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Structure Representation, What is a Representation?, Causal Models,…

- - - - Structure emerging from pixels
    - - Methodologically formulate a structured probabilistic graphical model and do posterior inference
- - - - What happens to floss purchases if we double the price of toothpaste?
      - P(floss∣do(toothpaste))
        
        The do-operator (from Judea Pearl’s causal framework) means: Performing a “mini-surgery” on the causal system while holding everything else the same
  - - - Parents of 𝑋 are direct causes of 𝑋
      - f is a deterministic function
      - UX is an exogenous noise term (unobserved factors)
    - - Toothpaste price affects floss purchases
        Health awareness affects both
      - Structural Equation
        
        Health=U1
        
        Price=f1(Health,U2)
        
        Floss=f2(Price,Health,U3)
    - - A regression like: Floss=β⋅Price+ϵ does not tell you what happens if you intervene on Price. A structural equation does.
    - - do(Price=2×Price)
      - Original: Price=𝑓1(Health,𝑈2) | After intervention: Price=2×Price_{baseline}. All other equations remain unchanged.
      - Before Intervention: Price=f(Cost,Demand,Tax,UPrice)
        General intervention: do(Price=p) and Ignore Cost, Demand, Tax, etc.
      - Common Types of Price Interventions
        
        Price Increase / Decrease
        
        Price Discount
        
        Price Cap
        
        Price Floor
        
        Randomized Price (A/B Test)
  - - - P(X1,X2,…,Xn)=∏_{i=1}^n P(Xi∣Parents(Xi))
    - - Suppose the causal relationships are:
      - Health → Price
      - Health → Floss
      - Price → Floss
      - Then the factorization is:
      - P(Health,Price,Floss)=P(Health)⋅P(Price∣Health)⋅P(Floss∣Price,Health)
      - Each term corresponds to one structural equation.
    - - Xi=fi(Parents(Xi),Ui)
      - With independent noise variables Ui, they induce the factorization:
      - P(X1,…,Xn)=∏_i P(Xi∣Parents(Xi))
      - Structural equations → causal graph → causal factorization
      - Statistical factorization: many valid forms
        Causal factorization: only one correct form (up to equivalent graphs)
      - Enables interventions
        
        do-operator
        
        When we intervene on Xk: do(Xk=x)
        
        We: Remove the factor 𝑃(𝑋𝑘∣Parents(𝑋𝑘))
        
        Replace it with 𝑋𝑘=𝑥
        
        Resulting interventional distribution:
        
        𝑃(𝑋∣𝑑𝑜(𝑋𝑘=𝑥))=𝛿(𝑋𝑘=𝑥)∏_{𝑖≠𝑘}𝑃(𝑋𝑖∣Parents(𝑋𝑖))P(X∣do(Xk=x))
        
        When we intervene on Xk: do(Xk=x)
        
        We: Remove the factor 𝑃(𝑋𝑘∣Parents(𝑋𝑘))
        
        Replace it with 𝑋𝑘=𝑥