| import OAI.Dynamics.ConditionalShuffle.ScheduledInstrument |
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| namespace OAI |
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| noncomputable section |
| open scoped Classical |
| namespace Revealed.Disintegration |
| open Thorp Thorp.Conditional |
| variable {Ω R E G : Type} [Fintype Ω] [Nonempty Ω] [Fintype R] [fintype_E : Fintype E] |
| [Fintype G] [Nonempty G] |
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| lemma fairMass_at_image_pos (a : Ω → E) (c : Ω) : 0 < fairMass a (a c) := by |
| let retained_fintype_E := fintype_E |
| unfold fairMass |
| apply div_pos _ (by exact_mod_cast Fintype.card_pos) |
| have h := Finset.single_le_sum (s:=Finset.univ) (f:=fun x : Ω => if a x = a c then (1 : ℝ) else 0) |
| (fun x _ => by split_ifs <;> norm_num) (Finset.mem_univ c) |
| simp only [ite_true] at h |
| linarith |
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| lemma resample_conditional (a : Ω → E) (b : Ω → G) (F : Ω → R → Ω) |
| (hobs : ∀ c r, a (F c r) = a c) |
| (hstat : ∀ f : Ω → ℝ, mean (fun c => mean (fun r => f (F c r))) = mean f) |
| (hrow : ∀ c c', a c = a c' → ∀ r, F c r = F c' r) (c : Ω) : |
| fairMass (fun r => b (F c r)) = conditional (fairMass (fun x => (a x,b x))) (a c) := by |
| funext g |
| have hK (x : Ω) (hx : a x = a c) : fairMass (fun r => b (F x r)) g = fairMass (fun r => b (F c r)) g := by |
| congr 1; funext r; rw [hrow x c hx r] |
| have hj : fairMass (fun x => (a x,b x)) (a c,g) = |
| fairMass a (a c) * fairMass (fun r => b (F c r)) g := by |
| rw [fairMass_eq_mean, ← hstat] |
| simp only [Prod.mk.injEq, hobs, ite_and, mean_indicator] |
| change mean (fun x : Ω => if a x = a c then fairMass (fun r => b (F x r)) g else 0) = _ |
| calc |
| _ = mean (fun x : Ω => (if a x = a c then (1 : ℝ) else 0) * fairMass (fun r => b (F c r)) g) := by |
| apply mean_congr; intro x |
| by_cases hx : a x = a c |
| · rw [ite_eq_left hx, ite_eq_left hx, one_mul, hK x hx] |
| · simp only [ite_eq_right hx, zero_mul] |
| _ = _ := by rw [mean_mul_const, ← fairMass_eq_mean] |
| rw [conditional, marginal_fairMass, ite_eq_right (ne_of_gt (fairMass_at_image_pos a c)), hj] |
| exact (mul_div_cancel_left₀ _ (ne_of_gt (fairMass_at_image_pos a c))).symm |
|
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| end Revealed.Disintegration |
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| end |
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| end OAI |
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