’94] ∆ , α ⊢ k τ : type ∆ ⊢ k µα.τ : type � False } � Lam = µα. { Bool { True type recursion � � App( α ∗ α ) � � Abs( α → α ) } τ = µα.σ Γ ⊢ t : σ [ α �→ τ ] Γ ⊢ t : τ Γ , x : σ [ α �→ τ ] ⊢ s : ρ Γ ⊢ τ. roll ( t ) : τ Γ ⊢ match t with roll x ⇒ s : ρ Ohad Kammar, Matthijs V´ ak´ ar, and Sam Staton A domain theory for quasi-Borel spaces
Iso-recursive types: FPC type variable contexts [Fiore-Plotkin’94] ∆ = { α 1 , . . . , α n } ∆ , α ⊢ k τ : type ∆ ⊢ k µα.τ : type ω Cpo -enriched category of type recursion domains ⟦ ∆ ⊢ k τ : type ⟧ : ( C op ) n × C n → C ⟦ ∆ ⊢ k µα.τ : type ⟧ = minimal invariants locally continuous functor [Freyd’91,92, Pitts’96] Ohad Kammar, Matthijs V´ ak´ ar, and Sam Staton A domain theory for quasi-Borel spaces
Challenge ▶ probabilistic powerdomain ▶ commutativity/Fubini continuous domains [Jones-Plotkin’89] open problem [Jung-Tix’98] ▶ domain theory ▶ higher-order functions traditional approach: domain �→ Scott-open sets �→ Borel sets �→ distributions/valuations following [Ehrhard-Pagani-Tasson’18] our approach: ( domain , quasi-Borel space ) �→ distributions separate but compatible Ohad Kammar, Matthijs V´ ak´ ar, and Sam Staton A domain theory for quasi-Borel spaces
▶ ω Qbs : a category of pre-domain quasi-Borel spaces ▶ M : commutative probabilistic powerdomain over ω Qbs Theorem (adequacy) M adequately interprets: ▶ Statistical FPC ▶ Untyped Statistical λ -calculus Plan ▶ ω Qbs ▶ a powerdomain over ω Qbs ▶ a domain theory for ω Qbs Ohad Kammar, Matthijs V´ ak´ ar, and Sam Staton A domain theory for quasi-Borel spaces
Quasi-Borel pre-domains set subset of functions partial order on X R → X ω -qbs : X = ( X, ≤ X , M X ) ω -cpo quasi-Borel space • λ .x ∈ M X �→ c pointwise pointwise ω -chain lub s.t.: ∨ ( α n ) ∈ M ω = ⇒ α n ∈ M X X n Morphisms f : X → Y : Scott continuous qbs maps monotone and ∀ α ∈ M X . f ∨ n x n = ∨ n f x n f ◦ α ∈ M Y Ohad Kammar, Matthijs V´ ak´ ar, and Sam Staton A domain theory for quasi-Borel spaces
Quasi-Borel pre-domains set subset of functions partial order on X R → X ω -qbs : X = ( X, ≤ X , M X ) ω -cpo quasi-Borel space ϕ R − → Borel R • λ .x ∈ M X ϕ α �− → �− → • α ∈ M X = ⇒ α ◦ ϕ ∈ M X pointwise pointwise ω -chain lub s.t.: ∨ ( α n ) ∈ M ω = ⇒ α n ∈ M X X n Morphisms f : X → Y : Scott continuous qbs maps monotone and ∀ α ∈ M X . f ∨ n x n = ∨ n f x n f ◦ α ∈ M Y Ohad Kammar, Matthijs V´ ak´ ar, and Sam Staton A domain theory for quasi-Borel spaces
Quasi-Borel pre-domains set subset of functions partial order on X R → X ω -qbs : X = ( X, ≤ X , M X ) ω -cpo quasi-Borel space ϕ R − → Borel R • λ .x ∈ M X [ S n .α n ] �− − − − → • α ∈ M X = ⇒ α ◦ ϕ ∈ M X • ( α n ∈ M X ) n ∈ N = ⇒ [ r ∈ S n .α ( r )] ∈ M X Borel measurable pointwise pointwise countable partition ω -chain lub R = ⊎ n ∈ N S n s.t.: ∨ ( α n ) ∈ M ω = ⇒ α n ∈ M X X n Morphisms f : X → Y : Scott continuous qbs maps monotone and ∀ α ∈ M X . f ∨ n x n = ∨ n f x n f ◦ α ∈ M Y Ohad Kammar, Matthijs V´ ak´ ar, and Sam Staton A domain theory for quasi-Borel spaces
