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92416a2 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 | import OAI.Dynamics.ConditionalShuffle.OverlayFiber
namespace OAI
noncomputable section
open scoped Classical
namespace Revealed.Instrument
open Thorp Thorp.Conditional Thorp.Fourier Revealed.Disintegration
variable {E S G Ω R : Type} [fintype_S : Fintype S] [Fintype G] [Group G] [Fintype Ω] [nonempty_Ω : Nonempty Ω]
[Fintype R] [nonempty_R : Nonempty R]
variable (I : Data E S G Ω)
def resampledRun (J : E → Ω → R → G) (e : E) :
(n : ℕ) → (Fin n → Ω) → (Fin n → R) → G
| 0, _, _ => 1
| n+1, c, r => J (base I e n (observed I e n (Fin.init c))) (c (Fin.last n)) (r (Fin.last n)) *
resampledRun J e n (Fin.init c) (Fin.init r)
lemma resampledRun_law (J : E → Ω → R → G)
(hJ : ∀ e c, fairMass (J e c) = kernel I e (I.obs e c))
(e : E) (n : ℕ) (c : Fin n → Ω) :
fairMass (resampledRun I J e n c) = realProduct n (kernels I e n (observed I e n c)) := by
let retained_fintype_S := fintype_S
let retained_nonempty_Ω := nonempty_Ω
let retained_nonempty_R := nonempty_R
induction n with
| zero =>
funext g
simp only [resampledRun, fairMass, realProduct, Fintype.card_fun,
Fintype.card_fin, pow_zero, Nat.cast_one, div_one]
simp [eq_comm]
| succ n ih =>
have hc := Thorp.FiniteLaw.conv_fairMass
(J (base I e n (observed I e n (Fin.init c))) (c (Fin.last n)))
(resampledRun I J e n (Fin.init c))
change realConv _ _ = _ at hc
rw [hJ, ih] at hc
simp only [observed, kernels, realProduct, Fin.init_snoc, Fin.snoc_last]
rw [hc]
funext g
rw [fairMass_eq_mean, fairMass_eq_mean, mean_prod, mean_function_snoc]
rw [mean_comm]
apply mean_congr; intro r
apply mean_congr; intro q
simp only [resampledRun, Fin.init_snoc, Fin.snoc_last]
lemma distance_mean (e : E) (n : ℕ) :
distance I e n = mean (fun c : Fin n → Ω =>
tv (realProduct n (kernels I e n (observed I e n c)))
(fun _ => (Fintype.card G : ℝ)⁻¹)) := by
unfold distance
have hm := pathJoint_marginal I e n
have hprob : fairMass (observed I e n) = weights I e n := by
rw [← marginal_fairMass (observed I e n) (groupRun I e n)]
exact funext hm
rw [← hprob, fairMass_test]
lemma resampled_distance (J : E → Ω → R → G)
(hJ : ∀ e c, fairMass (J e c) = kernel I e (I.obs e c)) (e : E) (n : ℕ) :
mean (fun c : Fin n → Ω => tv (fairMass (resampledRun I J e n c))
(fun _ => (Fintype.card G : ℝ)⁻¹)) = distance I e n := by
simp_rw [resampledRun_law I J hJ, distance_mean]
end Revealed.Instrument
end
end OAI
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