| import OAI.Probability.DirectionalWalk.PosteriorTails |
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| namespace OAI |
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| open MeasureTheory ProbabilityTheory Filter Preorder |
| open scoped ENNReal BigOperators Topology |
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| namespace DirectionalZeroOne |
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|
| open scoped Classical |
|
|
| noncomputable def atomInfo {α : Type*} [MeasurableSpace α] (μ : Measure α) (x : α) : ℝ := |
| -Real.log (μ.real {x}) |
|
|
| noncomputable def discreteEntropy {α : Type*} [MeasurableSpace α] (μ : Measure α) : ℝ := |
| ∫ x, atomInfo μ x ∂μ |
|
|
| lemma atomInfo_nonneg {α : Type*} [MeasurableSpace α] (μ : Measure α) |
| [IsProbabilityMeasure μ] (x : α) : 0 ≤ atomInfo μ x := by |
| exact neg_nonneg.mpr (Real.log_nonpos measureReal_nonneg measureReal_le_one) |
|
|
| lemma discreteEntropy_nonneg {α : Type*} [MeasurableSpace α] (μ : Measure α) |
| [IsProbabilityMeasure μ] : 0 ≤ discreteEntropy μ := |
| integral_nonneg (atomInfo_nonneg μ) |
|
|
| lemma ae_atom_real_pos {α : Type*} [Countable α] [MeasurableSpace α] [MeasurableSingletonClass α] |
| (μ : Measure α) [IsFiniteMeasure μ] : ∀ᵐ x ∂μ, 0 < μ.real {x} := by |
| filter_upwards [ae_atom_pos μ] with x hx |
| exact ENNReal.toReal_pos hx (by finiteness) |
|
|
| lemma summable_integral_ratio {α : Type*} [Countable α] [MeasurableSpace α] |
| [MeasurableSingletonClass α] (μ : Measure α) [IsFiniteMeasure μ] |
| (q : α → ℝ≥0∞) (_hq : (∑' x, q x) ≠ ∞) : |
| (∫⁻ x, q x / μ {x} ∂μ) ≤ ∑' x, q x := by |
| rw [lintegral_countable'] |
| apply ENNReal.tsum_le_tsum |
| intro x |
| by_cases hx : μ {x} = 0 |
| · simp [hx] |
| · rw [ENNReal.div_mul_cancel hx (by finiteness)] |
| |
| lemma integrable_atom_ratio {α : Type*} [Countable α] [MeasurableSpace α] |
| [MeasurableSingletonClass α] (μ : Measure α) [IsFiniteMeasure μ] |
| (q : α → ℝ≥0∞) (hq : (∑' x, q x) ≠ ∞) : |
| Integrable (fun x => (q x / μ {x}).toReal) μ := by |
| apply integrable_toReal_of_lintegral_ne_top (measurable_of_countable _).aemeasurable |
| exact ne_of_lt ((summable_integral_ratio μ q hq).trans_lt (lt_top_iff_ne_top.mpr hq)) |
|
|
| lemma integral_atom_ratio_le {α : Type*} [Countable α] [MeasurableSpace α] |
| [MeasurableSingletonClass α] (μ : Measure α) [IsFiniteMeasure μ] |
| (q : α → ℝ≥0∞) (hq : (∑' x, q x) ≠ ∞) : |
| (∫ x, (q x / μ {x}).toReal ∂μ) ≤ (∑' x, q x).toReal := by |
| rw [integral_toReal] |
| · exact ENNReal.toReal_mono hq (summable_integral_ratio μ q hq) |
| · exact (measurable_of_countable _).aemeasurable |
| · filter_upwards [ae_atom_pos μ] with x hx |
| exact lt_top_iff_ne_top.mpr (ENNReal.div_ne_top (ne_of_lt ((ENNReal.le_tsum x).trans_lt (lt_top_iff_ne_top.mpr hq))) hx) |
|
|
| lemma atomInfo_le_comparison {α : Type*} [Countable α] [MeasurableSpace α] |
| [MeasurableSingletonClass α] (μ : Measure α) [IsProbabilityMeasure μ] |
