| import OAI.Probability.LatticePassage.ProductVariance |
| import Mathlib.MeasureTheory.Integral.Pi |
|
|
| namespace OAI |
|
|
| namespace CompanionFPP |
| open MeasureTheory ProbabilityTheory Finset |
|
|
| variable {α : Type*} [MeasurableSpace α] (μ : Measure α) [IsProbabilityMeasure μ] |
|
|
| lemma measurable_finCons (n : ℕ) : |
| Measurable (fun z : α × (Fin n → α) => (Fin.cons z.1 z.2 : Fin (n+1) → α)) := by |
| apply Measurable.of_eval |
| intro i |
| refine Fin.cases ?_ (fun j => ?_) i |
| · simpa using (measurable_fst : Measurable (Prod.fst : α × (Fin n → α) → α)) |
| · simpa only [Function.comp_def, Fin.cons_succ] using (measurable_pi_apply j).comp (measurable_snd : |
| Measurable (Prod.snd : α × (Fin n → α) → (Fin n → α))) |
|
|
| lemma finCons_preserving (n : ℕ) : |
| MeasurePreserving (fun z : α × (Fin n → α) => Fin.cons z.1 z.2) |
| (μ.prod (Measure.pi (fun _ : Fin n => μ))) |
| (Measure.pi (fun _ : Fin (n+1) => μ)) := by |
| simpa only [MeasurableEquiv.piFinSuccAbove_symm_apply, Fin.insertNthEquiv, |
| Fin.insertNth_zero, Fin.zero_succAbove, Equiv.coe_fn_mk, cast_eq] using |
| (measurePreserving_piFinSuccAbove (fun _ : Fin (n+1) => μ) 0).symm |
|
|
| lemma integral_finCons {n : ℕ} {f : (Fin (n+1) → α) → ℝ} |
| (hf : Measurable f) {C : ℝ} (hC : ∀ v, |f v| ≤ C) : |
| (∫ v, f v ∂Measure.pi (fun _ : Fin (n+1) => μ)) = |
| ∫ a, ∫ w, f (Fin.cons a w) ∂Measure.pi (fun _ : Fin n => μ) ∂μ := by |
| have he := integral_map (finCons_preserving μ n).aemeasurable hf.aestronglyMeasurable |
| rw [(finCons_preserving μ n).map_eq] at he |
| rw [he] |
| exact integral_prod _ ((bounded_memLp_two (hf.comp (measurable_finCons n)) |
| C (fun z => hC (Fin.cons z.1 z.2))).integrable (by norm_num)) |
|
|
| noncomputable def coordinateVariance {ι : Type*} [DecidableEq ι] |
| (f : (ι → α) → ℝ) (i : ι) (v : ι → α) : ℝ := |
| variance (fun u => f (Function.update v i u)) μ |
|
|
| lemma measurable_coordinateVariance {ι : Type*} [Fintype ι] [DecidableEq ι] |
| {f : (ι → α) → ℝ} (hf : Measurable f) {C : ℝ} (hC : ∀ v, |f v| ≤ C) (i : ι) : |
| Measurable (coordinateVariance μ f i) := by |
| apply measurable_section_variance (C := C) (f := fun z => f (Function.update z.1 i z.2)) |
| · exact hf.comp (by fun_prop) |
| · exact fun z => hC _ |
|
|
| lemma coordinateVariance_bound {ι : Type*} [Fintype ι] [DecidableEq ι] |
| {f : (ι → α) → ℝ} (hf : Measurable f) {C : ℝ} (hC0 : 0 ≤ C) |
| (hC : ∀ v, |f v| ≤ C) (i : ι) (v : ι → α) : |
| |coordinateVariance μ f i v| ≤ C^2 := by |
| rw [coordinateVariance, abs_of_nonneg (variance_nonneg _ _)] |
| exact variance_le_bound_square (hf.comp (by fun_prop)) hC0 (fun _ => hC _) |
|
|
| lemma integrable_coordinateVariance {ι : Type*} [Fintype ι] [DecidableEq ι] |
| {f : (ι → α) → ℝ} (hf : Measurable f) {C : ℝ} (hC0 : 0 ≤ C) |
| (hC : ∀ v, |f v| ≤ C) (i : ι) : |
| Integrable (coordinateVariance μ f i) (Measure.pi (fun _ : ι => μ)) := |
| (bounded_memLp_two (measurable_coordinateVariance μ hf hC i) (C^2) |
| (coordinateVariance_bound μ hf hC0 hC i)).integrable (by norm_num) |
