| import Mathlib |
| import OAI.Probability.GaussianInformation.Dyadic.ProjectionTests |
|
|
| namespace OAI |
| open MeasureTheory ProbabilityTheory Set |
| open scoped BigOperators ENNReal |
| open MeasureTheory Metric Set |
| open scoped ENNReal NNReal |
| open MeasureTheory Set Filter |
| open scoped ENNReal Topology |
| open MeasureTheory ProbabilityTheory Set Filter |
| open scoped ENNReal Topology |
| open InnerProductSpace Matrix MeasureTheory Set |
| open scoped RealInnerProductSpace ENNReal |
| open MeasureTheory ProbabilityTheory Set |
| open scoped ENNReal NNReal |
| namespace RepeatedLocalization |
| noncomputable def xlog (x : ℝ) : ℝ := x * Real.log x |
|
|
| lemma continuous_xlog : Continuous xlog := Real.continuous_mul_log |
| lemma xlog_zero : xlog 0 = 0 := by simp [xlog] |
| lemma xlog_one : xlog 1 = 0 := by simp [xlog] |
| lemma xlog_mul (x y : ℝ) : xlog (x*y) = y*xlog x+x*xlog y := by |
| have h := Real.negMulLog_mul x y |
| simp only [Real.negMulLog_def, neg_mul] at h |
| dsimp [xlog] |
| linarith |
|
|
| lemma mul_log_div_le {x y : ℝ} (hx : 0 ≤ x) (hy : 0 < y) : |
| x * Real.log (x/y) ≤ x^2/y-x := by |
| by_cases hx0 : x = 0 |
| · simp [hx0] |
| have h := Real.log_le_sub_one_of_pos (div_pos (lt_of_le_of_ne hx (Ne.symm hx0)) hy) |
| have H := mul_le_mul_of_nonneg_left h hx |
| simpa only [mul_sub, mul_one, pow_two, mul_div_assoc] using H |
|
|
| lemma xlog_lower_tangent {x y : ℝ} (hx : 0 ≤ x) (hy : 0 < y) : |
| xlog y + (Real.log y+1)*(x-y) ≤ xlog x := by |
| have h := Real.self_sub_one_le_mul_log (div_nonneg hx hy.le) |
| by_cases hx0 : x = 0 |
| · simp only [hx0, xlog] |
| nlinarith |
| rw [Real.log_div hx0 hy.ne'] at h |
| have H := mul_le_mul_of_nonneg_left h hy.le |
| dsimp [xlog] |
| field_simp at H |
| nlinarith |
| |
| lemma integrable_bounded_measurable {α : Type*} [MeasurableSpace α] |
| (μ : Measure α) [IsFiniteMeasure μ] (f : α → ℝ) (hf : Measurable f) |
| {C : ℝ} (hC : ∀ x, |f x| ≤ C) : Integrable f μ := by |
| apply Integrable.mono' (integrable_const C) hf.aestronglyMeasurable |
| exact ae_of_all _ (fun x => by simpa only [Real.norm_eq_abs] using hC x) |
|
|
| lemma integrable_xlog_probability {α : Type*} [MeasurableSpace α] |
| (μ : Measure α) [IsFiniteMeasure μ] (p : α → ℝ) (hp : Measurable p) |
| (hp0 : ∀ x, 0 ≤ p x) (hp1 : ∀ x, p x ≤ 1) : Integrable (fun x => xlog (p x)) μ := by |
| have hr : Set.range p ⊆ Icc (0:ℝ) 1 := by rintro _ ⟨x,rfl⟩; exact ⟨hp0 x,hp1 x⟩ |
| obtain ⟨C,hC⟩ := (isCompact_Icc.image continuous_xlog).isBounded.exists_norm_le |
