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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 | import OAI.Probability.FreeIsing.Conditioning.PlantedInformation
import OAI.Probability.FreeIsing.Energy.ComponentCovariance
namespace OAI
noncomputable section
open MeasureTheory
open scoped BigOperators NNReal
namespace Problem367
abbrev ComponentIndex {d : ℕ} (c : Fin d) := Σ n : ℕ, ReducedTail c n
def componentIndexVertex {d : ℕ} {c : Fin d} (u : ComponentIndex c) : TreeVertex d :=
componentVertex u.2
theorem componentIndexVertex_injective {d : ℕ} (c : Fin d) :
Function.Injective (componentIndexVertex (c := c)) := by
rintro ⟨k, u⟩ ⟨l, v⟩ h
have hw : u.1 = v.1 := (List.cons.inj (congrArg Subtype.val h)).2
have hkl : k = l := by
simpa only [u.2.1, v.2.1] using congrArg List.length hw
subst l
have huv : u = v := Subtype.ext hw
subst v
rfl
theorem HasFreeIsingLaw.component_critical_information_lower {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) :
min (Real.sqrt (t : ℝ)) 1 / 5 ≤
∫ p, (condExp
(PlantedInformation.observationSigma (componentIndexVertex (c := c)) t)
(mu.prod (PlantedSymmetry.gaussianNoiseLaw (TreeVertex d) t))
(fun q => spinSign (q.1 ⟨[c], List.IsChain.singleton c⟩)) p) ^ 2
∂mu.prod (PlantedSymmetry.gaussianNoiseLaw (TreeVertex d) t) := by
classical
let := hmu.isProbabilityMeasure
apply PlantedInformation.critical_information_of_spin_sums mu
(componentIndexVertex (c := c)) (componentIndexVertex_injective c)
⟨[c], List.IsChain.singleton c⟩ ht
(componentIndexFinset c) (fun _ u => theta ^ u.1)
· intro n
simp only [componentIndexFinset, Finset.sum_sigma]
have hlen (k : ℕ) (u : ReducedTail c k) : u.1.length = k := u.2.1
simpa only [hlen] using sum_component_radial_sq_levels_critical theta c hcritical n
· intro n
simp only [componentIndexFinset, Finset.sum_sigma, componentIndexVertex]
exact hmu.component_signal_critical hlo hhi c hcritical n
· intro n
simp only [componentIndexFinset, Finset.sum_sigma, componentIndexVertex]
calc
_ = ∑ k ∈ Finset.range (n + 1), ∑ l ∈ Finset.range (n + 1),
∑ u : ReducedTail c k, ∑ v : ReducedTail c l,
theta ^ k * theta ^ l *
(∫ σ, spinSign (σ (componentVertex u)) * spinSign (σ (componentVertex v)) ∂mu) := by
apply Finset.sum_congr rfl
intro k hk
rw [Finset.sum_comm]
_ ≤ _ := hmu.component_covariance_sum_critical_le hlo hhi c hcritical n
theorem HasFreeIsingLaw.component_critical_information_lower_of_le
{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)
(m' : MeasurableSpace ((TreeVertex d → IsingSpin) × (TreeVertex d → ℝ)))
(hm' : m' ≤ MeasurableSpace.prod inferInstance inferInstance)
(hobs : PlantedInformation.observationSigma (componentIndexVertex (c := c)) t ≤ m') :
min (Real.sqrt (t : ℝ)) 1 / 5 ≤
∫ p, (condExp m'
(mu.prod (PlantedSymmetry.gaussianNoiseLaw (TreeVertex d) t))
(fun q => spinSign (q.1 ⟨[c], List.IsChain.singleton c⟩)) p) ^ 2
∂mu.prod (PlantedSymmetry.gaussianNoiseLaw (TreeVertex d) t) := by
let := hmu.isProbabilityMeasure
apply (hmu.component_critical_information_lower hlo hhi c hcritical ht).trans
apply posterior_information_mono hobs hm'
exact (PlantedInformation.spinSign_memLp mu ⟨[c], List.IsChain.singleton c⟩).comp_measurePreserving
measurePreserving_fst
end Problem367
end
end OAI
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