| import OAI.Probability.DilutedSpin.ExternalCovariance |
| import OAI.Probability.DilutedSpin.ExternalFreshLaw |
|
|
| namespace OAI |
|
|
| section |
| section |
| namespace DilutedSpinGlass.HeterogeneousMarks |
| open _root_.MeasureTheory _root_.OAI.MeasureTheory ProbabilityTheory |
| open scoped NNReal BigOperators |
| variable {Ω K Z X Y : Type} [Fintype Ω] |
| [Countable K] [MeasurableSpace K] [MeasurableSingletonClass K] |
| [Countable Z] [MeasurableSpace Z] [MeasurableSingletonClass Z] |
| [MeasurableSpace X] [MeasurableSpace Y] {L M : ℕ} |
| {A : K → Type} [∀ i, Fintype (A i)] |
| (ξ : Fin M → Measure Y) [∀ j, IsProbabilityMeasure (ξ j)] |
| (μ : Measure X) [IsProbabilityMeasure μ] (ν : Measure ((K × ℕ) × Z)) [IsProbabilityMeasure ν] |
| (τ : Measure Z) [IsProbabilityMeasure τ] (r s : ℝ≥0) (S : PrescribedTree L) (a : S.Leaf) |
| (T : KernelTower Ω L) (Q : (i : K) → Fin L → FiniteLaw (A i)) (m : Fin (L+1) → ℝ) |
| (base : RootPath Y M → (k : ℕ) → RootPath X k → FinitePath Ω L → ℝ) |
| (old : (i : (K × ℕ) × Z) → FinitePath Ω L → FinitePath (A i.1.1) L → ℝ) |
| (D E : (i : K) → Z → FinitePath Ω L → FinitePath (A i) L → ℝ) |
| (f : (S.Leaf → FinitePath Ω L) → ℝ) |
|
|
| omit [Countable K] [MeasurableSpace K] [MeasurableSingletonClass K] |
| [Countable Z] [MeasurableSpace Z] [MeasurableSingletonClass Z] |
| [MeasurableSpace X] [MeasurableSpace Y] in |
| /-- Changing only the fresh probe label does not change its alphabet, prior, |
| or observable. All old labels, factors, and counts remain literally fixed. -/ |
| lemma rootExternalTreeScore_probe (q : K) (j k : ℕ) (z : FullRootState Y X ((K×ℕ)×Z) M) |
| (v : Z) (t u : ℝ) : |
| rootExternalTreeScore S a T (fun i => Q i.1.1) (fun l => m l.succ) base old |
| (fun i x y => 1+t*D i.1.1 i.2 x y+u*E i.1.1 i.2 x y) |
| (fun i => E i.1.1 i.2) f z ((q,j),v) = |
| rootExternalTreeScore S a T (fun i => Q i.1.1) (fun l => m l.succ) base old |
| (fun i x y => 1+t*D i.1.1 i.2 x y+u*E i.1.1 i.2 x y) |
| (fun i => E i.1.1 i.2) f z ((q,k),v) := rfl |
|
|
| lemma externalAverage_probe |
| (hb : ∀ k y, Measurable (fun z : RootPath Y M × RootPath X k => base z.1 k z.2 y)) |
| {B : ℝ} (hf : ∀ x, |f x| ≤ B) |
| (hE : ∀ i v x y, |E i v x y| ≤ 1) (t u : ℝ) |
| (hA : ∀ i v x y, 1/2 ≤ 1+t*D i v x y+u*E i v x y) (q : K) (j k : ℕ) : |
| externalAverage ξ μ ν (τ.map (fun v => ((q,j),v))) r s S a T (fun i => Q i.1.1) m base old |
| (fun i => D i.1.1 i.2) (fun i => E i.1.1 i.2) f t u = |
| externalAverage ξ μ ν (τ.map (fun v => ((q,k),v))) r s S a T (fun i => Q i.1.1) m base old |
| (fun i => D i.1.1 i.2) (fun i => E i.1.1 i.2) f t u := by |
| rw [externalAverage_fresh_label ξ μ ν τ r s S a T (fun i => Q i.1.1) m base old (fun i => D i.1.1 i.2) (fun i => E i.1.1 i.2) f hb hf (fun i => hE i.1.1 i.2) t u (fun i => hA i.1.1 i.2) (fun v : Z => ((q,j),v)) (measurable_const.prodMk measurable_id), externalAverage_fresh_label ξ μ ν τ r s S a T (fun i => Q i.1.1) m base old (fun i => D i.1.1 i.2) (fun i => E i.1.1 i.2) f hb hf (fun i => hE i.1.1 i.2) t u (fun i => hA i.1.1 i.2) (fun v : Z => ((q,k),v)) (measurable_const.prodMk measurable_id)] |
| apply integral_congr_ae |
| filter_upwards [] with z |
| apply integral_congr_ae |
| filter_upwards [] with v |
| exact rootExternalTreeScore_probe S a T Q m base old D E f q j k z v t u |
|
|
| lemma externalCovariance_probe |
| (hb : ∀ k y, Measurable (fun z : RootPath Y M × RootPath X k => base z.1 k z.2 y)) |
| {B : ℝ} (hf : ∀ x, |f x| ≤ B) |
| (hE : ∀ i v x y, |E i v x y| ≤ 1) (t u : ℝ) |
| (hA : ∀ i v x y, 1/2 ≤ 1+t*D i v x y+u*E i v x y) (q : K) (j k : ℕ) : |
| externalCovariance ξ μ ν (τ.map (fun v => ((q,j),v))) r s S a T (fun i => Q i.1.1) m base old |
| (fun i => D i.1.1 i.2) (fun i => E i.1.1 i.2) f t u = |
| externalCovariance ξ μ ν (τ.map (fun v => ((q,k),v))) r s S a T (fun i => Q i.1.1) m base old |
| (fun i => D i.1.1 i.2) (fun i => E i.1.1 i.2) f t u := by |
| unfold externalCovariance |
| rw [externalAverage_probe ξ μ ν τ r s S a T Q m base old D E f hb hf hE t u hA q j k, externalAverage_probe ξ μ ν τ r s S a T Q m base old D E (fun _ => 1) hb (B := 1) (by intro; norm_num) hE t u hA q j k] |
|
|
| end DilutedSpinGlass.HeterogeneousMarks |
| end |
|
|
| end |
|
|
| end OAI |
|
|
