| import Mathlib |
| import OAI.Probability.SKGap.Posterior.GOEPointwisePosterior |
| import OAI.Probability.SKGap.Posterior.ConditionalObservationData |
| import OAI.Probability.SKGap.Stability.ConditionalMeanScalar |
|
|
| namespace OAI |
|
|
| section |
| noncomputable section |
| namespace SKGap |
| open Matrix Real Set MeasureTheory ProbabilityTheory |
| open scoped BigOperators Matrix.Norms.Frobenius |
| |
| def empiricalObservation {n : ℕ} [NeZero n] (j t σ : ℝ) (y : Fin n → ℝ) : Fin n → ℝ := |
| fun i=>y i-scalarD j (empiricalLaw y,t,σ)+j*scalarBMoment (empiricalLaw y)*tanh (y i) |
|
|
| def empiricalConditionalGoe {n : ℕ} [NeZero n] (j t σ : ℝ) (y : Fin n → ℝ) |
| (g : MatrixCoordinates (Fin n) → ℝ) : Matrix (Fin n) (Fin n) ℝ := |
| let p : ScalarPoint := (empiricalLaw y,t,σ) |
| let q := scalarQMoment p.1 |
| let s := scalarS j p |
| conditionalBlockMatrix (j/(n:ℝ)) j q s (s+j*q) (scalarA j p) |
| (j*scalarBMoment p.1) (conditionalKappa j p) (conditionalEll j p) |
| (empiricalUnit y q).ofLp (sqrt s • (empiricalGramColumns y (scalarD j p) s 2).ofLp) g |
|
|
| lemma empirical_observation_covariance {n : ℕ} [NeZero n] (j t σ : ℝ) (y : Fin n → ℝ) |
| (hq : 0 ≤ scalarQMoment (empiricalLaw y)) : |
| (j/(n:ℝ))*sqrt ((n:ℝ)*scalarQMoment (empiricalLaw y))^2+(t+σ^2)= |
| scalarS j (empiricalLaw y,t,σ) ∧ |
| (t+σ^2)+2*(j/(n:ℝ))*sqrt ((n:ℝ)*scalarQMoment (empiricalLaw y))^2= |
| scalarS j (empiricalLaw y,t,σ)+j*scalarQMoment (empiricalLaw y) := by |
| have hn : (n:ℝ) ≠ 0 := by exact_mod_cast NeZero.ne n |
| rw [sq_sqrt (mul_nonneg (Nat.cast_nonneg n) hq)] |
| dsimp [scalarS] |
| constructor < |
|
|
| lemma empirical_conditional_mean {n : ℕ} [NeZero n] {j t σ : ℝ} |
| (hj : 0 ≤ j) (ht : 0 < t) (y : Fin n → ℝ) |
| (hq : 0 < scalarQMoment (empiricalLaw y)) : |
| let p : ScalarPoint := (empiricalLaw y,t,σ) |
| let q := scalarQMoment p.1 |
| let s := scalarS j p |
| let u := (empiricalUnit y q).ofLp |
| let Z := sqrt s • (empiricalGramColumns y (scalarD j p) s 2).ofLp |
| (fun i k=>(goeRegressionCoefficient (j/(n:ℝ)) (t+σ^2) (fun l=>tanh (y l))*ᵥ |
| empiricalObservation j t σ y) (i,k))= |
| vecMulVec u ((j*sqrt (q/s)) • Z-(j*scalarA j p/s) • u)+ |
| vecMulVec ((j*sqrt (q/s)) • Z-(j*scalarA j p/s) • u) u+ |
| (2*j*(scalarA j p+j*scalarBMoment p.1*q)/(s+j*q)) • vecMulVec u u := by |
| dsimp only |
| let p : ScalarPoint := (empiricalLaw y,t,σ) |
| let q := scalarQMoment p.1 |
| let s := scalarS j p |
| let B := j*scalarBMoment p.1 |
| let c := sqrt ((n:ℝ)*q) |
| let u := (empiricalUnit y q).ofLp |
| let Z := sqrt s • (empiricalGramColumns y (scalarD j p) s 2).ofLp |
| have hn : 0 < (n:ℝ) := by exact_mod_cast NeZero.pos n |
| have hv : 0 < t+σ^2 := add_pos_of_pos_of_nonneg ht (sq_nonneg _) |