|
|
| omit [IsProbabilityMeasure μ] in |
| lemma coordinateVariance_cons_zero {n : ℕ} (f : (Fin (n+1) → α) → ℝ) |
| (a : α) (w : Fin n → α) : |
| coordinateVariance μ f 0 (Fin.cons a w) = variance (fun u => f (Fin.cons u w)) μ := by |
| simp only [coordinateVariance, Fin.update_cons_zero] |
|
|
| omit [IsProbabilityMeasure μ] in |
| lemma coordinateVariance_cons_succ {n : ℕ} (f : (Fin (n+1) → α) → ℝ) |
| (i : Fin n) (a : α) (w : Fin n → α) : |
| coordinateVariance μ f i.succ (Fin.cons a w) = |
| coordinateVariance μ (fun v => f (Fin.cons a v)) i w := by |
| simp only [coordinateVariance, ← Fin.cons_update] |
|
|
| lemma integral_coordinateVariance_zero {n : ℕ} {f : (Fin (n+1) → α) → ℝ} |
| (hf : Measurable f) {C : ℝ} (hC0 : 0 ≤ C) (hC : ∀ v, |f v| ≤ C) : |
| (∫ v, coordinateVariance μ f 0 v ∂Measure.pi (fun _ : Fin (n+1) => μ)) = |
| ∫ w, variance (fun u => f (Fin.cons u w)) μ ∂Measure.pi (fun _ : Fin n => μ) := by |
| rw [integral_finCons μ (measurable_coordinateVariance μ hf hC 0) |
| (coordinateVariance_bound μ hf hC0 hC 0)] |
| simp [coordinateVariance_cons_zero] |
|
|
| lemma integral_coordinateVariance_succ {n : ℕ} {f : (Fin (n+1) → α) → ℝ} |
| (hf : Measurable f) {C : ℝ} (hC0 : 0 ≤ C) (hC : ∀ v, |f v| ≤ C) (i : Fin n) : |
| (∫ v, coordinateVariance μ f i.succ v ∂Measure.pi (fun _ : Fin (n+1) => μ)) = |
| ∫ a, ∫ w, coordinateVariance μ (fun v => f (Fin.cons a v)) i w |
| ∂Measure.pi (fun _ : Fin n => μ) ∂μ := by |
| rw [integral_finCons μ (measurable_coordinateVariance μ hf hC i.succ) |
| (coordinateVariance_bound μ hf hC0 hC i.succ)] |
| simp only [coordinateVariance_cons_succ] |
|
|
| theorem variance_fin_le_sum_coordinateVariance (n : ℕ) |
| {f : (Fin n → α) → ℝ} (hf : Measurable f) |
| {C : ℝ} (hC0 : 0 ≤ C) (hC : ∀ v, |f v| ≤ C) : |
| variance f (Measure.pi (fun _ : Fin n => μ)) ≤ |
| ∑ i : Fin n, ∫ v, coordinateVariance μ f i v ∂Measure.pi (fun _ : Fin n => μ) := by |
| induction n with |
| | zero => |
| have he : f = fun _ => f (fun i => Fin.elim0 i) := by |
| funext v |
| congr 1 |
| funext i |
| exact Fin.elim0 i |
| rw [he] |
| simp [variance_eq_integral measurable_const.aemeasurable] |
| | succ n ih => |
| let F := fun z : α × (Fin n → α) => f (Fin.cons z.1 z.2) |
| have hFm : Measurable F := hf.comp (measurable_finCons n) |
| have hFb : ∀ z, |F z| ≤ C := fun z => hC _ |
| have he : variance f (Measure.pi (fun _ : Fin (n+1) => μ)) = |
| variance F (μ.prod (Measure.pi (fun _ : Fin n => μ))) := by |
| rw [← (finCons_preserving μ n).map_eq, |
| variance_map hf.aemeasurable (finCons_preserving μ n).aemeasurable] |
| rfl |
| rw [he] |
| have hv := product_variance_le (μ := μ) (ν := Measure.pi (fun _ : Fin n => μ)) |
| hFm hC0 hFb |
| apply hv.trans |
| have hI : ∀ i : Fin n, Integrable (fun a => ∫ w, |
| coordinateVariance μ (fun v => f (Fin.cons a v)) i w |
| ∂Measure.pi (fun _ : Fin n => μ)) μ := by |
| intro i |
| have hm := (measurable_coordinateVariance μ hf hC i.succ).comp (measurable_finCons n) |
| have hb := fun z : α × (Fin n → α) => |
