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a89d282 | 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.InvariantIsing.Gaussian.CountableGaussianComparison
import OAI.Probability.IsingPerceptron.MonomialGgAtMinimum
import OAI.Probability.IsingPerceptron.JointGGClosure
namespace OAI
/-! Stability of the actual Gaussian partition under bounded base-energy changes. -/
noncomputable section
open MeasureTheory ProbabilityTheory IsingPerceptron
namespace InvariantIsing
/-- An independent Gaussian perturbation changes the mean pressure by at
most half its maximal variance, including after a fixed base-energy tilt. -/
theorem countable_cylinder_log_mean_cost {X : Type*} [MeasurableSpace X] [Countable X]
[MeasurableSingletonClass X] (ν : Measure X) [IsProbabilityMeasure ν]
(H : X → ℝ) (hH : Integrable (fun x => Real.exp (H x)) ν)
(A : X → ℕ →₀ ℝ) {B : ℝ} (hA : ∀ x, (A x).sum (fun _ c => c ^ 2) ≤ B) :
let d := (∫ g : ℕ → ℝ, Real.log (∫ x, Real.exp (H x + cylinderField (A x) g) ∂ν)
∂gaussianCoordinates) - Real.log (∫ x, Real.exp (H x) ∂ν)
0 ≤ d ∧ d ≤ B / 2 := by
have : IsProbabilityMeasure (ν.tilted H) := isProbabilityMeasure_tilted hH
dsimp only
rw [cylinder_log_partition_mean_base ν H hH A hA, add_sub_cancel_left]
simpa only [one_pow, one_mul] using cylinder_cgf_mean_bounds (ν.tilted H) A hA 1
lemma cylinder_partition_exp_integrable_ae {X : Type*} [MeasurableSpace X] [Countable X]
[MeasurableSingletonClass X] (ν : Measure X) [IsProbabilityMeasure ν]
(H : X → ℝ) (hH : Integrable (fun x => Real.exp (H x)) ν)
(A : X → ℕ →₀ ℝ) {B : ℝ} (hA : ∀ x, (A x).sum (fun _ c => c ^ 2) ≤ B) :
∀ᵐ g : ℕ → ℝ ∂gaussianCoordinates,
Integrable (fun x => Real.exp (H x + cylinderField (A x) g)) ν := by
have : IsProbabilityMeasure (ν.tilted H) := isProbabilityMeasure_tilted hH
have hv (x : X) : (((A x).sum (fun _ c => c ^ 2)).toNNReal : ℝ) ≤ B := by
rw [Real.coe_toNNReal _ (show 0 ≤ (A x).sum (fun _ c => c ^ 2) from
Finset.sum_nonneg (fun _ _ => sq_nonneg _))]
exact hA x
have he := (gaussian_field_exp_integrable (ν := ν.tilted H)
(measurable_cylinderFields A) (fun x => cylinderField_law (A x)) hv 1).1.prod_right_ae
filter_upwards [he] with g hg
simpa only [one_mul, smul_eq_mul, ← Real.exp_add] using
(integrable_tilted_iff hH (fun x => Real.exp (1 * cylinderField (A x) g))).mp hg
/-- A uniform deterministic energy change bounds the mean logpartition
change, for every countable prior and bounded-variance Gaussian field. -/
theorem countable_cylinder_log_mean_base_compare {X : Type*} [MeasurableSpace X] [Countable X]
[MeasurableSingletonClass X] (ν : Measure X) [IsProbabilityMeasure ν]
(H J : X → ℝ) (hH : Integrable (fun x => Real.exp (H x)) ν)
(hJ : Integrable (fun x => Real.exp (J x)) ν)
(A : X → ℕ →₀ ℝ) {B K : ℝ} (hA : ∀ x, (A x).sum (fun _ c => c ^ 2) ≤ B)
(hK : ∀ x, |H x - J x| ≤ K) :
|(∫ g : ℕ → ℝ, Real.log (∫ x, Real.exp (H x + cylinderField (A x) g) ∂ν)
∂gaussianCoordinates) -
∫ g : ℕ → ℝ, Real.log (∫ x, Real.exp (J x + cylinderField (A x) g) ∂ν)
∂gaussianCoordinates| ≤ K := by
have hiH := (cylinder_log_partition_memLp_two ν H hH A hA 1).integrable (by norm_num)
have hiJ := (cylinder_log_partition_memLp_two ν J hJ A hA 1).integrable (by norm_num)
simp only [one_mul] at hiH hiJ
rw [← integral_sub hiH hiJ, ← Real.norm_eq_abs]
apply (norm_integral_le_of_norm_le_const (μ := gaussianCoordinates) (C := K) ?_).trans_eq (by simp)
filter_upwards [cylinder_partition_exp_integrable_ae ν H hH A hA,
cylinder_partition_exp_integrable_ae ν J hJ A hA] with g hgH hgJ
have h := logMean_abs_sub_le ν (b := 1) (by norm_num)
(by simpa only [one_mul] using hgH) (by simpa only [one_mul] using hgJ)
(fun x => by simpa only [add_sub_add_right_eq_sub] using hK x)
simpa only [logMean, div_one, one_mul, Real.norm_eq_abs] using h
end InvariantIsing
end
end OAI
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