| import Mathlib |
| import OAI.Probability.SKBarriers.Calculus.MarginalVariance |
| import OAI.Probability.SKBarriers.Calculus.BilinearSchur |
|
|
| namespace OAI |
|
|
| section |
| section |
| noncomputable section |
| open scoped BigOperators Topology |
| open MeasureTheory ProbabilityTheory Filter |
| noncomputable section |
| open MeasureTheory Set Filter |
| open scoped Topology Interval |
| noncomputable section |
| open MeasureTheory Set |
| open scoped Interval |
| namespace SK.Analytic |
| theorem hasDerivAt_score_comp |
| {E : Type} [NormedAddCommGroup E] [NormedSpace ℝ E] |
| (V : E → ℝ) (hV : ContDiff ℝ 2 V) {g : ℝ → E} {u : E} {t : ℝ} |
| (hg : HasDerivAt g u t) (v : E) : |
| HasDerivAt (fun s => fderiv ℝ V (g s) v) (Hessian V (g t) u v) t := by |
| have hd : Differentiable ℝ (fderiv ℝ V) := |
| (hV.fderiv_right (m := 1) (by norm_num)).differentiable (by norm_num) |
| have h := (hd (g t)).hasFDerivAt.comp_hasDerivAt t hg |
| simpa only [map_zero, add_zero, Function.comp_def, Hessian] using |
| h.clm_apply (hasDerivAt_const t v) |
|
|
| def marginalPotential {E : Type} (V : E × ℝ → ℝ) (a b : ℝ) (x : E) : ℝ := |
| -Real.log (intervalPartition (fun y => V (x,y)) a b) |
|
|
| theorem contDiff_marginalPotential |
| {E : Type} [NormedAddCommGroup E] [NormedSpace ℝ E] [FiniteDimensional ℝ E] |
| (V : E × ℝ → ℝ) (n : ℕ) (hV : ContDiff ℝ n V) {a b : ℝ} (hab : a < b) : |
| ContDiff ℝ n (marginalPotential V a b) := by |
| have hp : ContDiff ℝ n (fun z => Real.exp (-V z)) := hV.neg.exp |
| have hZ := contDiff_intervalIntegral n (fun z => Real.exp (-V z)) hp a b |
| exact (hZ.log (fun x => ne_of_gt (intervalPartition_pos |
| (hV.continuous.comp (continuous_const.prodMk continuous_id)) hab))).neg |
|
|
| theorem marginal_hessian_lower |
| {E : Type} [NormedAddCommGroup E] [NormedSpace ℝ E] [FiniteDimensional ℝ E] |
| (V : E × ℝ → ℝ) (hV : ContDiff ℝ 2 V) |
| (q : E → ℝ) (hq : q 0 = 0) {c : ℝ} (hc : 0 < c) |
| (hlower : ∀ z u s, c * (q u + s^2) ≤ Hessian V z (u,s) (u,s)) |
| {a b : ℝ} (hab : a < b) (x u : E) : |
| c*q u ≤ Hessian (marginalPotential V a b) x u u := by |
| let g : ℝ × ℝ → E × ℝ := fun z => (x+z.1 • u, z.2) |
| let e₁ : E × ℝ := (u,0) |
| let e₂ : E × ℝ := (0,1) |
| let v : ℝ → ℝ → ℝ := fun t y => V (g (t,y)) |
| let v₁ : ℝ → ℝ → ℝ := fun t y => fderiv ℝ V (g (t,y)) e₁ |
| let v₂ : ℝ → ℝ → ℝ := fun t y => Hessian V (g (t,y)) e₁ e₁ |
| let w₁ : ℝ → ℝ → ℝ := fun t y => fderiv ℝ V (g (t,y)) e₂ |
| let w₂ : ℝ → ℝ → ℝ := fun t y => Hessian V (g (t,y)) e₂ e₂ |
| let hxy : ℝ → ℝ → ℝ := fun t y => Hessian V (g (t,y)) e₂ e₁ |
| have hg : ContDiff ℝ 2 g := by dsimp only [g]; fun_prop |
| have hd : ContDiff ℝ 1 (fderiv ℝ V) := hV.fderiv_right (by norm_num) |
| have hdd : Continuous (fderiv ℝ (fderiv ℝ V)) := hd.continuous_fderiv (by norm_num) |
