| 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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|