| (q : α → ℝ≥0∞) (hq : ∀ x, q x ≠ 0 ∧ q x ≠ ∞) : |
| ∀ᵐ x ∂μ, atomInfo μ x ≤ -Real.log (q x).toReal + (q x / μ {x}).toReal - 1 := by |
| filter_upwards [ae_atom_real_pos μ] with x hx |
| have hqx : 0 < (q x).toReal := ENNReal.toReal_pos (hq x).1 (hq x).2 |
| have hh := Real.log_le_sub_one_of_pos (div_pos hqx hx) |
| rw [Real.log_div (ne_of_gt hqx) (ne_of_gt hx)] at hh |
| simp only [atomInfo,ENNReal.toReal_div,measureReal_def] at * |
| linarith |
|
|
| lemma integrable_atomInfo_of_comparison {α : Type*} [Countable α] [MeasurableSpace α] |
| [MeasurableSingletonClass α] (μ : Measure α) [IsProbabilityMeasure μ] |
| (q : α → ℝ≥0∞) (hq0 : ∀ x, q x ≠ 0) (hq : (∑' x, q x) ≠ ∞) |
| (hcost : Integrable (fun x => -Real.log (q x).toReal) μ) : Integrable (atomInfo μ) μ := by |
| have hqx x : q x ≠ 0 ∧ q x ≠ ∞ := ⟨hq0 x, |
| ne_of_lt ((ENNReal.le_tsum x).trans_lt (lt_top_iff_ne_top.mpr hq))⟩ |
| apply ((hcost.add (integrable_atom_ratio μ q hq)).sub (integrable_const 1)).mono' |
| (measurable_of_countable _).aestronglyMeasurable |
| filter_upwards [atomInfo_le_comparison μ q hqx] with x hx |
| simpa only [Pi.sub_apply,Pi.add_apply,Real.norm_eq_abs,abs_of_nonneg (atomInfo_nonneg μ x)] using hx |
|
|
| lemma discreteEntropy_le_comparison {α : Type*} [Countable α] [MeasurableSpace α] |
| [MeasurableSingletonClass α] (μ : Measure α) [IsProbabilityMeasure μ] |
| (q : α → ℝ≥0∞) (hq0 : ∀ x, q x ≠ 0) (hq : (∑' x, q x) ≠ ∞) |
| (hcost : Integrable (fun x => -Real.log (q x).toReal) μ) : |
| discreteEntropy μ ≤ (∫ x, -Real.log (q x).toReal ∂μ) + (∑' x, q x).toReal - 1 := by |
| have hqx x : q x ≠ 0 ∧ q x ≠ ∞ := ⟨hq0 x, |
| ne_of_lt ((ENNReal.le_tsum x).trans_lt (lt_top_iff_ne_top.mpr hq))⟩ |
| have hi := integral_mono_ae (integrable_atomInfo_of_comparison μ q hq0 hq hcost) |
| ((hcost.add (integrable_atom_ratio μ q hq)).sub (integrable_const 1)) |
| (atomInfo_le_comparison μ q hqx) |
| simp only [Pi.sub_apply,Pi.add_apply] at hi |
| have hsum : Integrable (fun x => -Real.log (q x).toReal + (q x / μ {x}).toReal) μ := |
| hcost.add (integrable_atom_ratio μ q hq) |
| rw [integral_sub hsum (integrable_const 1), |
| integral_add hcost (integrable_atom_ratio μ q hq)] at hi |
| simp only [integral_const,probReal_univ,smul_eq_mul,one_mul] at hi |
| exact hi.trans (by linarith [integral_atom_ratio_le μ q hq]) |
|
|
| lemma rnDeriv_atom {α : Type*} [MeasurableSpace α] [MeasurableSingletonClass α] |
| (μ ν : Measure α) [IsFiniteMeasure μ] [IsFiniteMeasure ν] (hμν : μ ≪ ν) |
| (x : α) (hx : ν {x} ≠ 0) : μ.rnDeriv ν x = μ {x} / ν {x} := by |
| have h := Measure.setLIntegral_rnDeriv hμν ({x} : Set α) |
| rw [lintegral_singleton] at h |
| exact (ENNReal.eq_div_iff hx (by finiteness)).mpr (by simpa only [mul_comm] using h) |
|
|
| lemma llr_atom {α : Type*} [Countable α] [MeasurableSpace α] [MeasurableSingletonClass α] |
| (μ ν : Measure α) [IsFiniteMeasure μ] [IsFiniteMeasure ν] (hμν : μ ≪ ν) : |
