| import OAI.Analysis.IrrationalRotation.Model.CircleDigitCoupling |
| import OAI.Analysis.IrrationalRotation.Model.CircleParameters |
| import OAI.Analysis.IrrationalRotation.Model.MixedRadixCircle |
| import Mathlib.Probability.ProductMeasure |
|
|
| namespace OAI |
|
|
| noncomputable section |
| open MeasureTheory ProbabilityTheory |
| namespace Invariant233.CircleCoupling |
| open CircleDigitCoupling CircleParameterData OneCoordinateCoupling |
|
|
| abbrev Sample := ℕ → CoordinateSample |
|
|
| def initialLaw : Measure CoordinateSample := |
| baseLaw.map (fun x => (none, (x, x))) |
|
|
| def tag (j s : ℕ) : ℝ := if s = j then 1 else 0 |
|
|
| def coordinateLaw (p : CircleParameterData) (j i : ℕ) : Measure CoordinateSample := |
| if i = 0 then initialLaw else |
| rawLaw (kappa (p.blockIndex i)) (p.m (p.blockIndex i)) |
| (p.a (p.blockIndex i)) (p.b (p.blockIndex i)) (tag j (p.blockIndex i)) |
|
|
| def law (p : CircleParameterData) (j : ℕ) : Measure Sample := |
| Measure.infinitePi (coordinateLaw p j) |
|
|
| def observedDigit (p : CircleParameterData) (hr : ∀ i, 0 < MixedRadix.radices p.q i) |
| (side : Bool) (i : ℕ) (z : CoordinateSample) : Fin (MixedRadix.radices p.q i) := by |
| letI : NeZero (MixedRadix.radices p.q i) := ⟨Nat.ne_of_gt (hr i)⟩ |
| exact UniformQuantization.quantize (MixedRadix.radices p.q i) (endpoint side z) |
|
|
| def digits (p : CircleParameterData) (hr : ∀ i, 0 < MixedRadix.radices p.q i) |
| (side : Bool) (ω : Sample) : MixedRadix.Digits (MixedRadix.radices p.q) := |
| fun i => observedDigit p hr side i (ω i) |
|
|
| def point (p : CircleParameterData) (hr : ∀ i, 0 < MixedRadix.radices p.q i) |
| (side : Bool) (ω : Sample) : Circle := MixedRadix.circleValue (digits p hr side ω) |
|
|
| def label (i : ℕ) (ω : Sample) : Label := (ω i).1 |
|
|
| def posterior (p : CircleParameterData) (j : ℕ) (side : Bool) (i : ℕ) : |
| Kernel (Fin (MixedRadix.radices p.q i)) Label := |
| if i = 0 then Kernel.const _ (Measure.dirac none) else |
| labelsGivenDigit side (MixedRadix.radices p.q i) (kappa (p.blockIndex i)) |
| (p.m (p.blockIndex i)) (p.a (p.blockIndex i)) (p.b (p.blockIndex i)) |
| (tag j (p.blockIndex i)) |
|
|
| end Invariant233.CircleCoupling |
|
|
| end |
|
|
| end OAI |
|
|