| apply integrable_bounded_measurable μ _ (continuous_xlog.measurable.comp hp) |
| intro x |
| simpa only [Real.norm_eq_abs, Function.comp_apply] using |
| hC _ (Set.mem_image_of_mem _ (hr (Set.mem_range_self x))) |
|
|
| noncomputable def scoreMass {α : Type*} [MeasurableSpace α] (μ : Measure α) (p : α → ℝ) : ℝ := |
| ∫ x, p x ∂μ |
|
|
| noncomputable def infoContribution {α : Type*} [MeasurableSpace α] |
| (μ : Measure α) (p : α → ℝ) : ℝ := |
| (∫ x, xlog (p x) ∂μ) - xlog (scoreMass μ p) |
|
|
| lemma scoreMass_nonneg {α : Type*} [MeasurableSpace α] |
| (μ : Measure α) (p : α → ℝ) (hp0 : ∀ x, 0 ≤ p x) : 0 ≤ scoreMass μ p := |
| integral_nonneg hp0 |
|
|
| lemma scoreMass_le_one {α : Type*} [MeasurableSpace α] |
| (μ : Measure α) [IsProbabilityMeasure μ] (p : α → ℝ) (hp : Measurable p) |
| (hp0 : ∀ x, 0 ≤ p x) (hp1 : ∀ x, p x ≤ 1) : scoreMass μ p ≤ 1 := by |
| have hi := integrable_bounded_measurable μ p hp (C := 1) (fun x => by simpa only [abs_of_nonneg (hp0 x)] using hp1 x) |
| simpa [scoreMass] using integral_mono hi (integrable_const (1:ℝ)) hp1 |
|
|
| lemma infoContribution_zero_mass {α : Type*} [MeasurableSpace α] |
| (μ : Measure α) [IsFiniteMeasure μ] (p : α → ℝ) (hp : Measurable p) |
| (hp0 : ∀ x, 0 ≤ p x) (hp1 : ∀ x, p x ≤ 1) (hP : scoreMass μ p = 0) : |
| infoContribution μ p = 0 := by |
| have hi := integrable_bounded_measurable μ p hp (C := 1) (fun x => by simpa only [abs_of_nonneg (hp0 x)] using hp1 x) |
| have hz : p =ᵐ[μ] 0 := (integral_eq_zero_iff_of_nonneg hp0 hi).mp hP |
| have h : (fun x => xlog (p x)) =ᵐ[μ] 0 := by filter_upwards [hz] with x hx; simp [hx, xlog_zero] |
| simp [infoContribution, hP, xlog_zero, integral_congr_ae h] |
|
|
| lemma infoContribution_nonneg {α : Type*} [MeasurableSpace α] |
| (μ : Measure α) [IsProbabilityMeasure μ] (p : α → ℝ) (hp : Measurable p) |
| (hp0 : ∀ x, 0 ≤ p x) (hp1 : ∀ x, p x ≤ 1) : 0 ≤ infoContribution μ p := by |
| have hP := scoreMass_nonneg μ p hp0 |
| by_cases hz : scoreMass μ p = 0 |
| · rw [infoContribution_zero_mass μ p hp hp0 hp1 hz] |
| have hpos : 0 < scoreMass μ p := lt_of_le_of_ne hP (Ne.symm hz) |
| have hip := integrable_bounded_measurable μ p hp (C := 1) (fun x => by simpa only [abs_of_nonneg (hp0 x)] using hp1 x) |
| have hil := integrable_xlog_probability μ p hp hp0 hp1 |
| have h := integral_mono ((integrable_const _).add |
| ((hip.sub (integrable_const _)).const_mul _)) hil (fun x => xlog_lower_tangent (hp0 x) hpos) |
| have hiq : Integrable (fun x => (Real.log (scoreMass μ p)+1)*(p x-scoreMass μ p)) μ := |
