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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 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 | import OAI.Probability.FreeIsing.Ising.ContinuousCovarianceIntegral
import OAI.Probability.FreeIsing.Ising.FiniteCovarianceEnergy
import OAI.Probability.FreeIsing.Ising.PosteriorIntegralRepresentation
import OAI.Probability.FreeIsing.Spectral.OrthogonalRowIdentification
namespace OAI
noncomputable section
open MeasureTheory ProbabilityTheory Filter
open scoped NNReal Topology BigOperators
namespace Problem367.IsingInnovation
variable {d : ℕ} {A : Type*} [MeasurableSpace A]
theorem posterior_martingale
{beta : ℝ} {μ : Measure (TreeVertex d → IsingSpin)}
(hμ : HasFreeIsingLaw (Real.tanh beta) μ)
(ν : Measure A) {B : ℝ≥0 → A → ℝ} (hB : IsPreBrownianReal B ν)
(hmB : ∀ t, Measurable (B t)) (hcB : ∀ a, Continuous (fun t => B t a))
(v : TreeVertex d) :
Martingale (posterior beta B v) (ArrayPosteriorStationarity.filtration B hmB)
(μ.prod (Measure.infinitePi (fun _ : TreeVertex d => ν))) := by
let : IsProbabilityMeasure μ := hμ.isProbabilityMeasure
let : IsProbabilityMeasure ν := hB.isGaussianProcess.isProbabilityMeasure
have h := martingale_condExp (fun q : State d A => spinSign (q.1 v))
(ArrayPosteriorStationarity.filtration B hmB)
(μ.prod (Measure.infinitePi (fun _ : TreeVertex d => ν)))
exact h.congr (posterior_progressive beta B hmB hcB v).stronglyAdapted
(fun t => (posterior_eq_condExp hμ ν hB hmB t v).symm)
theorem ae_eq_posterior_of_integer
{beta : ℝ} {μ : Measure (TreeVertex d → IsingSpin)}
(hμ : HasFreeIsingLaw (Real.tanh beta) μ)
(ν : Measure A) {B : ℝ≥0 → A → ℝ} (hB : IsPreBrownianReal B ν)
(hmB : ∀ t, Measurable (B t)) (hcB : ∀ a, Continuous (fun t => B t a))
(v : TreeVertex d) {Z : ℝ≥0 → State d A → ℝ}
(hZ : Martingale Z (ArrayPosteriorStationarity.filtration B hmB)
(μ.prod (Measure.infinitePi (fun _ : TreeVertex d => ν))))
(he : ∀ N : ℕ, 0 < N → Z N =ᵐ[μ.prod (Measure.infinitePi (fun _ : TreeVertex d => ν))]
posterior beta B v N) (t : ℝ≥0) :
Z t =ᵐ[μ.prod (Measure.infinitePi (fun _ : TreeVertex d => ν))]
posterior beta B v t := by
obtain ⟨N, hN⟩ := exists_nat_gt t
have hN0 : 0 < N := by
have hn : (0 : ℝ≥0) < (N : ℝ≥0) := lt_of_le_of_lt (show (0 : ℝ≥0) ≤ t from zero_le) hN
exact_mod_cast hn
exact (hZ.condExp_ae_eq hN.le).symm.trans
((condExp_congr_ae (he N hN0)).trans
((posterior_martingale hμ ν hB hmB hcB v).condExp_ae_eq hN.le))
theorem exists_continuous_posterior
(hd : 3 ≤ d) {beta : ℝ} {μ : Measure (TreeVertex d → IsingSpin)}
(hμ : HasFreeIsingLaw (Real.tanh beta) μ) (hb : 0 ≤ beta)
(hcrit : ((d - 1 : ℕ) : ℝ) * Real.tanh beta ^ 2 ≤ 1)
(ν : Measure A) {B : ℝ≥0 → A → ℝ} (hB : IsPreBrownianReal B ν)
(hmB : ∀ t, Measurable (B t)) (hcB : ∀ a, Continuous (fun t => B t a))
(v : TreeVertex d) :
let P := μ.prod (Measure.infinitePi (fun _ : TreeVertex d => ν))
let F := ArrayPosteriorStationarity.filtration B hmB
∃ Z : ℝ≥0 → State d A → ℝ,
IsStronglyPredictable F Z ∧ Martingale Z F P ∧
(∀ᵐ q ∂P, Continuous (fun t => Z t q)) ∧
(∀ t, MemLp (Z t) 2 P) ∧ Z 0 =ᵐ[P] 0 ∧
∀ t, Z t =ᵐ[P] posterior beta B v t := by
let : IsProbabilityMeasure μ := hμ.isProbabilityMeasure
let : IsProbabilityMeasure ν := hB.isGaussianProcess.isProbabilityMeasure
let P := μ.prod (Measure.infinitePi (fun _ : TreeVertex d => ν))
obtain ⟨J, Y, Z, hJ, hY, hZ, hYprop, hJlim, hJsum, hYE, hYdiff,
hZpred, hZM, hZcont, hZ0, hZmean, hYZ, hZE, hmax⟩ :=
exists_continuous_covariance_integral hd hμ hb hcrit ν hB hmB hcB v
refine ⟨Z, hZpred, hZM, hZcont, hZ, hZ0, ?_⟩
apply ae_eq_posterior_of_integer hμ ν hB hmB hcB v hZM
intro N hN
obtain ⟨X, I, hI, hM, hX, hXlim, hXI, hmI, htI, hImean, hsI, hfinite⟩ :=
exists_posterior_integral_representation hμ hd hb hcrit ν hB hmB hcB v N
have hJX (n : ℕ) (u : TreeVertex d) : J n u N =ᵐ[P] X n u := by
apply BrownianGrid.ae_eq_of_mean_square_limits _ (hJ n u N) (hX n u)
(hJlim n u N hN) (hXlim n u)
intro k
exact BrownianFiltrationGrid.uniformSum_memLp
(isBrownianReal_innovation hμ ν hB hmB hcB u).toIsPreBrownianReal
(innovation_martingale hμ ν hB hmB hcB u).stronglyAdapted
(fun s t hst => BrownianArrayFiltration.coordinate_increment_indep_past
(innovation_increment_indep_filtration hμ ν hB hmB hcB s t hst) u)
(finiteCovariance beta B v u n) (finiteCovariance_stronglyAdapted beta B hmB v u n)
(finiteCovariance_memLp beta B hmB P v u n) N k
have hJI (u : TreeVertex d) : Tendsto (fun n =>
∫ q, (J n u N q - I u q) ^ 2 ∂P) atTop (𝓝 0) := by
apply (hXI u).congr'
apply Eventually.of_forall
intro n
apply integral_congr_ae
filter_upwards [hJX n u] with q hq
rw [hq]
have ho (n : ℕ) : Pairwise fun u w =>
∫ q, J n u N q * J n w N q ∂P = 0 :=
finiteCovariance_integrals_orthogonal hμ ν hB hmB hcB v n N
(fun u => hJ n u N) (fun u => hJlim n u N hN)
have he : (∫ q, Z N q ^ 2 ∂P) = ∫ q, posterior beta B v N q ^ 2 ∂P :=
tendsto_nhds_unique (hZE N)
(finiteCovariance_timeEnergy_tendsto hμ ν hB hmB hcB v N)
exact FilteringRepresentationClosure.ae_eq_of_orthogonal_row_meanSquare_limits
(fun n u => hJ n u N) (fun n => hY n N) hI (hZ N) hM
(fun n => hJsum n N) ho hJI (hYZ N) hsI he
end Problem367.IsingInnovation
end
end OAI
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