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f61e547 | 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 | import OAI.Probability.FreeIsing.Energy.ArtanhResponseCoefficients
import OAI.Probability.FreeIsing.Ising.CanonicalPosteriorStops
import OAI.Probability.FreeIsing.Ising.ContinuousRootResponse
namespace OAI
noncomputable section
open MeasureTheory
open scoped NNReal
namespace Problem367.IsingObservationFactor
open IsingObservationInnovation IsingCopyEnergy ArtanhMartingaleFormula
variable {d : ℕ} {beta : ℝ}
theorem continuousPosterior_variance_pos (v : TreeVertex d)
(t : ℝ≥0) (x : Paths d) :
0 < 1 - continuousPosterior beta v t x ^ 2 :=
sub_pos.mpr (Real.tanh_sq_lt_one _)
theorem firstWeight_mul_continuousRootCovariance
(hb : 0 ≤ beta) (v : TreeVertex d) (t : ℝ≥0) (x : Paths d) :
firstWeight (continuousPosterior beta (ConnectedHull.root d) t x) *
continuousRootCovariance hb v t x = continuousRootResponse hb v t x := by
change (1 / (1 - continuousPosterior beta (ConnectedHull.root d) t x ^ 2)) *
((1 - continuousPosterior beta (ConnectedHull.root d) t x ^ 2) *
continuousRootResponse hb v t x) = _
rw [one_div, ← mul_assoc,
inv_mul_cancel₀ (continuousPosterior_variance_pos (ConnectedHull.root d) t x).ne', one_mul]
theorem continuousRootCovariance_div_variance
(hb : 0 ≤ beta) (v : TreeVertex d) (t : ℝ≥0) (x : Paths d) :
continuousRootCovariance hb v t x /
(1 - continuousPosterior beta (ConnectedHull.root d) t x ^ 2) =
continuousRootResponse hb v t x := by
rw [← ArtanhResponseCoefficients.firstWeight_mul]
exact firstWeight_mul_continuousRootCovariance hb v t x
theorem continuousRootCovariance_square_tsum
(hb : 0 ≤ beta) (t : ℝ≥0) (x : Paths d) :
(∑' v, continuousRootCovariance hb v t x ^ 2) =
(1 - continuousPosterior beta (ConnectedHull.root d) t x ^ 2) ^ 2 *
(∑' v, continuousRootResponse hb v t x ^ 2) := by
simp only [continuousRootCovariance, continuousPosterior, mul_pow, tsum_mul_left]
theorem secondWeight_mul_continuousRootCovariance_square_tsum
(hb : 0 ≤ beta) (t : ℝ≥0) (x : Paths d) :
secondWeight (continuousPosterior beta (ConnectedHull.root d) t x) *
(∑' v, continuousRootCovariance hb v t x ^ 2) =
continuousPosterior beta (ConnectedHull.root d) t x *
(∑' v, continuousRootResponse hb v t x ^ 2) := by
rw [ArtanhResponseCoefficients.secondWeight_mul_tsum_sq]
simp_rw [continuousRootCovariance_div_variance]
theorem continuousRootCovariance_square_tsum_le_response
(hb : 0 ≤ beta) (t : ℝ≥0) (x : Paths d) :
(∑' v, continuousRootCovariance hb v t x ^ 2) ≤
∑' v, continuousRootResponse hb v t x ^ 2 := by
rw [continuousRootCovariance_square_tsum]
have h0 := (continuousPosterior_variance_pos (beta := beta) (ConnectedHull.root d) t x).le
have h1 : 1 - continuousPosterior beta (ConnectedHull.root d) t x ^ 2 ≤ 1 :=
sub_le_self _ (sq_nonneg _)
exact mul_le_of_le_one_left (tsum_nonneg (fun _ => sq_nonneg _))
(pow_le_one₀ h0 h1)
theorem abs_continuousPosterior_mul_response_square_tsum_le
(hb : 0 ≤ beta) (t : ℝ≥0) (x : Paths d) :
|continuousPosterior beta (ConnectedHull.root d) t x *
(∑' v, continuousRootResponse hb v t x ^ 2)| ≤
∑' v, continuousRootResponse hb v t x ^ 2 := by
rw [abs_mul, abs_of_nonneg (show 0 ≤ ∑' v, continuousRootResponse hb v t x ^ 2
from tsum_nonneg (fun _ => sq_nonneg _))]
exact mul_le_of_le_one_left (tsum_nonneg (fun _ => sq_nonneg _))
(abs_lt.mpr (continuousPosterior_interior beta (ConnectedHull.root d) t x)).le
theorem firstWeight_mul_stopped_continuousRootCovariance
(hb : 0 ≤ beta) (v : TreeVertex d) (τ : Paths d → WithTop ℝ≥0)
(t : ℝ≥0) (x : Paths d) :
firstWeight (stoppedProcess (continuousPosterior beta (ConnectedHull.root d)) τ t x) *
stoppedProcess (continuousRootCovariance hb v) τ t x =
stoppedProcess (continuousRootResponse hb v) τ t x :=
firstWeight_mul_continuousRootCovariance hb v _ x
theorem firstWeight_stopped_mul_continuousRootCovariance_of_le
(hb : 0 ≤ beta) (v : TreeVertex d) (τ : Paths d → WithTop ℝ≥0)
(t : ℝ≥0) (x : Paths d) (ht : (t : WithTop ℝ≥0) ≤ τ x) :
firstWeight (stoppedProcess (continuousPosterior beta (ConnectedHull.root d)) τ t x) *
continuousRootCovariance hb v t x = continuousRootResponse hb v t x := by
rw [stoppedProcess_eq_of_le ht]
exact firstWeight_mul_continuousRootCovariance hb v t x
theorem firstWeight_stopped_pullback_mul_continuousRootCovariance_of_le
{Ω : Type*} (hb : 0 ≤ beta) (v : TreeVertex d) (φ : Ω → Paths d)
(τ : Ω → WithTop ℝ≥0) (t : ℝ≥0) (ω : Ω)
(ht : (t : WithTop ℝ≥0) ≤ τ ω) :
firstWeight (stoppedProcess
(fun s ω => continuousPosterior beta (ConnectedHull.root d) s (φ ω)) τ t ω) *
continuousRootCovariance hb v t (φ ω) = continuousRootResponse hb v t (φ ω) := by
rw [stoppedProcess_eq_of_le ht]
exact firstWeight_mul_continuousRootCovariance hb v t (φ ω)
theorem firstWeight_stopped_copy_mul_continuousRootCovariance_of_le
(hb : 0 ≤ beta) (v : TreeVertex d) (j : Bool)
(τ : Joint d → WithTop ℝ≥0) (t : ℝ≥0) (q : Joint d)
(ht : (t : WithTop ℝ≥0) ≤ τ q) :
firstWeight (stoppedProcess
(copyContinuousPosterior beta (ConnectedHull.root d) j) τ t q) *
continuousRootCovariance hb v t (projection j q) =
continuousRootResponse hb v t (projection j q) :=
firstWeight_stopped_pullback_mul_continuousRootCovariance_of_le hb v
(projection j) τ t q ht
end Problem367.IsingObservationFactor
end
end OAI
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