File size: 3,082 Bytes
b317718 | 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 | import OAI.Probability.FreeIsing.Posterior.FiniteIsingFilteringRepresentation
namespace OAI
noncomputable section
open MeasureTheory ProbabilityTheory
open scoped NNReal
namespace Problem367.FiniteIsingFiltering
variable {d : ℕ} {Ω : Type*} [MeasurableSpace Ω]
variable (B : TreeVertex d → ℝ≥0 → Ω → ℝ)
variable (hmB : ∀ i t, Measurable (B i t))
theorem observationFiltration_mono_of_subset
{D E : Finset (TreeVertex d)} (hDE : D ⊆ E) :
observationFiltration B hmB D ≤ observationFiltration B hmB E := by
intro t
change (⨆ i : D, Filtration.natural (plantedArrayObservation B signal i)
(fun s => (plantedArrayObservation_measurable hmB measurable_signal i s).stronglyMeasurable) t) ≤
⨆ i : E, Filtration.natural (plantedArrayObservation B signal i)
(fun s => (plantedArrayObservation_measurable hmB measurable_signal i s).stronglyMeasurable) t
exact iSup_le fun i => le_iSup_of_le ⟨i.1, hDE i.2⟩ le_rfl
theorem observationFiltration_le_arrayObservationFiltration
(D : Finset (TreeVertex d)) :
observationFiltration B hmB D ≤
arrayObservationFiltration B hmB signal measurable_signal := by
intro t
change (⨆ i : D, Filtration.natural (plantedArrayObservation B signal i)
(fun s => (plantedArrayObservation_measurable hmB measurable_signal i s).stronglyMeasurable) t) ≤
⨆ i : TreeVertex d, Filtration.natural (plantedArrayObservation B signal i)
(fun s => (plantedArrayObservation_measurable hmB measurable_signal i s).stronglyMeasurable) t
exact iSup_le fun i => le_iSup_of_le i.1 le_rfl
theorem observationFiltration_monotone
{D : ℕ → Finset (TreeVertex d)} (hD : Monotone D) :
Monotone (fun n => observationFiltration B hmB (D n)) := by
intro n m hnm
exact observationFiltration_mono_of_subset B hmB (hD hnm)
theorem observationFiltration_time_monotone
{D : ℕ → Finset (TreeVertex d)} (hD : Monotone D) (t : ℝ≥0) :
Monotone (fun n => observationFiltration B hmB (D n) t) := by
intro n m hnm
exact observationFiltration_mono_of_subset B hmB (hD hnm) t
theorem iSup_observationFiltration_eq_arrayObservationFiltration
{κ : Type*} (D : κ → Finset (TreeVertex d))
(hcover : ∀ v, ∃ n, v ∈ D n) (t : ℝ≥0) :
(⨆ n, observationFiltration B hmB (D n) t) =
arrayObservationFiltration B hmB signal measurable_signal t := by
apply le_antisymm
· exact iSup_le fun n => observationFiltration_le_arrayObservationFiltration B hmB (D n) t
· change (⨆ i : TreeVertex d, Filtration.natural (plantedArrayObservation B signal i)
(fun s => (plantedArrayObservation_measurable hmB measurable_signal i s).stronglyMeasurable) t) ≤ _
refine iSup_le fun i => ?_
obtain ⟨n, hn⟩ := hcover i
apply le_iSup_of_le n
change _ ≤ ⨆ j : D n, Filtration.natural (plantedArrayObservation B signal j)
(fun s => (plantedArrayObservation_measurable hmB measurable_signal j s).stronglyMeasurable) t
exact le_iSup_of_le ⟨i, hn⟩ le_rfl
end Problem367.FiniteIsingFiltering
end
end OAI
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