| import OAI.Probability.DirectionalWalk.PosteriorMax |
|
|
| namespace OAI |
|
|
| open MeasureTheory ProbabilityTheory Filter Preorder |
| open scoped ENNReal BigOperators Topology |
|
|
| namespace DirectionalZeroOne |
|
|
| open scoped Classical |
|
|
| noncomputable def posteriorSup {Ω ι : Type*} [Fintype ι] (w : ι → ℕ → Ω → ℝ) |
| (ω : Ω) : ℝ := ⨆ n : ℕ, posteriorMax w n ω |
|
|
| lemma posteriorSup_bounds {Ω ι : Type*} [Fintype ι] (w : ι → ℕ → Ω → ℝ) |
| (hpos : ∀ e i ω, 0 ≤ w e i ω) (hsum : ∀ i ω, ∑ e, w e i ω ≤ 1) (ω : Ω) : |
| 0 ≤ posteriorSup w ω ∧ posteriorSup w ω ≤ Fintype.card ι := by |
| have hb : BddAbove (Set.range (fun n => posteriorMax w n ω)) := |
| ⟨Fintype.card ι,by rintro _ ⟨n,rfl⟩; exact posteriorMax_le_card w hpos hsum n ω⟩ |
| exact ⟨(posteriorMax_nonneg w hpos 0 ω).trans (le_ciSup hb 0), |
| ciSup_le (fun n => posteriorMax_le_card w hpos hsum n ω)⟩ |
|
|
| lemma posteriorMax_tendsto {Ω ι : Type*} [Fintype ι] (w : ι → ℕ → Ω → ℝ) |
| (hpos : ∀ e i ω, 0 ≤ w e i ω) (hsum : ∀ i ω, ∑ e, w e i ω ≤ 1) (ω : Ω) : |
| Tendsto (fun n => posteriorMax w n ω) atTop (𝓝 (posteriorSup w ω)) := by |
| exact tendsto_atTop_ciSup (posteriorMax_mono w ω) |
| ⟨Fintype.card ι,by rintro _ ⟨n,rfl⟩; exact posteriorMax_le_card w hpos hsum n ω⟩ |
|
|
| lemma measurable_posteriorSup {Ω ι : Type*} [m : MeasurableSpace Ω] [Fintype ι] |
| (μ : Measure Ω) (ℱ : Filtration ℕ m) (w : ι → ℕ → Ω → ℝ) |
| (hw : ∀ e, Martingale (w e) ℱ μ) : Measurable (posteriorSup w) := |
| Measurable.iSup (fun n => (posteriorMax_adapted μ ℱ w hw n).mono (ℱ.le n) le_rfl) |
|
|
| lemma posteriorSup_exponential {Ω ι : Type*} [m : MeasurableSpace Ω] [Fintype ι] |
| (μ : Measure Ω) [IsProbabilityMeasure μ] (ℱ : Filtration ℕ m) |
| (w : ι → ℕ → Ω → ℝ) (hw : ∀ e, Martingale (w e) ℱ μ) |
| (hpos : ∀ e i ω, 0 ≤ w e i ω) (hsum : ∀ i ω, ∑ e, w e i ω ≤ 1) |
| (L : ℕ) (hL : (Fintype.card ι : ℝ)*(1/2 : ℝ)^L ≤ 1) : |
| (∫ ω, Real.exp (Real.log (3/2) / (2*(L+1)+1) * posteriorSup w ω) ∂μ) ≤ 12 := by |
| let c : ℝ := Real.log (3/2) / (2*(L+1)+1) |
| have hc : 0 ≤ c := div_nonneg (Real.log_nonneg (by norm_num)) (by positivity) |
| have hM (n : ℕ) : Measurable (posteriorMax w n) := (posteriorMax_adapted μ ℱ w hw n).mono (ℱ.le n) le_rfl |
| have hI (n : ℕ) : Integrable (fun ω => Real.exp (c*posteriorMax w n ω)) μ := |
| integrable_exp_of_bound μ _ (hM n) (Fintype.card ι) c (posteriorMax_le_card w hpos hsum n) hc |
| have hS : Integrable (fun ω => Real.exp (c*posteriorSup w ω)) μ := |
| integrable_exp_of_bound μ _ (measurable_posteriorSup μ ℱ w hw) (Fintype.card ι) c |
| (fun ω => (posteriorSup_bounds w hpos hsum ω).2) hc |
| have ht := integral_tendsto_of_tendsto_of_monotone hI hS |
| (Filter.Eventually.of_forall (fun ω n N hnN => Real.exp_le_exp.mpr |
| (mul_le_mul_of_nonneg_left (posteriorMax_mono w ω hnN) hc))) |
| (Filter.Eventually.of_forall (fun ω => Real.continuous_exp.continuousAt.tendsto.comp |
