| import OAI.Probability.FreeIsing.Posterior.GaussianEnergy |
| import OAI.Probability.FreeIsing.Posterior.SqrtObservationDerivative |
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| namespace OAI |
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|
| noncomputable section |
| open MeasureTheory ProbabilityTheory Set |
| open scoped NNReal |
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| namespace Problem367.FinitePosterior |
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| variable {S V : Type*} [Fintype S] [Nonempty S] [Fintype V] [DecidableEq V] |
| [MeasurableSpace S] [MeasurableSingletonClass S] [StandardBorelSpace S] |
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|
| theorem integral_finiteStandardEnergy |
| (μ : Measure S) [IsProbabilityMeasure μ] |
| (energy : S → ℝ) |
| (hprior : ∀ s, μ.real {s} = Real.exp (energy s) / ∑ a, Real.exp (energy a)) |
| (spin : S → V → ℝ) (observable : S → ℝ) |
| (A : ℝ) (hbound : ∀ s, |observable s| ≤ A) |
| (hcoords : ∀ s v, spin s v = -1 ∨ spin s v = 1) |
| {T : ℝ} (hT : 0 ≤ T) : |
| let C : ℝ → ℝ := fun t => ∫ x, ∑ v, fieldCovariance energy spin observable |
| (finiteStandardObservation spin t x) v ^ 2 |
| ∂μ.prod (Measure.pi (fun _ : V => gaussianReal 0 1)) |
| IntervalIntegrable C volume 0 T ∧ |
| (∫ t in 0..T, C t) = |
| (∫ x, fieldMean energy spin observable (finiteStandardObservation spin T x) ^ 2 |
| ∂μ.prod (Measure.pi (fun _ : V => gaussianReal 0 1))) - |
| fieldMean energy spin observable 0 ^ 2 ∧ |
| 0 ≤ (∫ t in 0..T, C t) ∧ (∫ t in 0..T, C t) ≤ A ^ 2 := by |
| let ρ := μ.prod (Measure.pi (fun _ : V => gaussianReal 0 1)) |
| let E : ℝ → ℝ := fun t => ∫ x, |
| fieldMean energy spin observable (finiteStandardObservation spin t x) ^ 2 ∂ρ |
| let C : ℝ → ℝ := fun t => ∫ x, ∑ v, fieldCovariance energy spin observable |
| (finiteStandardObservation spin t x) v ^ 2 ∂ρ |
| have hE : Continuous E := by |
| simpa only [E, finiteStandardObservation, gaussianObservationCurve, Pi.pow_apply, Function.comp_def] using |
| (continuous_integral_sqrt_observation (P := ρ) |
| (((measurable_of_finite spin).comp measurable_fst).aestronglyMeasurable) |
| (measurable_snd.aestronglyMeasurable) |
| ((continuous_fieldMean energy spin observable).pow 2) |
| (fun x => abs_fieldMean_square_le energy spin observable x A hbound)) |
| have hd (t : ℝ) (ht : 0 < t) : HasDerivAt E (C t) t := |
| hasDerivAt_finiteStandardEnergy μ energy hprior spin observable A |
| hbound hcoords ht |
| have hCpos (t : ℝ) : 0 ≤ C t := |
| integral_nonneg fun x => Finset.sum_nonneg fun v _ => sq_nonneg _ |
| have hCI : IntervalIntegrable C volume 0 T := by |
| apply intervalIntegral.intervalIntegrable_deriv_of_nonneg hE.continuousOn |
| · intro t ht |
| exact hd t (by simpa only [min_eq_left hT, max_eq_right hT] using ht.1) |
| · exact fun t _ => hCpos t |
| have hFTC : (∫ t in 0..T, C t) = E T - E 0 := |
| intervalIntegral.integral_eq_sub_of_hasDerivAt_of_le hT hE.continuousOn |
| (fun t ht => hd t ht.1) hCI |
| have hzero : E 0 = fieldMean energy spin observable 0 ^ 2 := by |
| simp [E, finiteStandardObservation, gaussianObservationCurve, ρ] |
| have hm : AEStronglyMeasurable |
| (fun x => fieldMean energy spin observable (finiteStandardObservation spin T x) ^ 2) ρ := |
| (((continuous_fieldMean energy spin observable).pow 2).measurable.comp |
| (measurable_finiteStandardObservation spin T)).aestronglyMeasurable |
| have hb (x : S × (V → ℝ)) : |
| |fieldMean energy spin observable (finiteStandardObservation spin T x) ^ 2| ≤ A ^ 2 := |
| abs_fieldMean_square_le energy spin observable _ A hbound |
| have hInt : Integrable |
| (fun x => fieldMean energy spin observable (finiteStandardObservation spin T x) ^ 2) ρ := |
| Integrable.of_bound hm (A ^ 2) (Filter.Eventually.of_forall fun x => by |
| simpa only [Real.norm_eq_abs] using hb x) |
| have hEbound : E T ≤ A ^ 2 := by |
| calc |
| E T ≤ ∫ _x, A ^ 2 ∂ρ := integral_mono hInt (integrable_const _) fun x => |
| (le_abs_self _).trans (hb x) |
| _ = A ^ 2 := by simp [ρ] |
| refine ⟨hCI, ?_, intervalIntegral.integral_nonneg_of_forall hT hCpos, ?_⟩ |
| · simpa only [hzero] using hFTC |
| · rw [hFTC, hzero] |
| exact sub_le_self _ (sq_nonneg _) |>.trans hEbound |
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| end Problem367.FinitePosterior |
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| end |
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| end OAI |
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