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openAI-math-dataset / lean /OAI /Probability /SKGap /Matrix /NormalLogDensityCentered.lean
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roseking
Add lean source and binary files (part 128)
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import Mathlib
import OAI.Probability.SKGap.Localization.ScalarSmallNoiseTests
namespace OAI
section
noncomputable section
open MeasureTheory ProbabilityTheory InformationTheory Real Set Filter
open scoped NNReal ENNReal Topology
noncomputable section
open Real Set
noncomputable section
open MeasureTheory ProbabilityTheory Real Set Filter
open scoped Topology NNReal ENNReal BoundedContinuousFunction
open MeasureTheory ProbabilityTheory Filter Set Topology Real
open scoped NNReal ENNReal BoundedContinuousFunction
noncomputable section
open Set Filter Topology
noncomputable section
open MeasureTheory ProbabilityTheory Filter Set Topology Real
open scoped NNReal ENNReal BoundedContinuousFunction
noncomputable section
open MeasureTheory ProbabilityTheory Filter Set Topology Real
open scoped NNReal ENNReal BoundedContinuousFunction
noncomputable section
open MeasureTheory ProbabilityTheory Filter Set Topology Real
open scoped NNReal ENNReal
namespace SKGap
lemma normalLogDensity_centered (d s x : ℝ) :
normalLogDensity d s x = -log (sqrt (2*Real.pi*s))-(x-d)^2/(2*s) := by
unfold normalLogDensity normalLogConst
ring
lemma gaussian_pointwise_envelope {ι : Type*} [Fintype ι] (y : ι → ℝ)
{d s j D smin smax ψ : ℝ} (hsmin : 0 < smin) (hsl : smin ≤ s) (hsu : s ≤ smax)
(hd : d^2 ≤ D^2)
(hψ : (Fintype.card ι:ℝ)*ψ ≤ 3/(8*s)*(∑ i, (y i-d)^2)+3*(Fintype.card ι:ℝ)*j/2) :
exp ((Fintype.card ι:ℝ)*ψ)*productNormal d s y ≤
exp ((Fintype.card ι:ℝ)*(|log (sqrt (2*Real.pi*smin))|+3*j/2+D^2/(8*smin))-
(1/(16*smax))*∑ i, (y i)^2) := by
have hs : 0 < s := hsmin.trans_le hsl
have hmax : 0 < smax := hs.trans_le hsu
have hn : (0:ℝ) ≤ Fintype.card ι := Nat.cast_nonneg _
have hQ : 0 ≤ ∑ i, (y i)^2 := Finset.sum_nonneg (fun i _ => sq_nonneg _)
have hc := sum_centered_sq_lower y d
have hnd := mul_le_mul_of_nonneg_left hd hn
have h1 : -(∑ i, (y i-d)^2)/(8*s) ≤
-(∑ i, (y i)^2)/(16*s)+(Fintype.card ι:ℝ)*D^2/(8*s) := by
have he : -(∑ i, (y i)^2)/(16*s)+(Fintype.card ι:ℝ)*D^2/(8*s) =
(-(∑ i, (y i)^2)/2+(Fintype.card ι:ℝ)*D^2)/(8*s) := by ring
rw [he]
apply div_le_div_of_nonneg_right _ (by positivity : 0 ≤ 8*s)
linarith
have h2 : -(∑ i, (y i)^2)/(16*s) ≤ -(∑ i, (y i)^2)/(16*smax) := by
rw [neg_div, neg_div, neg_le_neg_iff]
gcongr
have h3 : (Fintype.card ι:ℝ)*D^2/(8*s) ≤ (Fintype.card ι:ℝ)*D^2/(8*smin) := by
gcongr
have hp : 0 < sqrt (2*Real.pi*smin) := sqrt_pos.mpr (by positivity)
have hlog : log (sqrt (2*Real.pi*smin)) ≤ log (sqrt (2*Real.pi*s)) :=
log_le_log hp (sqrt_le_sqrt (by nlinarith [Real.pi_pos]))
have hlog' : -log (sqrt (2*Real.pi*s)) ≤ |log (sqrt (2*Real.pi*smin))| :=
(neg_le_neg hlog).trans (neg_le_abs _)
have hnlog := mul_le_mul_of_nonneg_left hlog' hn
have he : (∑ i, normalLogDensity d s (y i)) =
-(Fintype.card ι:ℝ)*log (sqrt (2*Real.pi*s))-(∑ i, (y i-d)^2)/(2*s) := by
simp only [normalLogDensity_centered, Finset.sum_sub_distrib, Finset.sum_const,
Finset.card_univ, nsmul_eq_mul, Finset.sum_div]
ring
rw [productNormal_eq_exp _ hs, ← exp_add, he]
apply exp_le_exp.mpr
calc
_ ≤ 3/(8*s)*(∑ i, (y i-d)^2)+3*(Fintype.card ι:ℝ)*j/2+
(-(Fintype.card ι:ℝ)*log (sqrt (2*Real.pi*s))-(∑ i, (y i-d)^2)/(2*s)) :=
by linarith only [hψ]
_ = (Fintype.card ι:ℝ)*(-log (sqrt (2*Real.pi*s)))+3*(Fintype.card ι:ℝ)*j/2+
(-(∑ i, (y i-d)^2)/(8*s)) := by ring
_ ≤ (Fintype.card ι:ℝ)*|log (sqrt (2*Real.pi*smin))|+3*(Fintype.card ι:ℝ)*j/2+
(-(∑ i, (y i)^2)/(16*smax)+(Fintype.card ι:ℝ)*D^2/(8*smin)) :=
add_le_add (by linarith only [hnlog]) (h1.trans (add_le_add h2 h3))
_ = _ := by ring
end SKGap
end
end
end
end
end
end
end
end
end OAI