| (hip.sub (integrable_const _)).const_mul _ |
| have hadd := integral_add (integrable_const (xlog (scoreMass μ p))) hiq |
| simp only [Pi.add_apply, Pi.sub_apply] at h hadd |
| rw [hadd, integral_const_mul] at h |
| have hsub := integral_sub hip (integrable_const (scoreMass μ p)) |
| rw [hsub] at h |
| simp only [integral_const, probReal_univ, smul_eq_mul, one_mul, scoreMass, |
| sub_self, mul_zero, add_zero] at h |
| dsimp [infoContribution, scoreMass] |
| linarith |
|
|
| lemma xlog_upper_quadratic {x a : ℝ} (hx : 0 ≤ x) (ha : 0 < a) : |
| xlog x ≤ x^2/a + (Real.log a-1)*x := by |
| by_cases hz : x = 0 |
| · simp [hz,xlog_zero] |
| have h := mul_log_div_le hx ha |
| rw [Real.log_div hz ha.ne'] at h |
| dsimp [xlog] |
| nlinarith |
| |
| lemma infoContribution_le_log_l2bound {α : Type*} [MeasurableSpace α] |
| (μ : Measure α) [IsProbabilityMeasure μ] (p : α → ℝ) (hp : Measurable p) |
| (hp0 : ∀ x, 0 ≤ p x) (hp1 : ∀ x, p x ≤ 1) |
| {L : ℝ} (hL : 0 < L) (hbound : (∫ x, (p x)^2 ∂μ) ≤ L^2) : |
| infoContribution μ p ≤ 2*scoreMass μ p*(Real.log L-Real.log (scoreMass μ p)) := by |
| have hP := scoreMass_nonneg μ p hp0 |
| by_cases hz : scoreMass μ p = 0 |
| · rw [infoContribution_zero_mass μ p hp hp0 hp1 hz,hz]; simp |
| have hpos : 0 < scoreMass μ p := lt_of_le_of_ne hP (Ne.symm hz) |
| let A := L^2 / scoreMass μ p |
| have ha : 0 < A := div_pos (sq_pos_of_pos hL) hpos |
| have hip := integrable_bounded_measurable μ p hp (C := 1) (fun x => by simpa only [abs_of_nonneg (hp0 x)] using hp1 x) |
| have hi2 : Integrable (fun x => (p x)^2) μ := by |
| apply integrable_bounded_measurable μ _ (hp.pow_const 2) (C := 1) |
| intro x |
| rw [abs_of_nonneg (sq_nonneg _)] |
| nlinarith [hp0 x,hp1 x] |
| have hil := integrable_xlog_probability μ p hp hp0 hp1 |
| have h := integral_mono hil ((hi2.div_const A).add (hip.const_mul (Real.log A-1))) |
| (fun x => xlog_upper_quadratic (hp0 x) ha) |
| have heq : (∫ x, (p x)^2/A+(Real.log A-1)*p x ∂μ) = |
| (∫ x, (p x)^2 ∂μ)/A+(Real.log A-1)*scoreMass μ p := by |
| rw [integral_add (hi2.div_const A) (hip.const_mul _),integral_div,integral_const_mul] |
| rfl |
| change (∫ x, xlog (p x) ∂μ) ≤ (∫ x, (p x)^2/A+(Real.log A-1)*p x ∂μ) at h |
| rw [heq] at h |
| have hb : (∫ x, (p x)^2 ∂μ)/A ≤ scoreMass μ p := by |
| have h := div_le_div_of_nonneg_right hbound ha.le |
| have he : L^2/A = scoreMass μ p := by dsimp [A]; field_simp |
| exact he ▸ h |
| have hlog : Real.log A = 2*Real.log L-Real.log (scoreMass μ p) := by |
