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c4cdbbd | 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 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 | import OAI.Probability.InvariantIsing.Fields.FieldBounds
import OAI.Probability.IsingPerceptron.CascadePair
namespace OAI
/-!
The finite-depth Gaussian marking input used in the one-site functional.
The logarithmic terminal is unbounded; the linear-growth cascade theorem
supplies its integrability. This is the Gaussian specialization of
Bolthausen--Sznitman (1998), Appendix Proposition A.2.
-/
noncomputable section
open MeasureTheory ProbabilityTheory
open scoped BigOperators NNReal
namespace InvariantIsing
lemma integral_gaussianReal_zero_eq_standard (v : ℝ≥0) {F : ℝ → ℝ}
(hF : Measurable F) :
(∫ g, F g ∂gaussianReal 0 v) =
∫ g, F (Real.sqrt (v : ℝ) * g) ∂gaussianReal 0 1 := by
have hm : (gaussianReal 0 1).map (fun g => Real.sqrt (v : ℝ) * g) =
gaussianReal 0 v := by
convert! gaussianReal_map_const_mul (μ := 0) (v := 1) (Real.sqrt (v : ℝ)) using 1
simp only [mul_zero, mul_one]
congr 1
ext
simp only [NNReal.coe_mk, Real.sq_sqrt v.coe_nonneg]
rw [← hm]
exact integral_map
(show AEMeasurable (fun g : ℝ => Real.sqrt (v : ℝ) * g) (gaussianReal 0 1) from
(measurable_id.const_mul _).aemeasurable) hF.aestronglyMeasurable
lemma gaussianOperator_eq_gaussian_logMean {a : ℝ} (ha : a ≠ 0) (v : ℝ≥0)
{F : ℝ → ℝ} (hF : Measurable F) (x : ℝ) :
gaussianOperator a v F x =
IsingPerceptron.logMean a (gaussianReal 0 v) (fun g => F (x + g)) := by
simp only [gaussianOperator, ha, ite_false, IsingPerceptron.logMean]
have hm : Measurable (fun g : ℝ => Real.exp (a * F (x + g))) :=
(((hF.comp (measurable_const.add measurable_id)).const_mul a).exp)
rw [integral_gaussianReal_zero_eq_standard v hm]
ring
lemma gaussianCascadeRecursion_eq_fold (n : ℕ) (b : ℕ → ℝ) (v : ℕ → ℝ≥0)
(hb : ∀ i < n, b i ≠ 0) {F : ℝ → ℝ} (hF : Measurable F) :
IsingPerceptron.cascadeRecursion n b
(fun i => ⟨gaussianReal 0 (v i), inferInstance⟩)
(fun _ p => p.1 + p.2) F =
(List.ofFn (fun i : Fin n => (b i, (v i : ℝ)))).foldr
(fun av U => gaussianOperator av.1 av.2 U) F := by
induction n generalizing b v with
| zero => rfl
| succ n ih =>
have ht := IsingPerceptron.measurable_cascadeRecursion n (fun i => b (i + 1))
(fun i => (⟨gaussianReal 0 (v (i + 1)), inferInstance⟩ : ProbabilityMeasure ℝ))
(fun _ => measurable_fst.add measurable_snd) hF
have he := ih (fun i => b (i + 1)) (fun i => v (i + 1))
(fun i hi => hb (i + 1) (by omega))
change Measurable (IsingPerceptron.cascadeRecursion n (fun i => b (i + 1))
(fun i => (⟨gaussianReal 0 (v (i + 1)), inferInstance⟩ : ProbabilityMeasure ℝ))
(fun _ p => p.1 + p.2) F) at ht
rw [he] at ht
rw [List.ofFn_succ, List.foldr_cons]
funext x
change IsingPerceptron.logMean (b 0) (gaussianReal 0 (v 0))
(fun g => IsingPerceptron.cascadeRecursion n (fun i => b (i + 1))
(fun i => ⟨gaussianReal 0 (v (i + 1)), inferInstance⟩)
(fun _ p => p.1 + p.2) F (x + g)) = _
rw [he]
exact (gaussianOperator_eq_gaussian_logMean (hb 0 (by omega)) (v 0) ht x).symm
def gaussianCascadeLaw (n : ℕ) (b : ℕ → ℝ) (v : ℕ → ℝ≥0) :
Measure (IsingPerceptron.NoiseTree ℝ n) :=
IsingPerceptron.noiseCascadeLaw ℝ n b
(fun i => ⟨gaussianReal 0 (v i), inferInstance⟩)
def gaussianCascadeLog (n : ℕ) (x : ℝ) (T : IsingPerceptron.NoiseTree ℝ n) : ℝ :=
Real.log (∫ leaf, Real.exp (IsingPerceptron.noiseLeafTerminal n
(fun _ z => Real.log (Real.cosh z)) (fun _ p => p.1 + p.2) x leaf)
∂IsingPerceptron.noiseLeafKernel ℝ n T)
/-- Exact finite Gaussian marking recursion, with the integrability needed
to take the outer expectation. No boundedness of `log cosh` is assumed. -/
theorem gaussianCascadeLog_recursion (n : ℕ) (b : ℕ → ℝ) (v : ℕ → ℝ≥0)
(hb : IsingPerceptron.CascadeExponents n b) (x : ℝ) :
Integrable (gaussianCascadeLog n x) (gaussianCascadeLaw n b v) ∧
(∫ T, gaussianCascadeLog n x T ∂gaussianCascadeLaw n b v) =
((List.ofFn (fun i : Fin n => (b i, (v i : ℝ)))).foldr
(fun av U => gaussianOperator av.1 av.2 U)
(fun z => Real.log (Real.cosh z))) x := by
have h := IsingPerceptron.gaussian_logCosh_cascade n b hb (fun _ => 0) v x
have he := IsingPerceptron.noise_leaf_log_eq_ratio n b hb
(fun i => (⟨gaussianReal 0 (v i), inferInstance⟩ : ProbabilityMeasure ℝ))
(fun _ => measurable_fst.add measurable_snd) IsingPerceptron.measurable_logCosh x
refine ⟨h.1.congr (Filter.EventuallyEq.symm he), ?_⟩
calc
_ = _ := integral_congr_ae he
_ = _ := h.2.1
_ = _ := congrFun (gaussianCascadeRecursion_eq_fold n b v
(fun i hi => (hb.1 i hi).1.ne') IsingPerceptron.measurable_logCosh) x
end InvariantIsing
end
end OAI
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