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7d9f027 | 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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 | 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 <;> field_simp <;> ring
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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