File size: 3,272 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 | import OAI.Probability.FreeIsing.Conditioning.ComponentInformation
import OAI.Probability.FreeIsing.Conditioning.Pullback
namespace OAI
noncomputable section
open MeasureTheory
open scoped NNReal
namespace Problem367
section Transport
variable {Ω Ξ : Type*} [mΩ : MeasurableSpace Ω] [mΞ : MeasurableSpace Ξ]
(μ : Measure Ω) (ν : Measure Ξ) [IsFiniteMeasure μ] [IsFiniteMeasure ν]
theorem posterior_information_comp_measurePreserving
(e : Ω → Ξ) (he : MeasurePreserving e μ ν)
(m : MeasurableSpace Ξ) (hm : m ≤ mΞ) {f : Ξ → ℝ} (hf : Integrable f ν) :
(∫ ω, (condExp (m.comap e) μ (f ∘ e) ω) ^ 2 ∂μ) =
∫ ξ, (condExp m ν f ξ) ^ 2 ∂ν := by
let : MeasurableSpace Ξ := mΞ
calc
_ = ∫ ω, (condExp m ν f (e ω)) ^ 2 ∂μ := by
apply integral_congr_ae
filter_upwards [condExp_comp_measurePreserving μ ν e he m hm hf] with ω hω
exact congrArg (fun z : ℝ => z ^ 2) hω
_ = _ := by
have hp : StronglyMeasurable (condExp m ν f) := stronglyMeasurable_condExp.mono hm
exact he.hasLaw.integral_comp (f := fun ξ => (condExp m ν f ξ) ^ 2)
(hp.pow 2).aestronglyMeasurable
theorem posterior_information_le_of_measurePreserving
(e : Ω → Ξ) (he : MeasurePreserving e μ ν)
(m : MeasurableSpace Ξ) (hm : m ≤ mΞ)
(M : MeasurableSpace Ω) (hM : M ≤ mΩ) (hobs : m.comap e ≤ M)
{f : Ξ → ℝ} (hf : MemLp f 2 ν) :
(∫ ξ, (condExp m ν f ξ) ^ 2 ∂ν) ≤
∫ ω, (condExp M μ (f ∘ e) ω) ^ 2 ∂μ := by
let : MeasurableSpace Ω := mΩ
let : MeasurableSpace Ξ := mΞ
rw [← posterior_information_comp_measurePreserving μ ν e he m hm
(hf.integrable (by norm_num))]
exact posterior_information_mono hobs hM (hf.comp_measurePreserving he)
end Transport
theorem HasFreeIsingLaw.component_critical_information_lower_transport
{d : ℕ} {theta : ℝ} {mu : Measure (TreeVertex d → IsingSpin)}
(hmu : HasFreeIsingLaw theta mu) (hlo : -1 ≤ theta) (hhi : theta ≤ 1)
(c : Fin d) (hcritical : ((d - 1 : ℕ) : ℝ) * theta ^ 2 = 1)
{t : ℝ≥0} (ht : 0 < t)
{Ω : Type*} [mΩ : MeasurableSpace Ω] (P : Measure Ω) [IsFiniteMeasure P]
(e : Ω → (TreeVertex d → IsingSpin) × (TreeVertex d → ℝ))
(he : MeasurePreserving e P (mu.prod (PlantedSymmetry.gaussianNoiseLaw (TreeVertex d) t)))
(M : MeasurableSpace Ω) (hM : M ≤ mΩ)
(hobs : (PlantedInformation.observationSigma (componentIndexVertex (c := c)) t).comap e ≤ M) :
min (Real.sqrt (t : ℝ)) 1 / 5 ≤
∫ ω, (condExp M P (fun η => spinSign ((e η).1 ⟨[c], List.IsChain.singleton c⟩)) ω) ^ 2 ∂P := by
let : MeasurableSpace Ω := mΩ
let := hmu.isProbabilityMeasure
apply (hmu.component_critical_information_lower hlo hhi c hcritical ht).trans
apply posterior_information_le_of_measurePreserving P
(mu.prod (PlantedSymmetry.gaussianNoiseLaw (TreeVertex d) t)) e he
(PlantedInformation.observationSigma (componentIndexVertex (c := c)) t)
(PlantedInformation.observationSigma_le _ _) M hM hobs
exact (PlantedInformation.spinSign_memLp mu ⟨[c], List.IsChain.singleton c⟩).comp_measurePreserving
measurePreserving_fst
end Problem367
end
end OAI
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