File size: 6,784 Bytes
f61e547 | 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 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 | import OAI.Probability.FreeIsing.Brownian.Bridge
import OAI.Probability.FreeIsing.Gaussian.IncrementLikelihood
namespace OAI
noncomputable section
open MeasureTheory ProbabilityTheory
open scoped ENNReal NNReal
namespace Problem367
theorem preBrownian_past_drift_withDensity
{Omega : Type*} [MeasurableSpace Omega]
{B : ℝ≥0 → Omega → ℝ} {mu : Measure Omega}
(hB : IsPreBrownianReal B mu) (hmB : ∀ t, Measurable (B t))
{T : ℝ≥0} (hT : T ≠ 0) (s : ℝ) :
Measure.map (fun w (t : Set.Iic T) => (t.val : ℝ) * s + B t w) mu =
(Measure.map (fun w (t : Set.Iic T) => B t w) mu).withDensity
(fun y : Set.Iic T → ℝ => ENNReal.ofReal
(gaussianSignalLikelihood T s (y ⟨T, by simp⟩))) := by
let : IsProbabilityMeasure mu := hB.isGaussianProcess.isProbabilityMeasure
let R : Omega → Set.Iic T → ℝ := fun w t => brownianBridge B T t w
let E : Omega → (Set.Iic T → ℝ) × ℝ := fun w => (R w, B T w)
let Es : Omega → (Set.Iic T → ℝ) × ℝ :=
fun w => (R w, (T : ℝ) * s + B T w)
let F : ((Set.Iic T → ℝ) × ℝ) → Set.Iic T → ℝ :=
fun z t => z.1 t + ((t.val : ℝ) / (T : ℝ)) * z.2
let L : ℝ → ℝ≥0∞ := fun x => ENNReal.ofReal (gaussianSignalLikelihood T s x)
have hR : Measurable R := by
apply Measurable.of_eval
intro t
exact (hmB t).sub ((hmB T).const_mul _)
have hE : Measurable E := hR.prodMk (hmB T)
have hEs : Measurable Es := hR.prodMk (measurable_const.add (hmB T))
have hF : Measurable F := by unfold F; fun_prop
have hL : Measurable L := by unfold L gaussianSignalLikelihood; fun_prop
have hind : IndepFun R (B T) mu := preBrownian_bridge_indep_terminal hB hT
have hlaw : Measure.map E mu = (Measure.map R mu).prod (gaussianReal 0 T) := by
rw [IndepFun.map_prod_eq_prod_map_map hR.aemeasurable (hmB T).aemeasurable hind,
(hB.hasLaw_eval T).map_eq]
have hlaws : Measure.map Es mu =
(Measure.map R mu).prod (gaussianReal ((T : ℝ) * s) T) := by
have hshift : Measurable (fun x : ℝ => (T : ℝ) * s + x) := by fun_prop
have hbt : Measurable (fun w => (T : ℝ) * s + B T w) := by fun_prop
have hinds : IndepFun R (fun w => (T : ℝ) * s + B T w) mu := by
simpa only [Function.comp_def, id_eq] using
hind.comp measurable_id hshift
change Measure.map (fun w => (R w, (T : ℝ) * s + B T w)) mu = _
rw [IndepFun.map_prod_eq_prod_map_map hR.aemeasurable
hbt.aemeasurable hinds]
congr 1
simpa only [zero_add] using
(gaussianReal_const_add (hB.hasLaw_eval T) ((T : ℝ) * s)).map_eq
have hchange : Measure.map Es mu =
(Measure.map E mu).withDensity (fun z => L z.2) := by
rw [hlaws, hlaw, gaussianReal_signal_withDensity T s]
exact prod_withDensity_right hL
have hFE : F ∘ E = fun w (t : Set.Iic T) => B t w := by
funext w t
dsimp [F, E, R, brownianBridge]
ring
have hFEs : F ∘ Es = fun w (t : Set.Iic T) => (t.val : ℝ) * s + B t w := by
have hT' : (T : ℝ) ≠ 0 := by exact_mod_cast hT
funext w t
dsimp [F, Es, R, brownianBridge]
field_simp
ring
have hpast : Measurable (fun w (t : Set.Iic T) => B t w) :=
Measurable.of_eval (fun t => hmB t)
calc
Measure.map (fun w (t : Set.Iic T) => (t.val : ℝ) * s + B t w) mu =
