File size: 3,439 Bytes
c4cdbbd | 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 70 71 72 73 74 75 76 77 78 79 80 81 82 | import OAI.Probability.InvariantIsing.Fields.FieldAffineScalarBridge
namespace OAI
/-! Propagation of a covariance derivative through the unchanged earlier
levels. The propagated tangent is the actual conditional expectation. -/
noncomputable section
open MeasureTheory ProbabilityTheory IsingPerceptron
open scoped NNReal
namespace InvariantIsing
def fieldConstantPrefix {I : Set ℝ} (hI : IsOpen I) (F : FieldSmoothFamily I) :
(L : List (ℝ × ℝ≥0)) → (∀ av ∈ L, 0 < (av.2 : ℝ)) → FieldSmoothFamily I
| [], _ => F
| av :: L, hL =>
(fieldConstantPrefix hI F L (fun bv hb => hL bv (List.mem_cons_of_mem av hb))).transform
hI av.2 0 av.1 (fun _ _ => by simpa only [zero_mul, add_zero] using hL av List.mem_cons_self)
theorem fieldConstantPrefix_value {I : Set ℝ} (hI : IsOpen I) (F : FieldSmoothFamily I)
(L : List (ℝ × ℝ≥0)) (hL : ∀ av ∈ L, 0 < (av.2 : ℝ)) (t : ℝ) :
(fun z => (fieldConstantPrefix hI F L hL).U (t, z)) =
fieldScalarValue L (fun z => F.U (t, z)) := by
induction L with
| nil => rfl
| cons av L ih =>
let htail := fun bv hb => hL bv (List.mem_cons_of_mem av hb)
let G := fieldConstantPrefix hI F L htail
change gaussianTransform ((av.2 : ℝ) + 0 * t) av.1 (fun z => G.U (t, z)) = _
rw [field_gaussianTransform_eq_operator]
simp only [zero_mul, add_zero, Real.toNNReal_coe]
rw [show (fun z => G.U (t, z)) = fieldScalarValue L (fun z => F.U (t, z)) from ih htail]
rfl
theorem fieldConstantPrefix_tangent {I : Set ℝ} (hI : IsOpen I) (F : FieldSmoothFamily I)
(L : List (ℝ × ℝ≥0)) (hL : ∀ av ∈ L, 0 < (av.2 : ℝ)) (t : ℝ) :
(fun z => (fieldConstantPrefix hI F L hL).T (t, z)) =
fieldScalarMean L (fun z => F.U (t, z)) (fun z => F.T (t, z)) := by
induction L with
| nil => rfl
| cons av L ih =>
let htail := fun bv hb => hL bv (List.mem_cons_of_mem av hb)
let G := fieldConstantPrefix hI F L htail
have hU := fieldConstantPrefix_value hI F L htail t
have hT := ih htail
have hmU : Measurable (fun z => G.U (t, z)) := G.mU.comp (by fun_prop)
have hmT : Measurable (fun z => G.T (t, z)) := G.mT.comp (by fun_prop)
change (fun z => G.tangent av.2 0 av.1 (t, z)) = _
simp only [FieldSmoothFamily.tangent, zero_mul, zero_div, add_zero]
rw [field_gaussianTiltAverage_eq_transition _ _ hmU hmT]
simp only [Real.toNNReal_coe]
rw [hU, hT]
rfl
theorem fieldConstantPrefix_mean {I : Set ℝ} (hI : IsOpen I) (F : FieldSmoothFamily I)
(L : List (ℝ × ℝ≥0)) (hL : ∀ av ∈ L, 0 < (av.2 : ℝ)) (t : ℝ) :
(fun z => (fieldConstantPrefix hI F L hL).X (t, z)) =
fieldScalarMean L (fun z => F.U (t, z)) (fun z => F.X (t, z)) := by
induction L with
| nil => rfl
| cons av L ih =>
let htail := fun bv hb => hL bv (List.mem_cons_of_mem av hb)
let G := fieldConstantPrefix hI F L htail
have hU := fieldConstantPrefix_value hI F L htail t
have hX := ih htail
have hmU : Measurable (fun z => G.U (t, z)) := G.mU.comp (by fun_prop)
have hmX : Measurable (fun z => G.X (t, z)) := G.mX.comp (by fun_prop)
change gaussianTiltAverage ((av.2 : ℝ) + 0 * t) av.1
(fun z => G.U (t, z)) (fun z => G.X (t, z)) = _
simp only [zero_mul, add_zero]
rw [field_gaussianTiltAverage_eq_transition _ _ hmU hmX]
simp only [Real.toNNReal_coe]
rw [hU, hX]
rfl
end InvariantIsing
end
end OAI
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