Quasi-Borel pre-domains X = ( X, ≤ X , M X ) • λ .x ∈ M X • α ∈ M X = ⇒ α ◦ ϕ ∈ M X • ( α n ∈ M X ) n ∈ N = ⇒ [ r ∈ S n .α ( r )] ∈ M X s.t.: ∨ ( α n ) ∈ M ω = ⇒ α n ∈ M X X n Example S = ( S, Σ S ) measurable space ( ) S, = , { α : R → S | α Borel measurable } so R ∈ ω Qbs Ohad Kammar, Matthijs V´ ak´ ar, and Sam Staton A domain theory for quasi-Borel spaces
Quasi-Borel pre-domains X = ( X, ≤ X , M X ) • λ .x ∈ M X • α ∈ M X = ⇒ α ◦ ϕ ∈ M X • ( α n ∈ M X ) n ∈ N = ⇒ [ r ∈ S n .α ( r )] ∈ M X s.t.: ∨ ( α n ) ∈ M ω = ⇒ α n ∈ M X X n Example P = ( P, ≤ P ) ω -cpo � � ∨ [ ∈ S k n .a k ⊎ S k � P, ≤ P , n ] ∀ k. R = � n � n k � ( }) { � α : R → [0 , ∞ ] so L = [0 , ∞ ] , ≤ , � α Borel measurable ∈ ω Qbs Ohad Kammar, Matthijs V´ ak´ ar, and Sam Staton A domain theory for quasi-Borel spaces
Quasi-Borel pre-domains X = ( X, ≤ X , M X ) • λ .x ∈ M X • α ∈ M X = ⇒ α ◦ ϕ ∈ M X • ( α n ∈ M X ) n ∈ N = ⇒ [ r ∈ S n .α ( r )] ∈ M X s.t.: ∨ ( α n ) ∈ M ω = ⇒ α n ∈ M X X n Example X ω -qbs ( }) � { [ S. ⊥ , S ∁ .α ] X ⊥ := {⊥} + X, ⊥ ≤ X, � α ∈ M X , S Borel � Ohad Kammar, Matthijs V´ ak´ ar, and Sam Staton A domain theory for quasi-Borel spaces
Quasi-Borel pre-domains Theorem ω Qbs → ω Cpo × Qbs creates limits Products X 1 × X 2 = X 1 × X 2 x ≤ y ⇐ ⇒ ∀ i.x i ≤ y i � { } M X 1 × X 2 = ( α 1 , α 2 ) : R → X 1 × X 2 � ∀ i.α i ∈ M X i correlated Exponentials random elements ▶ Y X = { f : X → Y | f Scott continuous qbs morphism } = Qbs ( X, Y ) ▶ f ≤ g ⇐ ⇒ ∀ x ∈ X.f ( x ) ≤ g ( x ) { � } uncurry α : R × X → Y � ▶ M Y X = α : R → Y X � � Scott continuous qbs morphism so Y R = M Y � Ohad Kammar, Matthijs V´ ak´ ar, and Sam Staton A domain theory for quasi-Borel spaces
Characterising ω Qbs Yoneda [ Sbs op , Set ] cpp Sbs = countable Yoneda product preserving SepSh [Staton et al.’16] ( ) r ← − 1 ) F ( R F : Sbs op → Set separated: F R → ( F 1 ) R injective r ∈ R − − − − − − − − − − Thm: Qbs ≃ SepSh [Heunen et al.’17] Yoneda [ Sbs op , ω Cpo ] cpp Sbs = Yoneda ω SepSh f ( x ) ≤ f ( y ) = ⇒ x ≤ y ( ) r ← − 1 ) F ( R F : Sbs op → ω Cpo ω -separated: F R → ( F 1 ) R full r ∈ R − − − − − − − − − − Thm: ω Qbs ≃ ω Cpo ω SepSh Ohad Kammar, Matthijs V´ ak´ ar, and Sam Staton A domain theory for quasi-Borel spaces
Characterising ω Qbs Grothendieck quasi-topos Qbs strong subobject classifier: M Ω = 2 R Ω = 2 strong monos: P 2 ≤ P f − − → Ω X Y ( f ◦ ) − 1 [ M Y ] = M X ∨ qbs ω -chain ( P ) − → P Internal ω -cpo P: ( P, ≤ P , ∨) + internal quasi-topos logic ω -cpo axioms Theorem ω Qbs ≃ ω Cpo ( Qbs ) Ohad Kammar, Matthijs V´ ak´ ar, and Sam Staton A domain theory for quasi-Borel spaces
Characterising ω Qbs By local presentability: ω Cpo ≃ Mod( ω cpo , Set ) Qbs ≃ Mod( qbs , Set ) essentially algebraic theories ω qbs : ω cpo ∪ qbs ∪ compatibility axiom Theorem ω Qbs ≃ Mod( ω qbs , Set ) so ω Qbs locally presentable, hence cocomplete Ohad Kammar, Matthijs V´ ak´ ar, and Sam Staton A domain theory for quasi-Borel spaces
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