File size: 4,096 Bytes
f61e547 | 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 | import OAI.Probability.FreeIsing.Posterior.PlantedSymmetry
import OAI.Probability.FreeIsing.Spins.Moments
namespace OAI
noncomputable section
open MeasureTheory ProbabilityTheory
open scoped NNReal
namespace Problem367
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]
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
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
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]
end Problem367
end
end OAI
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