| import OAI.Probability.FreeIsing.Gaussian.LinearSignalTilt |
| import OAI.Probability.FreeIsing.Spins.PosteriorTilt |
|
|
| namespace OAI |
|
|
| noncomputable section |
|
|
| open MeasureTheory |
|
|
| namespace Problem367 |
|
|
| theorem map_cond_spin_observation_eq_linearSignalTilt |
| {Ω Y : Type*} {mΩ : MeasurableSpace Ω} {mY : MeasurableSpace Y} |
| {mu : Measure Ω} [IsFiniteMeasure mu] |
| {S : Ω → IsingSpin} (hS : Measurable S) |
| {X : Ω → Y} (hX : Measurable X) {q : Y → ℝ} (hq : Measurable q) |
| (hpost : mu[fun x => spinSign (S x) | mY.comap X] =ᵐ[mu] fun x => q (X x)) |
| (a : IsingSpin) (hfair : mu {x | S x = a} = (1 / 2 : ENNReal)) : |
| Measure.map X (ProbabilityTheory.cond mu {x | S x = a}) = |
| linearSignalTilt (Measure.map X mu) q (spinSign a) := by |
| have hQ : Measurable (fun x => q (X x)) := hq.comp hX |
| have hSb : ∀ᵐ x ∂mu, ‖spinSign (S x)‖ ≤ (1 : ℝ) := by |
| apply Filter.Eventually.of_forall |
| intro x |
| cases hs : S x <;> norm_num [spinSign, hs] |
| have hQb : ∀ᵐ x ∂mu, |q (X x)| ≤ 1 := by |
| have hb := ae_bdd_norm_condExp_of_ae_bdd_norm (m := mY.comap X) hSb |
| filter_upwards [hb, hpost] with x hx hp |
| simpa only [hp, Real.norm_eq_abs] using hx |
| have hQi : Integrable (fun x => q (X x)) mu := |
| Integrable.of_bound hQ.aestronglyMeasurable 1 (by simpa only [Real.norm_eq_abs] using hQb) |
| have ha : |spinSign a| ≤ 1 := by cases a <;> norm_num [spinSign] |
| let : IsProbabilityMeasure (ProbabilityTheory.cond mu {x | S x = a}) := |
| ProbabilityTheory.cond_isProbabilityMeasure (by rw [hfair]; norm_num) |
| have hdi := (integrable_const (1 : ℝ)).add (hQi.const_mul (spinSign a)) |
| let : IsFiniteMeasure (linearSignalTilt mu (fun x => q (X x)) (spinSign a)) := |
| isFiniteMeasure_withDensity_ofReal hdi.hasFiniteIntegral |
| have hmap : Measure.map X (linearSignalTilt mu (fun x => q (X x)) (spinSign a)) = |
| linearSignalTilt (Measure.map X mu) q (spinSign a) := by |
| unfold linearSignalTilt |
| exact map_withDensity_comp (mu := mu) (f := X) |
| (w := fun y => ENNReal.ofReal (1 + spinSign a * q y)) hX |
| (ENNReal.measurable_ofReal.comp (measurable_const.add (measurable_const.mul hq))) |
| rw [← hmap] |
| ext t ht |
| apply (ENNReal.toReal_eq_toReal_iff' (measure_ne_top _ _) (measure_ne_top _ _)).mp |
| let f : Y → ℝ := t.indicator (fun _ => 1) |
| have hfm : Measurable f := measurable_const.indicator ht |
| have hfint : Integrable (fun x => f (X x)) mu := by |
| refine Integrable.of_bound (hfm.comp hX).aestronglyMeasurable 1 ?_ |
| exact Filter.Eventually.of_forall (fun x => by |
| by_cases hx : X x ∈ t <;> simp [f, hx]) |
| have hfsmall : StronglyMeasurable[mY.comap X] (fun x => f (X x)) := |
| (hfm.comp (measurable_iff_comap_le.mpr le_rfl)).stronglyMeasurable |
| have hint : (∫ y, f y ∂Measure.map X (ProbabilityTheory.cond mu {x | S x = a})) = |
| ∫ y, f y ∂Measure.map X (linearSignalTilt mu (fun x => q (X x)) (spinSign a)) := by |
| rw [integral_map hX.aemeasurable hfm.aestronglyMeasurable, |
| integral_map hX.aemeasurable hfm.aestronglyMeasurable, |
| integral_cond_spin_posterior_tilt hX.comap_le hS hfsmall hfint a hfair, |
| integral_linearSignalTilt hQ hQb ha hfint, |
| ← integral_const_mul, |
| ← integral_add hfint ((integrable_boundedSignal_mul hQ hQb hfint).const_mul (spinSign a))] |
| apply integral_congr_ae |
| filter_upwards [hpost] with x hx |
| rw [hx] |
| ring |
| simpa [f, integral_indicator ht, measureReal_def] using hint |
| |
| end Problem367 |
| |
| end |
| |
| end OAI |
| |