| simp [A,Real.log_div (pow_ne_zero _ hL.ne') hz,Real.log_pow] |
| change (∫ x, xlog (p x) ∂μ) - scoreMass μ p * Real.log (scoreMass μ p) ≤ _ |
| rw [hlog] at h |
| nlinarith |
|
|
| lemma log_ratio_lower {a b P : ℝ} (ha : 0 ≤ a) (hb : 0 ≤ b) |
| (hP : 0 < P) (hz : b = 0 → a = 0) : |
| a*(Real.log P+1)-P*b ≤ a*Real.log (a/b) := by |
| by_cases hb0 : b = 0 |
| · simp [hb0,hz hb0] |
| have hbp : 0 < b := lt_of_le_of_ne hb (Ne.symm hb0) |
| have h := xlog_lower_tangent (div_nonneg ha hb) hP |
| have H := mul_le_mul_of_nonneg_left h hb |
| dsimp [xlog] at H |
| field_simp at H |
| nlinarith |
|
|
| lemma finite_log_sum {ι : Type*} [Fintype ι] (a b : ι → ℝ) |
| (ha : ∀ i, 0 ≤ a i) (hb : ∀ i, 0 ≤ b i) (hb1 : ∑ i,b i = 1) |
| (hz : ∀ i, b i = 0 → a i = 0) : |
| xlog (∑ i,a i) ≤ ∑ i,a i * Real.log (a i/b i) := by |
| have hA : 0 ≤ ∑ i,a i := Finset.sum_nonneg (fun i _ => ha i) |
| by_cases h0 : ∑ i,a i = 0 |
| · have hall : ∀ i, a i = 0 := fun i => (Finset.sum_eq_zero_iff_of_nonneg (fun j _ => ha j)).mp h0 i (Finset.mem_univ i) |
| simp [hall,xlog_zero] |
| have hAp : 0 < ∑ i,a i := lt_of_le_of_ne hA (Ne.symm h0) |
| have h := Finset.sum_le_sum (s := Finset.univ) (fun i _ => log_ratio_lower (ha i) (hb i) hAp (hz i)) |
| simp only [Finset.sum_sub_distrib,← Finset.sum_mul,← Finset.mul_sum,hb1,mul_one] at h |
| dsimp [xlog] |
| nlinarith |
|
|
| lemma log_ratio_mul {a b : ℝ} (hb : b ≠ 0) : |
| a*Real.log (a/b) = xlog a-a*Real.log b := by |
| by_cases ha : a = 0 |
| · simp [ha,xlog_zero] |
| rw [Real.log_div ha hb] |
| simp [xlog,mul_sub] |
|
|
| lemma neg_xlog_le_one {x : ℝ} (hx : 0 ≤ x) : -xlog x ≤ 1 := by |
| have h := Real.self_sub_one_le_mul_log hx |
| dsimp [xlog] |
| linarith |
|
|
| lemma xlog_nonpos {x : ℝ} (hx : 0 ≤ x) (hx1 : x ≤ 1) : xlog x ≤ 0 := by |
| exact mul_nonpos_of_nonneg_of_nonpos hx (Real.log_nonpos hx hx1) |
|
|
| lemma finite_entropy_mass_bound {ι : Type*} [Fintype ι] [Nonempty ι] |
| (a : ι → ℝ) (ha : ∀ i, 0 ≤ a i) : |
| -(∑ i,xlog (a i)) ≤ (∑ i,a i)*Real.log (Fintype.card ι)-xlog (∑ i,a i) := by |
| have hN : (0:ℝ) < Fintype.card ι := by exact_mod_cast Fintype.card_pos |
| have h := finite_log_sum a (fun _ => 1/(Fintype.card ι:ℝ)) ha |
| (fun _ => by positivity) (by simp [hN.ne']) (fun _ hz => (ne_of_gt (one_div_pos.mpr hN) hz).elim) |
| simp_rw [log_ratio_mul (by positivity : 1/(Fintype.card ι:ℝ) ≠ 0),Real.log_div one_ne_zero hN.ne',Real.log_one,zero_sub, |
| mul_neg,sub_neg_eq_add,Finset.sum_add_distrib,← Finset.sum_mul] at h |
| linarith |
|
|
| end RepeatedLocalization |
|
|
| end OAI |
|
|
