| import OAI.Analysis.IrrationalRotation.Model.OneCoordinateCoupling |
| import OAI.Analysis.IrrationalRotation.Model.FiniteKernelMixture |
| import OAI.Analysis.IrrationalRotation.Model.UniformQuantization |
|
|
| namespace OAI |
|
|
| noncomputable section |
| open MeasureTheory ProbabilityTheory |
| open scoped ENNReal BigOperators |
| namespace Invariant233.CircleDigitCoupling |
| open CouplingMarginalBounds CouplingMixture OneCoordinateCoupling |
|
|
| abbrev Label := Option (Fin 5) |
| abbrev CoordinateSample := Label × (ℝ × ℝ) |
|
|
| def endpointLaw (side : Bool) (m a b : ℝ) : Label → Measure ℝ |
| | none => baseLaw |
| | some i => if side then rightLaw (2 * m) a b i else leftLaw (2 * m) a b i |
|
|
| def endpointDensity (side : Bool) (m a b : ℝ) : Label → ℝ → ℝ |
| | none => fun _ => 1 |
| | some i => if side then rightDensity (2 * m) a b i else leftDensity (2 * m) a b i |
|
|
| def marginalDensity (side : Bool) (κ m a b t x : ℝ) : ℝ := |
| diagonalMass κ m a b t + |
| ∑ i, componentProbabilities κ m a b t i * endpointDensity side m a b (some i) x |
|
|
| def labelMass (κ m a b t : ℝ) : Label → ℝ≥0∞ |
| | none => ENNReal.ofReal (diagonalMass κ m a b t) |
| | some i => ENNReal.ofReal (componentProbabilities κ m a b t i) |
|
|
| def rawLaw (κ m a b t : ℝ) : Measure CoordinateSample := |
| labelledMixture baseLaw (ENNReal.ofReal (diagonalMass κ m a b t)) |
| (fun i => ENNReal.ofReal (componentProbabilities κ m a b t i)) |
| (leftLaw (2 * m) a b) (rightLaw (2 * m) a b) |
|
|
| def endpoint (side : Bool) (z : CoordinateSample) : ℝ := |
| if side then z.2.2 else z.2.1 |
|
|
| def bin (r : ℕ) (d : Fin r) : Set ℝ := |
| Set.Ico ((d : ℝ) / r) (((d : ℝ) + 1) / r) |
|
|
| def digitJoint (side : Bool) (r : ℕ) [NeZero r] (κ m a b t : ℝ) : |
| Measure (Fin r × Label) := |
| (rawLaw κ m a b t).map (fun z => |
| (UniformQuantization.quantize r (endpoint side z), z.1)) |
|
|
| def digitWeight (side : Bool) (r : ℕ) (κ m a b t : ℝ) |
| (l : Label) (d : Fin r) : ℝ≥0∞ := |
| labelMass κ m a b t l * endpointLaw side m a b l (bin r d) |
|
|
| def labelsGivenDigit (side : Bool) (r : ℕ) (κ m a b t : ℝ) : Kernel (Fin r) Label := |
| FiniteKernelMixture.normalizedKernel |
| (fun l => Kernel.const (Fin r) (Measure.dirac l)) |
| (digitWeight side r κ m a b t) |
| (fun _ => measurable_of_countable _) |
|
|
| def width (a b : ℝ) (which : Bool) : ℝ := if which then b else a |
|
|
| def topLabel (which : Bool) : Label := some (if which then 2 else 0) |
| def bottomLabel (which : Bool) : Label := some (if which then 3 else 1) |
|
|
| end Invariant233.CircleDigitCoupling |
|
|
| end |
|
|
| end OAI |
|
|
