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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 | import OAI.Probability.InvariantIsing.Fields.FieldCanonicalEndpoint
import OAI.Probability.InvariantIsing.Fields.FieldEndpointSampling
import Mathlib.MeasureTheory.Integral.Pi
namespace OAI
/-! The full endpoint kernel preserves independence across the sites. -/
noncomputable section
open MeasureTheory ProbabilityTheory IsingPerceptron Set
open scoped BigOperators NNReal
namespace InvariantIsing
lemma kernel_comp_real {X Y : Type*} [MeasurableSpace X] [MeasurableSpace Y]
(κ : Kernel X Y) [IsMarkovKernel κ] (μ : Measure X) [IsProbabilityMeasure μ]
(s : Set Y) (hs : MeasurableSet s) :
(κ ∘ₘ μ).real s = ∫ x, (κ x).real s ∂μ := by
have hi : Integrable (s.indicator (fun _ : Y => (1 : ℝ))) (κ ∘ₘ μ) :=
(integrable_const 1).indicator hs
have he := integral_indicator_one (μ := κ ∘ₘ μ) hs
rw [Measure.comp_eq_comp_const_apply] at hi
have hc := Kernel.integral_comp hi
simp only [Kernel.const_apply] at hc
rw [← Measure.comp_eq_comp_const_apply] at hc
have ht (x : X) : (∫ y, s.indicator (fun _ : Y => (1 : ℝ)) y ∂κ x) =
(κ x).real s := integral_indicator_one (μ := κ x) hs
simp_rw [ht] at hc
exact he.symm.trans hc
lemma product_kernel_comp {N : ℕ} (κ : Kernel ℝ ℝ) [IsMarkovKernel κ]
(K : Kernel (Fin N → ℝ) (Fin N → ℝ)) [IsMarkovKernel K]
(hK : ∀ z, K z = Measure.pi (fun i => κ (z i)))
(ν : Fin N → Measure ℝ) [∀ i, IsProbabilityMeasure (ν i)] :
K ∘ₘ Measure.pi ν = Measure.pi (fun i => κ ∘ₘ ν i) := by
apply (Measure.pi_eq (μ := fun i => κ ∘ₘ ν i) ?_).symm
intro s hs
suffices hreal : ((K ∘ₘ Measure.pi ν) (univ.pi s)).toReal =
(∏ i, (κ ∘ₘ ν i) (s i)).toReal by
have he := congrArg ENNReal.ofReal hreal
rw [ENNReal.ofReal_toReal (measure_ne_top _ _),
ENNReal.ofReal_toReal (ENNReal.prod_ne_top (fun i _ => measure_ne_top _ _))] at he
exact he
rw [← measureReal_def, kernel_comp_real K (Measure.pi ν) _ (MeasurableSet.univ_pi hs)]
have he (z : Fin N → ℝ) : (K z).real (univ.pi s) =
∏ i, (κ (z i)).real (s i) := by
rw [hK, measureReal_def, Measure.pi_pi, ENNReal.toReal_prod]
rfl
simp_rw [he]
change (∫ z : Fin N → ℝ, ∏ i,
(fun y : ℝ => (κ y).real (s i)) (z i) ∂Measure.pi ν) = _
rw [integral_fintype_prod_eq_prod (fun i y => (κ y).real (s i)), ENNReal.toReal_prod]
apply Finset.prod_congr rfl
intro i _
exact (kernel_comp_real κ (ν i) (s i) (hs i)).symm
theorem fieldVectorTailEndpointKernel_product (N : ℕ) (L : List (ℝ × ℝ≥0))
(hL : ∀ av ∈ L, 0 < av.1) (z : Fin N → ℝ) :
fieldVectorTailEndpointKernel N L hL z =
Measure.pi (fun i => fieldTailEndpointKernel L hL (z i)) := by
induction L generalizing z with
| nil =>
change Measure.dirac z = Measure.pi (fun i => Measure.dirac (z i))
apply (Measure.pi_eq (μ := fun i => Measure.dirac (z i)) ?_).symm
intro s hs
simp only [Measure.dirac_apply' _ (MeasurableSet.univ_pi hs),
Measure.dirac_apply' _ (hs _)]
have hz : (z ∈ univ.pi s) ↔ (∀ i ∈ (Finset.univ : Finset (Fin N)), z i ∈ s i) := by
change (∀ i ∈ (Set.univ : Set (Fin N)), z i ∈ s i) ↔ _
simp only [Set.mem_univ, Finset.mem_univ, true_implies]
simp only [Set.indicator, Pi.one_apply, Finset.prod_boole]
by_cases h : z ∈ univ.pi s
· simp only [ite_eq_left h, ite_eq_left (hz.mp h)]
· have hn : ¬ ∀ i ∈ (Finset.univ : Finset (Fin N)), z i ∈ s i := fun hh => h (hz.mpr hh)
simp only [ite_eq_right h, ite_eq_right hn]
| cons av L ih =>
have ht := fun bv hb => hL bv (List.mem_cons_of_mem av hb)
have hreg := fieldScalarValue_regular L ht measurable_logCosh logCosh_linearGrowth
change (fieldVectorTailEndpointKernel N L ht ∘ₖ
fieldVectorTransitionKernel N av.1 av.2 _ hreg.1) z = _
rw [Kernel.comp_apply]
change fieldVectorTailEndpointKernel N L ht ∘ₘ
Measure.pi (fun i => fieldTransitionKernel av.1 av.2 _ hreg.1 (z i)) = _
rw [product_kernel_comp (fieldTailEndpointKernel L ht)
(fieldVectorTailEndpointKernel N L ht) (ih ht)]
rfl
end InvariantIsing
end
end OAI
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