| ((posteriorMax_tendsto w hpos hsum ω).const_mul c))) |
| apply le_of_tendsto ht |
| filter_upwards [] with n |
| exact geometric_tail_exponential μ (posteriorMax w n) (hM n) (Fintype.card ι) |
| (posteriorMax_le_card w hpos hsum n) (2*(L+1)+1) (by have := Nat.cast_nonneg (α := ℝ) L; linarith) |
| (posteriorMax_geometric_tail μ ℱ w hw hpos hsum L hL n) |
|
|
| noncomputable def goodPosteriorSet {Ω ι : Type*} [Fintype ι] |
| (w : ι → ℕ → Ω → ℝ) (n : ℕ) : Set Ω := {ω | (∀ e, 0 ≤ w e n ω) ∧ ∑ e, w e n ω ≤ 1} |
|
|
| noncomputable def clippedPosterior {Ω ι : Type*} [Fintype ι] |
| (w : ι → ℕ → Ω → ℝ) (e : ι) (n : ℕ) : Ω → ℝ := (goodPosteriorSet w n).indicator (w e n) |
|
|
| lemma measurableSet_goodPosterior {Ω ι : Type*} [m : MeasurableSpace Ω] [Fintype ι] |
| (μ : Measure Ω) (ℱ : Filtration ℕ m) (w : ι → ℕ → Ω → ℝ) |
| (hw : ∀ e, Martingale (w e) ℱ μ) (n : ℕ) : MeasurableSet[ℱ n] (goodPosteriorSet w n) := by |
| let : MeasurableSpace Ω := ℱ n |
| unfold goodPosteriorSet |
| rw [Set.ofPred_and,Set.ofPred_forall] |
| apply MeasurableSet.inter |
| · |
| exact MeasurableSet.iInter (fun e => measurableSet_le measurable_const ((hw e).stronglyMeasurable n).measurable) |
| · exact measurableSet_le (Finset.measurable_sum _ (fun e _ => ((hw e).stronglyMeasurable n).measurable)) measurable_const |
|
|
| lemma clippedPosterior_bounds {Ω ι : Type*} [Fintype ι] (w : ι → ℕ → Ω → ℝ) : |
| (∀ e n ω, 0 ≤ clippedPosterior w e n ω) ∧ (∀ n ω, ∑ e, clippedPosterior w e n ω ≤ 1) := by |
| constructor |
| · intro e n ω |
| by_cases hω : ω ∈ goodPosteriorSet w n |
| · simpa only [clippedPosterior,Set.indicator_of_mem hω] using hω.1 e |
| · simp only [clippedPosterior,Set.indicator_of_notMem hω,le_refl] |
| · intro n ω |
| by_cases hω : ω ∈ goodPosteriorSet w n |
| · simpa only [clippedPosterior,Set.indicator_of_mem hω] using hω.2 |
| · simp only [clippedPosterior,Set.indicator_of_notMem hω,Finset.sum_const_zero]; norm_num |
|
|
| lemma clippedPosterior_ae_eq {Ω ι : Type*} [MeasurableSpace Ω] [Fintype ι] |
| (μ : Measure Ω) (w : ι → ℕ → Ω → ℝ) |
| (hpos : ∀ e n, ∀ᵐ ω ∂μ, 0 ≤ w e n ω) (hsum : ∀ n, ∀ᵐ ω ∂μ, ∑ e, w e n ω ≤ 1) |
| (e : ι) (n : ℕ) : w e n =ᵐ[μ] clippedPosterior w e n := by |
| filter_upwards [(ae_all_iff.mpr (fun e => hpos e n)),hsum n] with ω hω hsumω |
| exact (Set.indicator_of_mem (show ω ∈ goodPosteriorSet w n from ⟨hω,hsumω⟩) _).symm |
|
|
| lemma clippedPosterior_martingale {Ω ι : Type*} [m : MeasurableSpace Ω] [Fintype ι] |
| (μ : Measure Ω) (ℱ : Filtration ℕ m) (w : ι → ℕ → Ω → ℝ) |
| (hw : ∀ e, Martingale (w e) ℱ μ) |
| (hpos : ∀ e n, ∀ᵐ ω ∂μ, 0 ≤ w e n ω) (hsum : ∀ n, ∀ᵐ ω ∂μ, ∑ e, w e n ω ≤ 1) (e : ι) : |
| Martingale (clippedPosterior w e) ℱ μ := by |
| apply (hw e).congr |
| · intro n |
| exact ((hw e).stronglyMeasurable n).indicator (measurableSet_goodPosterior μ ℱ w hw n) |
| · exact clippedPosterior_ae_eq μ w hpos hsum e |
|
|