| llr μ ν =ᵐ[μ] fun x => Real.log (μ.real {x}) - Real.log (ν.real {x}) := by |
| filter_upwards [ae_atom_pos μ,hμν (ae_atom_pos ν)] with x hx hy |
| rw [llr,rnDeriv_atom μ ν hμν x hy,ENNReal.toReal_div,Real.log_div] |
| · rfl |
| · exact (ENNReal.toReal_pos hx (by finiteness)).ne' |
| · exact (ENNReal.toReal_pos hy (by finiteness)).ne' |
|
|
| lemma absolutelyContinuous_of_atoms {α : Type*} [Countable α] [MeasurableSpace α] |
| [MeasurableSingletonClass α] (μ ν : Measure α) |
| (h : ∀ x, ν {x} = 0 → μ {x} = 0) : μ ≪ ν := by |
| intro s hs |
| rw [← Set.biUnion_of_singleton s] |
| exact (measure_biUnion_null_iff (Set.to_countable s)).mpr fun x hx => h x |
| (measure_mono_null (Set.singleton_subset_iff.mpr hx) hs) |
|
|
| lemma atom_logRatio_nonneg {α : Type*} [Countable α] [MeasurableSpace α] |
| [MeasurableSingletonClass α] (μ ν : Measure α) |
| [IsProbabilityMeasure μ] [IsProbabilityMeasure ν] (hμν : μ ≪ ν) |
| (hi : Integrable (fun x => Real.log (μ.real {x}) - Real.log (ν.real {x})) μ) : |
| 0 ≤ ∫ x, Real.log (μ.real {x}) - Real.log (ν.real {x}) ∂μ := by |
| have hllr := llr_atom μ ν hμν |
| have h := InformationTheory.integral_llr_add_sub_measure_univ_nonneg hμν |
| (hi.congr hllr.symm) |
| simpa only [integral_congr_ae hllr,probReal_univ,add_sub_cancel_right] using h |
|
|
| lemma atom_klDiv_formula {α : Type*} [Countable α] [MeasurableSpace α] |
| [MeasurableSingletonClass α] (μ ν : Measure α) |
| [IsProbabilityMeasure μ] [IsProbabilityMeasure ν] (hμν : μ ≪ ν) |
| (hi : Integrable (fun x => Real.log (μ.real {x}) - Real.log (ν.real {x})) μ) : |
| InformationTheory.klDiv μ ν = ENNReal.ofReal |
| (∫ x, Real.log (μ.real {x}) - Real.log (ν.real {x}) ∂μ) := by |
| have hllr := llr_atom μ ν hμν |
| rw [InformationTheory.klDiv_of_ac_of_integrable hμν (hi.congr hllr.symm), |
| integral_congr_ae hllr,probReal_univ,probReal_univ,add_sub_cancel_right] |
|
|
| noncomputable def conditionalInfo {α β : Type*} [MeasurableSpace α] [MeasurableSpace β] |
| (ρ : Measure (α × β)) (p : α × β) : ℝ := |
| -Real.log (ρ.real {p} / ρ.fst.real {p.1}) |
|
|
| noncomputable def conditionalEntropy {α β : Type*} [MeasurableSpace α] [MeasurableSpace β] |
| (ρ : Measure (α × β)) : ℝ := ∫ p, conditionalInfo ρ p ∂ρ |
|
|
| lemma atom_le_fst {α β : Type*} [MeasurableSpace α] [MeasurableSpace β] |
| [MeasurableSingletonClass α] (ρ : Measure (α × β)) (p : α × β) : |
| ρ {p} ≤ ρ.fst {p.1} := by |
| rw [Measure.fst_apply (measurableSet_singleton _)] |
| exact measure_mono (by intro x hx; obtain rfl := Set.mem_singleton_iff.mp hx; rfl) |
|
|
| lemma conditionalInfo_nonneg {α β : Type*} [MeasurableSpace α] [MeasurableSpace β] |
| [MeasurableSingletonClass α] (ρ : Measure (α × β)) [IsFiniteMeasure ρ] (p : α × β) : |
| 0 ≤ conditionalInfo ρ p := by |
| apply neg_nonneg.mpr |
| apply Real.log_nonpos (div_nonneg measureReal_nonneg measureReal_nonneg) |
| by_cases h : ρ.fst.real {p.1} = 0 |