Measure.map F (Measure.map Es mu) := by rw [Measure.map_map hF hEs, hFEs]
_ = Measure.map F ((Measure.map E mu).withDensity (fun z => L z.2)) := by
rw [hchange]
_ = Measure.map F (Measure.map E (mu.withDensity (fun w => L (B T w)))) := by
congr 1
exact (map_withDensity_comp_observation mu E hE (fun z => L z.2)
(hL.comp measurable_snd)).symm
_ = Measure.map (fun w (t : Set.Iic T) => B t w)
(mu.withDensity (fun w => L (B T w))) := by rw [Measure.map_map hF hE, hFE]
_ = _ := map_withDensity_comp_observation mu _ hpast
(fun y : Set.Iic T → ℝ => L (y ⟨T, by simp⟩)) (hL.comp (measurable_pi_apply _))
theorem preBrownian_past_vector_drift_withDensity
{Omega V : Type*} [MeasurableSpace Omega] [Fintype V]
{B : ℝ≥0 → Omega → ℝ} {mu : Measure Omega}
(hB : IsPreBrownianReal B mu) (hmB : ∀ t, Measurable (B t))
{T : ℝ≥0} (hT : T ≠ 0) (s : V → ℝ) :
Measure.pi (fun v : V =>
Measure.map (fun w (t : Set.Iic T) => (t.val : ℝ) * s v + B t w) mu) =
(Measure.pi (fun _ : V =>
Measure.map (fun w (t : Set.Iic T) => B t w) mu)).withDensity
(fun y => ENNReal.ofReal
(gaussianVectorLikelihood T s (fun v => y v ⟨T, by simp⟩))) := by
classical
let : IsProbabilityMeasure mu := hB.isGaussianProcess.isProbabilityMeasure
have hpast : Measurable (fun w (t : Set.Iic T) => B t w) :=
Measurable.of_eval (fun t => hmB t)
let : IsProbabilityMeasure
(Measure.map (fun w (t : Set.Iic T) => B t w) mu) :=
(Measure.isProbabilityMeasure_map_iff hpast.aemeasurable).mpr inferInstance
have hshift (v : V) :
Measurable (fun w (t : Set.Iic T) => (t.val : ℝ) * s v + B t w) := by
apply Measurable.of_eval
intro t
exact measurable_const.add (hmB t)
let (v : V) : IsProbabilityMeasure
(Measure.map (fun w (t : Set.Iic T) => (t.val : ℝ) * s v + B t w) mu) :=
(Measure.isProbabilityMeasure_map_iff (hshift v).aemeasurable).mpr inferInstance
rw [pi_withDensity_of_probability
(fun _ : V => Measure.map (fun w (t : Set.Iic T) => B t w) mu)
(fun v : V => Measure.map (fun w (t : Set.Iic T) => (t.val : ℝ) * s v + B t w) mu)
(fun v (y : Set.Iic T → ℝ) => ENNReal.ofReal
(gaussianSignalLikelihood T (s v) (y ⟨T, by simp⟩)))
(fun v => by unfold gaussianSignalLikelihood; fun_prop)
(fun v => preBrownian_past_drift_withDensity hB hmB hT (s v))]
apply withDensity_congr_ae
filter_upwards [] with y
rw [← ENNReal.ofReal_prod_of_nonneg (fun v hv => by
unfold gaussianSignalLikelihood; positivity), prod_gaussianSignalLikelihood]
theorem preBrownian_past_spin_drift_withDensity
{Omega V : Type*} [MeasurableSpace Omega] [Fintype V]
{B : ℝ≥0 → Omega → ℝ} {mu : Measure Omega}
(hB : IsPreBrownianReal B mu) (hmB : ∀ t, Measurable (B t))
{T : ℝ≥0} (hT : T ≠ 0) (s : V → ℝ) (hs : ∀ v, s v ^ 2 = 1) :
Measure.pi (fun v : V =>
Measure.map (fun w (t : Set.Iic T) => (t.val : ℝ) * s v + B t w) mu) =
(Measure.pi (fun _ : V =>
Measure.map (fun w (t : Set.Iic T) => B t w) mu)).withDensity
(fun y => ENNReal.ofReal
(Real.exp ((∑ v, s v * y v ⟨T, by simp⟩) -
(T : ℝ) * Fintype.card V / 2))) := by
simpa only [gaussianVectorLikelihood_of_sq_eq_one _ _ _ hs] using
preBrownian_past_vector_drift_withDensity hB hmB hT s
end Problem367
end
end OAI
|