| have hvc : Continuous (Function.uncurry v) := hV.continuous.comp hg.continuous |
| have hdcont (e : E × ℝ) : Continuous (fun z => fderiv ℝ V (g z) e) := |
| (hd.continuous.comp hg.continuous).clm_apply continuous_const |
| have hddcont (e e' : E × ℝ) : Continuous (fun z => Hessian V (g z) e e') := |
| ((hdd.comp hg.continuous).clm_apply continuous_const).clm_apply continuous_const |
| have hg₁ (t y : ℝ) : HasDerivAt (fun s => g (s,y)) e₁ t := by |
| simpa only [g, e₁, one_smul, id_eq] using |
| (((hasDerivAt_id t).smul_const u).const_add x).prodMk (hasDerivAt_const t y) |
| have hg₂ (t y : ℝ) : HasDerivAt (fun s => g (t,s)) e₂ y := by |
| simpa only [g, e₂, id_eq] using |
| (hasDerivAt_const y (x+t • u)).prodMk (hasDerivAt_id y) |
| have hvd (t y : ℝ) : HasDerivAt (fun s => v s y) (v₁ t y) t := by |
| exact (hV.differentiable (by norm_num) _).hasFDerivAt.comp_hasDerivAt t (hg₁ t y) |
| have hv₁d (t y : ℝ) : HasDerivAt (fun s => v₁ s y) (v₂ t y) t := |
| hasDerivAt_score_comp V hV (hg₁ t y) e₁ |
| have hwd (t y : ℝ) : HasDerivAt (v t) (w₁ t y) y := by |
| exact (hV.differentiable (by norm_num) _).hasFDerivAt.comp_hasDerivAt y (hg₂ t y) |
| have hw₁d (t y : ℝ) : HasDerivAt (w₁ t) (w₂ t y) y := |
| hasDerivAt_score_comp V hV (hg₂ t y) e₂ |
| have hv₁yd (t y : ℝ) : HasDerivAt (v₁ t) (hxy t y) y := |
| hasDerivAt_score_comp V hV (hg₂ t y) e₁ |
| have hs (t y : ℝ) : 0 < w₂ t y ∧ c*q u ≤ v₂ t y - hxy t y^2/w₂ t y := by |
| apply bilinear_schur_lower (fderiv ℝ (fderiv ℝ V) (g (t,y))) |
| (hV.contDiffAt.isSymmSndFDerivAt (by norm_num)) q hq hc |
| (hlower (g (t,y))) u |
| have hBL := brascampLieb_interval_schur hab (v 0) (w₁ 0) (w₂ 0) (v₁ 0) |
| (hxy 0) (v₂ 0) (hwd 0) (hw₁d 0) (hv₁yd 0) |
| ((hddcont e₂ e₂).comp (continuous_const.prodMk continuous_id)) |
| ((hddcont e₂ e₁).comp (continuous_const.prodMk continuous_id)) |
| ((hddcont e₁ e₁).comp (continuous_const.prodMk continuous_id)) |
| (fun y => (hs 0 y).1) (fun y => (hs 0 y).2) |
| have hW := contDiff_marginalPotential V 2 hV hab |
| have hfirst (t : ℝ) := hasDerivAt_neg_log_intervalPartition v v₁ hvc |
| (hdcont e₁) hvd hab t |
| have hsecond := hasDerivAt_interval_score v v₁ v₂ hvc (hdcont e₁) |
| (hddcont e₁ e₁) hvd hv₁d hab 0 |
| have hscore : (fun t => fderiv ℝ (marginalPotential V a b) (x+t • u) u) = |
| (fun t => intervalMean (v t) (v₁ t) a b) := by |
| ext t |
| exact (hasDerivAt_affine_comp (marginalPotential V a b) |
| (hW.differentiable (by norm_num)) x u t).unique (hfirst t) |
| have hsecond' := hasDerivAt_affine_score (marginalPotential V a b) hW x u u 0 |
| rw [hscore] at hsecond' |
| have heq := hsecond'.unique hsecond |
| simp only [zero_smul, add_zero] at heq |
| rw [heq] |
| exact hBL |
| end SK.Analytic |
| |
| end |
| end |
| end |
| end |
| end |
| |
| end OAI |
| |