| · simp [h] |
| · apply (div_le_one (lt_of_le_of_ne measureReal_nonneg (Ne.symm h))).mpr |
| exact ENNReal.toReal_mono (by finiteness) (atom_le_fst ρ p) |
|
|
| lemma conditionalInfo_log {α β : Type*} [Countable α] [Countable β] |
| [MeasurableSpace α] [MeasurableSpace β] [MeasurableSingletonClass α] [MeasurableSingletonClass β] |
| (ρ : Measure (α × β)) [IsFiniteMeasure ρ] : |
| conditionalInfo ρ =ᵐ[ρ] fun p => -Real.log (ρ.real {p}) + Real.log (ρ.fst.real {p.1}) := by |
| filter_upwards [ae_atom_real_pos ρ] with p hp |
| have hfst : 0 < ρ.fst.real {p.1} := hp.trans_le (ENNReal.toReal_mono (by finiteness) (atom_le_fst ρ p)) |
| dsimp [conditionalInfo] |
| rw [Real.log_div hp.ne' hfst.ne'] |
| ring |
|
|
| lemma compProd_atom {α β : Type*} [MeasurableSpace α] [MeasurableSpace β] |
| [MeasurableSingletonClass α] [MeasurableSingletonClass β] |
| (μ : Measure α) [SFinite μ] (κ : Kernel α β) [IsSFiniteKernel κ] (p : α × β) : |
| (μ ⊗ₘ κ) {p} = μ {p.1} * κ p.1 {p.2} := by |
| have hset : ({p} : Set (α × β)) = {p.1} ×ˢ {p.2} := by ext x; simp [Prod.ext_iff] |
| rw [hset,Measure.compProd_apply_prod (measurableSet_singleton _) (measurableSet_singleton _), |
| lintegral_singleton,mul_comm] |
|
|
| lemma integrable_conditionalInfo {α β : Type*} [Countable α] [Countable β] |
| [MeasurableSpace α] [MeasurableSpace β] [MeasurableSingletonClass α] [MeasurableSingletonClass β] |
| (ρ : Measure (α × β)) [IsProbabilityMeasure ρ] (κ : Kernel α β) [IsMarkovKernel κ] |
| (hac : ρ ≪ ρ.fst ⊗ₘ κ) |
| (hi : Integrable (fun p : α × β => atomInfo (κ p.1) p.2) ρ) : |
| Integrable (conditionalInfo ρ) ρ := by |
| let ν := ρ.fst ⊗ₘ κ |
| have hsum : (∑' p, ν {p}) = 1 := by |
| have hh := lintegral_countable' (μ := ν) (fun _ => (1 : ℝ≥0∞)) |
| simpa using hh.symm |
| have hr := integrable_atom_ratio ρ (fun p => ν {p}) (by rw [hsum]; exact ENNReal.one_ne_top) |
| apply ((hi.add hr).sub (integrable_const 1)).mono' (measurable_of_countable _).aestronglyMeasurable |
| filter_upwards [ae_atom_real_pos ρ,hac (ae_atom_real_pos ν),conditionalInfo_log ρ] with p hp hν he |
| have hfst : 0 < ρ.fst.real {p.1} := hp.trans_le (ENNReal.toReal_mono (by finiteness) (atom_le_fst ρ p)) |
| have hn : ν.real {p} = ρ.fst.real {p.1} * (κ p.1).real {p.2} := by |
| simp only [ν,measureReal_def,compProd_atom,ENNReal.toReal_mul] |
| have hk : 0 < (κ p.1).real {p.2} := (mul_pos_iff_of_pos_left hfst).mp (hn ▸ hν) |
| have hh := Real.log_le_sub_one_of_pos (div_pos hν hp) |
| rw [Real.log_div hν.ne' hp.ne',hn,Real.log_mul hfst.ne' hk.ne'] at hh |
| rw [Real.norm_eq_abs,abs_of_nonneg (conditionalInfo_nonneg ρ p)] |
| simp only [Pi.sub_apply,Pi.add_apply,he,atomInfo,ENNReal.toReal_div,← measureReal_def,hn] |
| linarith |
| |
| lemma conditionalEntropy_le_crossEntropy {α β : Type*} [Countable α] [Countable β] |
| [MeasurableSpace α] [MeasurableSpace β] [MeasurableSingletonClass α] [MeasurableSingletonClass β] |