| lemma posteriorSup_congr_ae {Ω ι : Type*} [MeasurableSpace Ω] [Fintype ι] |
| (μ : Measure Ω) (w v : ι → ℕ → Ω → ℝ) (h : ∀ e n, w e n =ᵐ[μ] v e n) : |
| posteriorSup w =ᵐ[μ] posteriorSup v := by |
| filter_upwards [ae_all_iff.mpr (fun e => ae_all_iff.mpr (h e))] with ω hω |
| unfold posteriorSup |
| congr 1 |
| funext n |
| apply Finset.sum_congr rfl |
| intro e _ |
| unfold runningMax |
| congr 1 |
| funext i |
| exact hω e i |
|
|
| noncomputable def posteriorLevel (m : ℕ) : ℕ := ⌈Real.log (m+1) / Real.log 2⌉₊ |
|
|
| lemma posteriorLevel_dyadic (m : ℕ) : (m : ℝ) * (1/2 : ℝ)^(posteriorLevel m) ≤ 1 := by |
| have h2 : 0 < Real.log 2 := Real.log_pos (by norm_num) |
| have hceil := Nat.le_ceil (Real.log (m+1) / Real.log 2) |
| have hlog : Real.log (m+1) ≤ (posteriorLevel m : ℝ) * Real.log 2 := |
| (div_le_iff₀ h2).mp hceil |
| have he := Real.exp_le_exp.mpr hlog |
| rw [Real.exp_log (by positivity),Real.exp_nat_mul,Real.exp_log (by norm_num)] at he |
| have hpow : (2 : ℝ)^posteriorLevel m * (1/2 : ℝ)^posteriorLevel m = 1 := by |
| rw [← mul_pow]; norm_num |
| have hh := mul_le_mul_of_nonneg_right he (by positivity : 0 ≤ (1/2 : ℝ)^posteriorLevel m) |
| rw [hpow] at hh |
| have hp : 0 ≤ (1/2 : ℝ)^posteriorLevel m := by positivity |
| nlinarith |
|
|
| lemma posteriorLevel_scale (m : ℕ) (hm : 1 ≤ m) : |
| 2*((posteriorLevel m : ℝ)+1)+1 ≤ 7 * Real.log (m+1) / Real.log 2 := by |
| have h2 : 0 < Real.log 2 := Real.log_pos (by norm_num) |
| have hB : Real.log 2 ≤ Real.log (m+1) := |
| Real.log_le_log (by norm_num) (by exact_mod_cast (show 2 ≤ m+1 by omega)) |
| have hq : 0 ≤ Real.log (m+1) / Real.log 2 := div_nonneg (h2.le.trans hB) h2.le |
| have hceil : (posteriorLevel m : ℝ) < Real.log (m+1)/Real.log 2 + 1 := Nat.ceil_lt_add_one hq |
| have hh : 1 ≤ Real.log (m+1)/Real.log 2 := (le_div_iff₀ h2).mpr (by simpa using hB) |
| have he : 7*Real.log (m+1)/Real.log 2 = 7*(Real.log (m+1)/Real.log 2) := by ring |
| rw [he] |
| linarith |
|
|
| noncomputable def posteriorExponent : ℝ := Real.log (3/2) * Real.log 2 / 7 |
|
|
| lemma posteriorExponent_pos : 0 < posteriorExponent := by |
| exact div_pos (mul_pos (Real.log_pos (by norm_num)) (Real.log_pos (by norm_num))) (by norm_num) |
|
|
| lemma posteriorExponent_le (m : ℕ) (hm : 1 ≤ m) : |
| posteriorExponent / Real.log (m+1) ≤ Real.log (3/2) / (2*((posteriorLevel m : ℝ)+1)+1) := by |
| have h2 : 0 < Real.log 2 := Real.log_pos (by norm_num) |
| have hB : 0 < Real.log (m+1) := Real.log_pos (by exact_mod_cast (show 1 < m+1 by omega)) |
| have hb : 0 < 2*((posteriorLevel m : ℝ)+1)+1 := by positivity |
| have hscale := (le_div_iff₀ h2).mp (posteriorLevel_scale m hm) |
| apply (div_le_div_iff₀ hB hb).mpr |
| unfold posteriorExponent |
| have hl : 0 ≤ Real.log (3/2) := Real.log_nonneg (by norm_num) |
| have hh := mul_le_mul_of_nonneg_left hscale hl |
| nlinarith |
|
|
| lemma setIntegral_entropy_of_exp {Ω : Type*} [MeasurableSpace Ω] (μ : Measure Ω) |