| coordinateVariance_bound μ hf hC0 hC i.succ (Fin.cons z.1 z.2) |
| have hi := (bounded_memLp_two (μ := μ.prod (Measure.pi (fun _ : Fin n => μ))) |
| hm (C^2) hb).integrable (by norm_num) |
| simpa only [Function.comp_def, coordinateVariance_cons_succ] using hi.integral_prod_left |
| have hstep := integral_mono (integrable_section_variance hFm hC0 hFb) |
| (integrable_finsetSum Finset.univ (fun i _ => hI i)) (fun a => |
| ih (hf.comp ((measurable_finCons n).comp (measurable_const.prodMk measurable_id))) |
| (fun v => hC (Fin.cons a v))) |
| rw [integral_finsetSum _ (fun i _ => hI i)] at hstep |
| rw [Fin.sum_univ_succ, integral_coordinateVariance_zero μ hf hC0 hC] |
| simp_rw [integral_coordinateVariance_succ μ hf hC0 hC] |
| dsimp [F] at hstep ⊢ |
| linarith |
|
|
| lemma reindex_preserving {ι κ : Type*} [Fintype ι] [Fintype κ] (e : κ ≃ ι) : |
| MeasurePreserving (fun v : κ → α => fun i => v (e.symm i)) |
| (Measure.pi (fun _ : κ => μ)) (Measure.pi (fun _ : ι => μ)) := by |
| convert measurePreserving_piCongrLeft (fun _ : ι => μ) e using 1 |
| funext v i |
| simp [MeasurableEquiv.coe_piCongrLeft, Equiv.piCongrLeft_apply] |
|
|
| omit [IsProbabilityMeasure μ] in |
| lemma coordinateVariance_reindex {ι κ : Type*} [Fintype ι] [Fintype κ] |
| [DecidableEq ι] [DecidableEq κ] (e : κ ≃ ι) (f : (ι → α) → ℝ) (j : κ) (v : κ → α) : |
| coordinateVariance μ (fun w => f (fun i => w (e.symm i))) j v = |
| coordinateVariance μ f (e j) (fun i => v (e.symm i)) := by |
| unfold coordinateVariance |
| congr 1 |
| funext u |
| dsimp only |
| congr 1 |
| funext i |
| by_cases hi : i = e j |
| · subst i |
| simp |
| · have hj : e.symm i ≠ j := by |
| intro h |
| apply hi |
| simpa using congrArg e h |
| simp [Function.update_of_ne hi, Function.update_of_ne hj] |
|
|
| theorem variance_pi_le_sum_coordinateVariance {ι : Type*} [Fintype ι] [DecidableEq ι] |
| {f : (ι → α) → ℝ} (hf : Measurable f) |
| {C : ℝ} (hC0 : 0 ≤ C) (hC : ∀ v, |f v| ≤ C) : |
| variance f (Measure.pi (fun _ : ι => μ)) ≤ |
| ∑ i : ι, ∫ v, coordinateVariance μ f i v ∂Measure.pi (fun _ : ι => μ) := by |
| let e : Fin (Fintype.card ι) ≃ ι := (Fintype.equivFin ι).symm |
| have hmp := reindex_preserving μ e |
| have hg := hf.comp hmp.measurable |
| have ht := variance_fin_le_sum_coordinateVariance μ (Fintype.card ι) hg hC0 |
| (fun w => hC (fun i => w (e.symm i))) |
| simp only [Function.comp_def] at ht |
| rw [hmp.variance_fun_comp hf.aemeasurable] at ht |
| simp only [coordinateVariance_reindex] at ht |
| have he : ∀ j : Fin (Fintype.card ι), |
| (∫ w, coordinateVariance μ f (e j) (fun i => w (e.symm i)) |
| ∂Measure.pi (fun _ : Fin (Fintype.card ι) => μ)) = |
| ∫ v, coordinateVariance μ f (e j) v ∂Measure.pi (fun _ : ι => μ) := by |
| intro j |
| have h := integral_map hmp.aemeasurable |
| (measurable_coordinateVariance μ hf hC (e j)).aestronglyMeasurable |
| rw [hmp.map_eq] at h |
| exact h.symm |
| simp_rw [he] at ht |
| rwa [e.sum_comp (fun i => ∫ v, coordinateVariance μ f i v ∂Measure.pi (fun _ : ι => μ))] at ht |
|
|
| end CompanionFPP |
|
|
| end OAI |
|
|