| (ρ : Measure (α × β)) [IsProbabilityMeasure ρ] (κ : Kernel α β) [IsMarkovKernel κ] |
| (hac : ρ ≪ ρ.fst ⊗ₘ κ) |
| (hi : Integrable (fun p : α × β => atomInfo (κ p.1) p.2) ρ) : |
| conditionalEntropy ρ ≤ ∫ p, atomInfo (κ p.1) p.2 ∂ρ := by |
| have hc := integrable_conditionalInfo ρ κ hac hi |
| have he : (fun p : α × β => atomInfo (κ p.1) p.2 - conditionalInfo ρ p) =ᵐ[ρ] |
| fun p => Real.log (ρ.real {p}) - Real.log ((ρ.fst ⊗ₘ κ).real {p}) := by |
| filter_upwards [ae_atom_real_pos ρ,hac (ae_atom_real_pos (ρ.fst ⊗ₘ κ)),conditionalInfo_log ρ] with p hp hν he |
| have hfst : 0 < ρ.fst.real {p.1} := hp.trans_le (ENNReal.toReal_mono (by finiteness) (atom_le_fst ρ p)) |
| have hn : (ρ.fst ⊗ₘ κ).real {p} = ρ.fst.real {p.1} * (κ p.1).real {p.2} := by |
| simp only [measureReal_def,compProd_atom,ENNReal.toReal_mul] |
| have hk : 0 < (κ p.1).real {p.2} := (mul_pos_iff_of_pos_left hfst).mp (hn ▸ hν) |
| rw [hn,Real.log_mul hfst.ne' hk.ne',he] |
| dsimp [atomInfo] |
| ring |
| have hh := atom_logRatio_nonneg ρ (ρ.fst ⊗ₘ κ) hac ((hi.sub hc).congr he) |
| rw [← integral_congr_ae he,integral_sub hi hc] at hh |
| exact sub_nonneg.mp hh |
| |
| lemma joint_ac_marginal_product {α β : Type*} [Countable α] [Countable β] |
| [MeasurableSpace α] [MeasurableSpace β] [MeasurableSingletonClass α] [MeasurableSingletonClass β] |
| (ρ : Measure (α × β)) [IsProbabilityMeasure ρ] : ρ ≪ ρ.fst.prod ρ.snd := by |
| apply absolutelyContinuous_of_atoms |
| intro p hp |
| have he : (ρ.fst.prod ρ.snd) {p} = ρ.fst {p.1} * ρ.snd {p.2} := by |
| rw [← Set.singleton_prod_singleton,Measure.prod_prod] |
| rw [he,mul_eq_zero] at hp |
| rcases hp with hp | hp |
| · exact le_antisymm ((atom_le_fst ρ p).trans hp.le) bot_le |
| · have hh : ρ {p} ≤ ρ.snd {p.2} := by |
| rw [Measure.snd_apply (measurableSet_singleton _)] |
| exact measure_mono (by intro x hx; obtain rfl := Set.mem_singleton_iff.mp hx; rfl) |
| exact le_antisymm (hh.trans hp.le) bot_le |
| |
| lemma conditionalEntropy_le_entropy_snd {α β : Type*} [Countable α] [Countable β] |
| [MeasurableSpace α] [MeasurableSpace β] [MeasurableSingletonClass α] [MeasurableSingletonClass β] |
| (ρ : Measure (α × β)) [IsProbabilityMeasure ρ] |
| (hi : Integrable (atomInfo ρ.snd) ρ.snd) : conditionalEntropy ρ ≤ discreteEntropy ρ.snd := by |
| have hmap : MeasurePreserving Prod.snd ρ ρ.snd := ⟨measurable_snd,rfl⟩ |
| have hc : Integrable (fun p : α × β => atomInfo ρ.snd p.2) ρ := hmap.integrable_comp_of_integrable hi |
| have hh := conditionalEntropy_le_crossEntropy ρ (Kernel.const α ρ.snd) |
| (by simpa only [Measure.compProd_const] using joint_ac_marginal_product ρ) hc |
| have he : (∫ p, atomInfo ρ.snd p.2 ∂ρ) = discreteEntropy ρ.snd := by |
| dsimp [discreteEntropy,Measure.snd] |
| rw [integral_map_of_stronglyMeasurable measurable_snd (measurable_of_countable _).stronglyMeasurable] |
| exact hh.trans_eq he |
| |
| end DirectionalZeroOne |
| |
| end OAI |
| |