| have hs : 0 < s := add_pos_of_pos_of_nonneg hv (mul_nonneg hj hq.le) |
| have hu : u⬝ᵥu=1 := empiricalUnit_unit y hq |
| have hscale : c • u=(fun i=>tanh (y i)) := empirical_unit_scale y hq |
| have hc := empirical_observation_covariance j t σ y hq.le |
| have hdot : u⬝ᵥempiricalObservation j t σ y=(n:ℝ)*(scalarA j p+B*q)/c := |
| empirical_unit_observation_dot j t σ B y hq |
| have hmean := goeRegressionCoefficient_block_mean (c := c) (div_nonneg hj hn.le) hv hu |
| (empiricalObservation j t σ y) |
| rw [hscale] at hmean |
| rw [hc.1,hc.2,hdot] at hmean |
| have hperp : ((j/(n:ℝ))*c/s) • (empiricalObservation j t σ y- |
| ((n:ℝ)*(scalarA j p+B*q)/c) • u)= |
| (j*sqrt (q/s)) • Z-(j*scalarA j p/s) • u := by |
| ext i |
| have hzi := congrFun (empirical_Z_formula y (scalarD j p) hs) i |
| change Z i=(y i-scalarD j p)/(sqrt (n:ℝ)*sqrt s) at hzi |
| have hmi := congrFun hscale i |
| change c*u i=tanh (y i) at hmi |
| change (j/(n:ℝ)*c/s)*(y i-scalarD j p+B*tanh (y i)- |
| ((n:ℝ)*(scalarA j p+B*q)/c)*u i)= |
| j*sqrt (q/s)*Z i-j*scalarA j p/s*u i |
| rw [← hmi,hzi] |
| convert conditional_perp_mean_scalar hn hq hs j (scalarA j p) B |
| (y i-scalarD j p) (u i) using 1 |
| ring |
| rw [hperp] at hmean |
| have hparallel := conditional_parallel_mean_scalar (S := s+j*q) hn hq j (scalarA j p) B |
| rw [hparallel] at hmean |
| exact hmean |
|
|
| theorem empirical_conditional_law {n : ℕ} [NeZero n] {j t σ : ℝ} |
| (hj : 0 ≤ j) (ht : 0 < t) (y : Fin n → ℝ) |
| (hq : 0 < scalarQMoment (empiricalLaw y)) : |
| conditionalGoeLaw (j/(n:ℝ)) (t+σ^2) (fun i=>tanh (y i)) (empiricalObservation j t σ y)= |
| (gaussianCoordinates (MatrixCoordinates (Fin n))).map |
| (fun g p=>empiricalConditionalGoe j t σ y g p.1 p.2) := by |
| let q := scalarQMoment (empiricalLaw y) |
| have hn : 0 < (n:ℝ) := by exact_mod_cast NeZero.pos n |
| have hv : 0 < t+σ^2 := add_pos_of_pos_of_nonneg ht (sq_nonneg _) |
| have hh := conditionalGoeLaw_block (c := sqrt ((n:ℝ)*q)) (div_nonneg hj hn.le) hv |
| (empiricalUnit_unit y hq) (empiricalObservation j t σ y) |
| rw [empirical_unit_scale y hq] at hh |
| rw [(empirical_observation_covariance j t σ y hq.le).1, |
| (empirical_observation_covariance j t σ y hq.le).2] at hh |
| rw [hh] |
| congr 1 |
| funext g p |
| have hmean := congrArg (fun M : Matrix (Fin n) (Fin n) ℝ=>M p.1 p.2) |
| (empirical_conditional_mean (σ := σ) hj ht y hq) |
| change _+_=empiricalConditionalGoe j t σ y g p.1 p.2 |
| dsimp only [empiricalConditionalGoe,conditionalBlockMatrix] |
| dsimp only at hmean |
| rw [hmean] |
| simp only [Matrix.add_apply,conditionalKappa,conditionalEll] |
| ring |
| end SKGap |
| end |
| end |
|
|
| end OAI |
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