Causal Theories: A Categorical Approach to Bayesian Networks Brendan Fong, University of Oxford PSU Applied Algebra and Network Theory Seminar. 22 April 2015
Task: formalise how Bayesian networks guide our reasoning about a collection of random variables.
� Test: Simpson’s paradox What should we conclude from the following data? TB ¬ t , b t , ¬ b ¬ t , ¬ b t , b r 30 40 9 2 ¬ r R 10 40 51 18 T ¬ t t r 39 42 ¬ r R 61 58
� Test: Simpson’s paradox What should we conclude from the following data? TB ¬ t , b t , ¬ b ¬ t , ¬ b t , b r 30 40 9 2 ¬ r R 10 40 51 18 T ¬ t t r 39 42 ¬ r R 61 58
The big picture Network-style diagrammatic languages have developed to represent and reason about many different sciences: Why? Can we formalise their key features and relationships? Can we unify them? I work with John Baez and his team to understand these questions from a category-theoretic viewpoint.
The big picture Network-style diagrammatic languages have developed to represent and reason about many different sciences: Why? Can we formalise their key features and relationships? Can we unify them? I work with John Baez and his team to understand these questions from a category-theoretic viewpoint.
Outline 1. Categories 2. Lawvere theories 3. Bayesian networks 4. Causal theories 5. An application to Simpson’s paradox
Categories ○ Categories are a great algebraic framework for discussing interconnection, or composition. They first arose in the 1940s in algebraic topology. ○ From the 1980s, it became clear they had a role to play in formalising uses of string/network diagrams, such as Feynman diagrams:
Categories ○ Categories are a great algebraic framework for discussing interconnection, or composition. They first arose in the 1940s in algebraic topology. ○ From the 1980s, it became clear they had a role to play in formalising uses of string/network diagrams, such as Feynman diagrams:
Categories ○ Categories are a great algebraic framework for discussing interconnection, or composition. They first arose in the 1940s in algebraic topology. ○ From the 1980s, it became clear they had a role to play in formalising uses of string/network diagrams, such as Feynman diagrams:
Categories ○ A category C is the structure of one-dimensional flow charts. ○ They comprise objects , or types : X , Y , etc . together with morphisms between these types: f X Y ○ We can compose morphisms of matching types to get new morphisms: g f Y Z h X W The composition rule must have such pictures unambiguously describe a morphism.
Categories ○ A category C is the structure of one-dimensional flow charts. ○ They comprise objects , or types : X , Y , etc . together with morphisms between these types: f X Y ○ We can compose morphisms of matching types to get new morphisms: g f Y Z h X W The composition rule must have such pictures unambiguously describe a morphism.
Categories ○ A category C is the structure of one-dimensional flow charts. ○ They comprise objects , or types : X , Y , etc . together with morphisms between these types: f X Y ○ We can compose morphisms of matching types to get new morphisms: g f Y Z h X W The composition rule must have such pictures unambiguously describe a morphism.
Categories ○ There are various types of categories that allow extra operations. ○ A monoidal category (C , ⊗) is the structure of two-dimensional flow charts. f W X X h X Z k g V Y V The key point is that we have a notion of ‘parallel’ or tensor composition. ○ A symmetric monoidal category further allows you to cross wires: X Y Y X
Categories ○ There are various types of categories that allow extra operations. ○ A monoidal category (C , ⊗) is the structure of two-dimensional flow charts. f W X X h X Z k g V Y V The key point is that we have a notion of ‘parallel’ or tensor composition. ○ A symmetric monoidal category further allows you to cross wires: X Y Y X
Categories ○ There are various types of categories that allow extra operations. ○ A monoidal category (C , ⊗) is the structure of two-dimensional flow charts. f W X X h X Z k g V Y V The key point is that we have a notion of ‘parallel’ or tensor composition. ○ A symmetric monoidal category further allows you to cross wires: X Y Y X
Categories ○ A functor F ∶C → D is a map between categories. ○ It turns morphisms f X Y in C into morphisms Ff FX FY in D . This assignment must preserve composition. That is, the diagram Ff FY Fg FX FZ must be unambiguous. ○ A monoidal functor F ∶(C , ×) → (D , ⊠) also preserves the tensor.
