File size: 3,319 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 | import OAI.Probability.FreeIsing.Cavity.LogOdds
import OAI.Probability.FreeIsing.Estimates.ComponentFieldRecursion
namespace OAI
noncomputable section
open MeasureTheory ProbabilityTheory
open scoped NNReal BigOperators
namespace Problem367
local instance : MeasurableSingletonClass IsingSpin := ⟨fun _ => trivial⟩
theorem HasFreeIsingLaw.gaussian_component_field_sub_observation_le
{d : ℕ} (hd : 2 ≤ d) {beta : ℝ}
{μ : Measure (TreeVertex d → IsingSpin)}
(hμ : HasFreeIsingLaw (Real.tanh beta) μ) (hb : 0 ≤ beta)
(t : ℝ≥0) (c : Fin d) :
let ρ := μ.prod (PlantedSymmetry.gaussianNoiseLaw (TreeVertex d) t)
let F : IsingSpin → ℝ → ℝ := fun s x => (t : ℝ) * spinSign s + x
∀ᵐ z ∂ρ,
|Real.artanh (selectedPosterior ρ F (componentIndexVertex (c := c))
(RootedBroadcast.branchRoot c) z) -
PlantedSymmetry.observation t z (RootedBroadcast.branchRoot c)| ≤
((d - 1 : ℕ) : ℝ) * beta := by
classical
filter_upwards [hμ.gaussian_component_field_recursion hd hb t c] with z hz
rw [hz, add_sub_cancel_left]
have hcard : Fintype.card {a : Fin d // c ≠ a} = d - 1 := by
simp [Fintype.card_subtype_compl]
calc
_ ≤ ∑ a : {a : Fin d // c ≠ a},
|cavityMessage (Real.tanh beta)
(Real.artanh (selectedPosterior
(μ.prod (PlantedSymmetry.gaussianNoiseLaw (TreeVertex d) t))
(fun s x => (t : ℝ) * spinSign s + x) (childComponentIndexVertex c a.1)
(prefixFlip c (RootedBroadcast.branchRoot a.1)) z))| :=
Finset.abs_sum_le_sum_abs _ _
_ ≤ ∑ _a : {a : Fin d // c ≠ a}, beta :=
Finset.sum_le_sum (fun _ _ => abs_cavityMessage_tanh_le hb _)
_ = ((d - 1 : ℕ) : ℝ) * beta := by simp [hcard]
theorem HasFreeIsingLaw.gaussian_component_field_recursion_cond_spin
{d : ℕ} (hd : 2 ≤ d) {beta : ℝ}
{μ : Measure (TreeVertex d → IsingSpin)}
(hμ : HasFreeIsingLaw (Real.tanh beta) μ) (hb : 0 ≤ beta)
(t : ℝ≥0) (c : Fin d) (s : IsingSpin) :
let ρ := μ.prod (PlantedSymmetry.gaussianNoiseLaw (TreeVertex d) t)
let F : IsingSpin → ℝ → ℝ := fun a x => (t : ℝ) * spinSign a + x
∀ᵐ z ∂cond ρ {z | z.1 (RootedBroadcast.branchRoot c) = s},
Real.artanh (selectedPosterior ρ F (componentIndexVertex (c := c))
(RootedBroadcast.branchRoot c) z) =
(t : ℝ) * spinSign s + z.2 (RootedBroadcast.branchRoot c) +
∑ a : {a : Fin d // c ≠ a}, cavityMessage (Real.tanh beta)
(Real.artanh (selectedPosterior ρ F (childComponentIndexVertex c a.1)
(prefixFlip c (RootedBroadcast.branchRoot a.1)) z)) := by
have hm : Measurable (fun z : (TreeVertex d → IsingSpin) × (TreeVertex d → ℝ) =>
z.1 (RootedBroadcast.branchRoot c)) :=
(measurable_pi_apply _).comp measurable_fst
have hset : MeasurableSet {z : (TreeVertex d → IsingSpin) × (TreeVertex d → ℝ) |
z.1 (RootedBroadcast.branchRoot c) = s} := hm (measurableSet_singleton s)
filter_upwards [cond_absolutelyContinuous.ae_le
(hμ.gaussian_component_field_recursion hd hb t c), ae_cond_mem hset] with z hz hs
rw [hz]
change z.1 (RootedBroadcast.branchRoot c) = s at hs
simp only [PlantedSymmetry.observation, hs]
end Problem367
end
end OAI
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