| import OAI.Combinatorics.BooleanFunctions.Blocks |
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| namespace OAI |
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| open scoped BigOperators Topology NNReal ENNReal |
| open MeasureTheory ProbabilityTheory Filter Real |
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| noncomputable section |
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| namespace SquareRootDegree |
| namespace Projection |
| attribute [local instance] Classical.propDecidable |
| variable {α β γ κ : Type*} [Fintype α] [Nonempty α] [Fintype β] [Fintype γ] |
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| omit [Nonempty α] [Fintype β] in |
| /-- Conditional averages are characterized by their cell moments. -/ |
| theorem conditional_eq_of_projection (F : α → β) (g h : α → ℝ) |
| (hp : ∀ p : β → ℝ, average (fun x => p (F x)*g x) = average (fun x => p (F x)*h x)) : |
| conditionalAverage F g = conditionalAverage F h := by |
| funext x |
| unfold conditionalAverage conditionalValue cellMoment |
| congr 1 |
| exact hp (cell id (F x)) |
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| omit [Fintype γ] in |
| theorem conditional_nested (F : α → β) (H : β → γ) (g : α → ℝ) : |
| conditionalAverage (H ∘ F) g = conditionalAverage (H ∘ F) (conditionalAverage F g) := by |
| apply conditional_eq_of_projection |
| intro p |
| exact (conditionalAverage_projection F g (p ∘ H)).symm |
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| omit [Fintype γ] in |
| theorem pi_projection [Fintype κ] [DecidableEq κ] (F : α → β) (g : α → ℝ) |
| (p : (κ → β) → ℝ) (j : κ) : |
| average (fun x : κ → α => p (fun i => F (x i))*g (x j)) = |
| average (fun x : κ → α => p (fun i => F (x i))*conditionalAverage F g (x j)) := by |
| rw [← average_update _ j,← average_update (fun x : κ → α => |
| p (fun i => F (x i))*conditionalAverage F g (x j)) j] |
| congr 1 |
| funext x |
| have he (a : α) : (fun i => F (Function.update x j a i)) = |
| Function.update (fun i => F (x i)) j (F a) := by |
| funext i |
| by_cases hi : i = j <;> simp [hi] |
| simp only [he,Function.update_self] |
| exact (conditionalAverage_projection F g |
| (fun b => p (Function.update (fun i => F (x i)) j b))).symm |
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| omit [Fintype γ] in |
| theorem pi_conditional_sum [Fintype κ] [DecidableEq κ] |
| (F : α → β) (H : (κ → β) → γ) (g : α → ℝ) : |
| conditionalAverage (fun x : κ → α => H (fun i => F (x i))) |
| (fun x => ∑ i, g (x i)) = |
| conditionalAverage (fun x : κ → α => H (fun i => F (x i))) |
| (fun x => ∑ i, conditionalAverage F g (x i)) := by |
| apply conditional_eq_of_projection |
| intro p |
| simp only [Finset.mul_sum,average_sum] |
| apply Finset.sum_congr rfl |
| intro j _ |
| exact pi_projection F g (p ∘ H) j |
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| omit [Nonempty α] [Fintype β] [Fintype γ] in |
| theorem conditional_equiv {δ : Type*} [Fintype δ] (e : α ≃ δ) |
| {ε : Type*} (F : δ → ε) (g : δ → ℝ) : |
| conditionalAverage (F ∘ e) (g ∘ e) = conditionalAverage F g ∘ e := by |
| funext x |
| unfold conditionalAverage conditionalValue cellMoment cellMass |
| have hm := average_equiv e (cell F (F (e x))) |
| have hg := average_equiv e (fun y => cell F (F (e x)) y*g y) |
| exact congrArg₂ (fun a b : ℝ => a/b) hg hm |
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| end Projection |
| end SquareRootDegree |
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| namespace SquareRootDegree |
| namespace Projection |
| attribute [local instance] Classical.propDecidable |
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| variable {α β γ : Type*} [Fintype α] [Nonempty α] [Fintype β] |
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| omit [Fintype β] in |
| theorem cellMass_pos (F : α → β) (x : α) : 0 < cellMass F (F x) := by |
| unfold cellMass average |
| apply mul_pos (inv_pos.mpr (Nat.cast_pos.mpr Fintype.card_pos)) |
| apply Finset.sum_pos' |
| · intro y _ |
| unfold cell |
| split_ifs <;> positivity |
| · exact ⟨x,Finset.mem_univ x,by simp [cell]⟩ |
| |
| omit [Nonempty α] [Fintype β] in |
| theorem conditional_const_mul (F : α → β) (g : α → ℝ) (c : ℝ) : |
| conditionalAverage F (fun x => c*g x) = fun x => c*conditionalAverage F g x := by |
| funext x |
| unfold conditionalAverage conditionalValue cellMoment |
| simp_rw [show ∀ y, cell F (F x) y*(c*g y) = c*(cell F (F x) y*g y) by intros; ring] |
| rw [average_const_mul] |
| ring |
| |
| omit [Fintype β] in |
| theorem conditional_comp (F : α → β) (g : β → ℝ) : |
| conditionalAverage F (fun x => g (F x)) = fun x => g (F x) := by |
| funext x |
| unfold conditionalAverage conditionalValue |
| have he : cellMoment F (fun y => g (F y)) (F x) = g (F x)*cellMass F (F x) := by |
| unfold cellMoment cellMass |
| rw [← average_const_mul] |
| congr 1 |
| funext y |
| by_cases h : F y = F x <;> simp [cell,h] |
| rw [he,mul_div_cancel_right₀ _ (cellMass_pos F x).ne'] |
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| omit [Nonempty α] [Fintype β] in |
| theorem conditional_add (F : α → β) (g h : α → ℝ) : |
| conditionalAverage F (fun x => g x+h x) = |
| fun x => conditionalAverage F g x+conditionalAverage F h x := by |
| funext x |
| unfold conditionalAverage conditionalValue cellMoment |
| simp_rw [mul_add] |
| rw [average_add,add_div] |
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| variable {ι : Type*} [Fintype ι] [DecidableEq ι] |
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| theorem conditional_tuple_sum (F : α → β) (g : α → ℝ) : |
| conditionalAverage (fun x : ι → α => fun i => F (x i)) (fun x => ∑ i, g (x i)) = |
| fun x => ∑ i, conditionalAverage F g (x i) := by |
| have hp := pi_conditional_sum (κ := ι) F id g |
| simp only [id_eq] at hp |
| rw [hp] |
| exact conditional_comp (fun x : ι → α => fun i => F (x i)) |
| (fun b => ∑ i, conditionalValue F g (b i)) |
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| end Projection |
| end SquareRootDegree |
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| end |
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| end OAI |
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