Categories ○ A functor F ∶C → D is a map between categories. ○ It turns morphisms f X Y in C into morphisms Ff FX FY in D . This assignment must preserve composition. That is, the diagram Ff FY Fg FX FZ must be unambiguous. ○ A monoidal functor F ∶(C , ×) → (D , ⊠) also preserves the tensor.
Categories ○ A functor F ∶C → D is a map between categories. ○ It turns morphisms f X Y in C into morphisms Ff FX FY in D . This assignment must preserve composition. That is, the diagram Ff FY Fg FX FZ must be unambiguous. ○ A monoidal functor F ∶(C , ×) → (D , ⊠) also preserves the tensor.
Lawvere theories ○ Lawvere theories were developed by William Lawvere in his 1963 doctoral thesis as a categorical approach to universal algebra. ○ Let S be a set; we call the elements of this set sorts . A (multisorted) Lawvere theory T is a category with finite products such that every object is a finite product of sorts. ○ Each Lawvere theory T is a monoidal category ( T , ×) with tensor the product × . ○ If (C , ⊗) is a monoidal category, and T is a Lawvere theory, a model of T in (C , ⊗) is a monoidal functor F ∶( T , ×) � → ( C , ⊗) .
Lawvere theories ○ Lawvere theories were developed by William Lawvere in his 1963 doctoral thesis as a categorical approach to universal algebra. ○ Let S be a set; we call the elements of this set sorts . A (multisorted) Lawvere theory T is a category with finite products such that every object is a finite product of sorts. ○ Each Lawvere theory T is a monoidal category ( T , ×) with tensor the product × . ○ If (C , ⊗) is a monoidal category, and T is a Lawvere theory, a model of T in (C , ⊗) is a monoidal functor F ∶( T , ×) � → ( C , ⊗) .
Lawvere theories ○ Lawvere theories were developed by William Lawvere in his 1963 doctoral thesis as a categorical approach to universal algebra. ○ Let S be a set; we call the elements of this set sorts . A (multisorted) Lawvere theory T is a category with finite products such that every object is a finite product of sorts. ○ Each Lawvere theory T is a monoidal category ( T , ×) with tensor the product × . ○ If (C , ⊗) is a monoidal category, and T is a Lawvere theory, a model of T in (C , ⊗) is a monoidal functor F ∶( T , ×) � → ( C , ⊗) .
Lawvere theories ○ Lawvere theories were developed by William Lawvere in his 1963 doctoral thesis as a categorical approach to universal algebra. ○ Let S be a set; we call the elements of this set sorts . A (multisorted) Lawvere theory T is a category with finite products such that every object is a finite product of sorts. ○ Each Lawvere theory T is a monoidal category ( T , ×) with tensor the product × . ○ If (C , ⊗) is a monoidal category, and T is a Lawvere theory, a model of T in (C , ⊗) is a monoidal functor F ∶( T , ×) � → ( C , ⊗) .
Lawvere theories Example: the theory of groups ○ Let T G have objects X n for n ∈ N , and morphisms generated by µ ∶ X × X → X , η ∶ 1 → X , and ι ∶ X → X subject to: µ µ µ (Assoc) = µ η µ µ (Id) = = η ι µ η (Inv) = ι ι = ○ A model in ( Set , ×) is a group. ○ A model in ( Top , ×) is a topological group.
Lawvere theories Example: the theory of groups ○ Let T G have objects X n for n ∈ N , and morphisms generated by µ ∶ X × X → X , η ∶ 1 → X , and ι ∶ X → X subject to: µ µ µ (Assoc) = µ η µ µ (Id) = = η ι µ η (Inv) = ι ι = ○ A model in ( Set , ×) is a group. ○ A model in ( Top , ×) is a topological group.
Lawvere theories Example: the theory of groups ○ Let T G have objects X n for n ∈ N , and morphisms generated by µ ∶ X × X → X , η ∶ 1 → X , and ι ∶ X → X subject to: µ µ µ (Assoc) = µ η µ µ (Id) = = η ι µ η (Inv) = ι ι = ○ A model in ( Set , ×) is a group. ○ A model in ( Top , ×) is a topological group.
Outline 1. Categories 2. Lawvere theories 3. Bayesian networks 4. Causal theories 5. An application to Simpson’s paradox
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