| import OAI.Probability.FreeIsing.Posterior.PlantedSymmetry |
| import OAI.Probability.FreeIsing.Spins.Moments |
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| namespace OAI |
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| noncomputable section |
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| open MeasureTheory ProbabilityTheory |
| open scoped NNReal |
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| namespace Problem367 |
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| theorem cond_prod_first_event {Ω A : Type*} [MeasurableSpace Ω] [MeasurableSpace A] |
| (μ : Measure Ω) (ν : Measure A) [SFinite μ] [IsProbabilityMeasure ν] |
| (E : Set Ω) : |
| (μ.prod ν)[|E ×ˢ Set.univ] = (μ[|E]).prod ν := by |
| rw [ProbabilityTheory.cond, ProbabilityTheory.cond, Measure.prod_prod, measure_univ, mul_one, |
| ← Measure.restrict_prod_eq_prod_univ, Measure.prod_smul_left] |
|
|
| theorem HasFreeIsingLaw.measurePreserving_noise_cond_spin |
| {d : ℕ} {θ : ℝ} {A : Type*} [MeasurableSpace A] |
| {μ : Measure (TreeVertex d → IsingSpin)} (hμ : HasFreeIsingLaw θ μ) |
| (ν : TreeVertex d → Measure A) [∀ v, IsProbabilityMeasure (ν v)] |
| (r w : TreeVertex d) (s : IsingSpin) : |
| MeasurePreserving (fun p : (TreeVertex d → IsingSpin) × (TreeVertex d → A) => p.2 w) |
| (μ.prod (Measure.infinitePi ν))[|{p | p.1 r = s}] (ν w) := by |
| let := hμ.isProbabilityMeasure |
| have hmass : μ {σ | σ r = s} ≠ 0 := by rw [hμ.marginal]; norm_num |
| let : IsProbabilityMeasure μ[|{σ | σ r = s}] := cond_isProbabilityMeasure hmass |
| have hset : {p : (TreeVertex d → IsingSpin) × (TreeVertex d → A) | p.1 r = s} = |
| {σ : TreeVertex d → IsingSpin | σ r = s} ×ˢ Set.univ := by ext p; simp |
| rw [hset, cond_prod_first_event] |
| refine ⟨(measurable_pi_apply w).comp measurable_snd, ?_⟩ |
| change Measure.map ((fun z : TreeVertex d → A => z w) ∘ Prod.snd) _ = _ |
| rw [← Measure.map_map (measurable_pi_apply w) measurable_snd, |
| Measure.map_snd_prod, measure_univ, one_smul, Measure.infinitePi_map_eval] |
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| theorem HasFreeIsingLaw.gaussian_noise_hasLaw_cond_spin |
| {d : ℕ} {θ : ℝ} {μ : Measure (TreeVertex d → IsingSpin)} |
| (hμ : HasFreeIsingLaw θ μ) (t : ℝ≥0) (r w : TreeVertex d) (s : IsingSpin) : |
| HasLaw (fun p : (TreeVertex d → IsingSpin) × (TreeVertex d → ℝ) => p.2 w) |
| (gaussianReal 0 t) |
| (μ.prod (PlantedSymmetry.gaussianNoiseLaw (TreeVertex d) t))[|{p | p.1 r = s}] := by |
| exact (hμ.measurePreserving_noise_cond_spin (fun _ => gaussianReal 0 t) r w s).hasLaw |
|
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| theorem HasFreeIsingLaw.standard_gaussian_hasLaw_cond_spin |
| {d : ℕ} {θ : ℝ} {μ : Measure (TreeVertex d → IsingSpin)} |
| (hμ : HasFreeIsingLaw θ μ) (t : ℝ≥0) (ht : 0 < t) |
| (r w : TreeVertex d) (s : IsingSpin) : |
| HasLaw (fun p : (TreeVertex d → IsingSpin) × (TreeVertex d → ℝ) => |
| p.2 w / Real.sqrt t) (gaussianReal 0 1) |
| (μ.prod (PlantedSymmetry.gaussianNoiseLaw (TreeVertex d) t))[|{p | p.1 r = s}] := by |
| have h := gaussianReal_div_const (hμ.gaussian_noise_hasLaw_cond_spin t r w s) (Real.sqrt t) |
| have hs : t / NNReal.mk ((Real.sqrt t) ^ 2) (sq_nonneg _) = 1 := by |
| apply NNReal.eq |
| change (t : ℝ) / (Real.sqrt (t : ℝ)) ^ 2 = 1 |
| rw [Real.sq_sqrt (show (0 : ℝ) ≤ t from t.2), |
| div_self (ne_of_gt (show (0 : ℝ) < t from ht))] |
| simpa only [zero_div, hs] using h |
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| theorem HasFreeIsingLaw.own_gaussian_observation_cond_pos |
| {d : ℕ} {θ : ℝ} {μ : Measure (TreeVertex d → IsingSpin)} |
| (_hμ : HasFreeIsingLaw θ μ) (t : ℝ≥0) (ht : 0 < t) (r : TreeVertex d) : |
| ∀ᵐ p ∂(μ.prod (PlantedSymmetry.gaussianNoiseLaw (TreeVertex d) t))[|{p | p.1 r = .pos}], |
| PlantedSymmetry.observation t p r = |
| (t : ℝ) + Real.sqrt t * (p.2 r / Real.sqrt t) := by |
| have hs : MeasurableSet {p : (TreeVertex d → IsingSpin) × (TreeVertex d → ℝ) | |
| p.1 r = .pos} := |
| (show MeasurableSet ({IsingSpin.pos} : Set IsingSpin) from trivial).preimage |
| ((measurable_pi_apply r).comp measurable_fst) |
| filter_upwards [ae_cond_mem hs] with p hp |
| have hroot : p.1 r = .pos := hp |
| have hsqrt : Real.sqrt (t : ℝ) ≠ 0 := |
| (Real.sqrt_pos.2 (show (0 : ℝ) < t from ht)).ne' |
| simp only [PlantedSymmetry.observation, hroot, spinSign, mul_one] |
| rw [mul_div_cancel₀ _ hsqrt] |
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| end Problem367 |
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| end |
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| end OAI |
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