| [IsProbabilityMeasure μ] (X : Ω → ℝ) (hX : Integrable X μ) (c : ℝ) (hc : 0 < c) |
| (hE : Integrable (fun ω => Real.exp (c*X ω)) μ) |
| (hbound : (∫ ω, Real.exp (c*X ω) ∂μ) ≤ 12) (V : Set Ω) : |
| (∫ ω in V, X ω ∂μ) ≤ (12/c) * μ.real V * (1-Real.log (μ.real V)) := by |
| let p := μ.real V |
| by_cases hp : p = 0 |
| · change μ.real V = 0 at hp |
| have hz : μ V = 0 := (measureReal_eq_zero_iff (by finiteness)).mp hp |
| simp only [Measure.restrict_eq_zero.mpr hz,integral_zero_measure,hp,mul_zero,zero_mul,le_refl] |
| · have hp0 : 0 < p := lt_of_le_of_ne measureReal_nonneg (Ne.symm hp) |
| have hp1 : p ≤ 1 := measureReal_le_one |
| have hlog : Real.log p ≤ 0 := Real.log_nonpos hp0.le hp1 |
| have hpoint (ω : Ω) : c*X ω ≤ p*Real.exp (c*X ω) - Real.log p := by |
| have h := Real.add_one_le_exp (c*X ω + Real.log p) |
| rw [Real.exp_add,Real.exp_log hp0] at h |
| nlinarith |
| have hi : (∫ ω in V, c*X ω ∂μ) ≤ ∫ ω in V, (p*Real.exp (c*X ω)-Real.log p) ∂μ := |
| integral_mono (hX.restrict.const_mul c) |
| ((hE.restrict.const_mul p).sub (integrable_const (Real.log p))) hpoint |
| rw [integral_const_mul,integral_sub (hE.restrict.const_mul p) (integrable_const _), |
| integral_const_mul,integral_const,measureReal_restrict_apply_univ,smul_eq_mul] at hi |
| have he : (∫ ω in V, Real.exp (c*X ω) ∂μ) ≤ 12 := |
| (setIntegral_le_integral hE (Filter.Eventually.of_forall (fun ω => Real.exp_nonneg _))).trans hbound |
| have hcore : c * (∫ ω in V, X ω ∂μ) ≤ 12*p*(1-Real.log p) := by |
| change c * (∫ ω in V, X ω ∂μ) ≤ _ at hi |
| have hh := mul_le_mul_of_nonneg_left he hp0.le |
| have hl := mul_nonpos_of_nonneg_of_nonpos hp0.le hlog |
| dsimp [p] at hi ⊢ |
| dsimp [p] at hh hl |
| nlinarith |
| calc |
| _ ≤ 12*p*(1-Real.log p)/c := (le_div_iff₀ hc).mpr (by simpa [mul_comm] using hcore) |
| _ = _ := by dsimp [p]; ring |
|
|
| lemma posteriorSup_exponential_log {Ω ι : Type*} [m : MeasurableSpace Ω] [Fintype ι] [Nonempty ι] |
| (μ : Measure Ω) [IsProbabilityMeasure μ] (ℱ : Filtration ℕ m) |
| (w : ι → ℕ → Ω → ℝ) (hw : ∀ e, Martingale (w e) ℱ μ) |
| (hpos : ∀ e i ω, 0 ≤ w e i ω) (hsum : ∀ i ω, ∑ e, w e i ω ≤ 1) : |
| (∫ ω, Real.exp (posteriorExponent / Real.log (Fintype.card ι+1) * posteriorSup w ω) ∂μ) ≤ 12 := by |
| have hm : 1 ≤ Fintype.card ι := Fintype.card_pos_iff.mpr ‹Nonempty ι› |
| have hB : 0 < Real.log (Fintype.card ι+1) := Real.log_pos (by exact_mod_cast (show 1 < Fintype.card ι+1 by omega)) |
| let c := posteriorExponent / Real.log (Fintype.card ι+1) |
| let c' := Real.log (3/2) / (2*((posteriorLevel (Fintype.card ι) : ℝ)+1)+1) |
| have hle : c ≤ c' := posteriorExponent_le _ hm |
| have hc : 0 ≤ c := (div_pos posteriorExponent_pos hB).le |
| have hc' : 0 ≤ c' := hc.trans hle |
| have hmeas := measurable_posteriorSup μ ℱ w hw |
| have hb := fun ω => (posteriorSup_bounds w hpos hsum ω).2 |
| have hi := integral_mono (integrable_exp_of_bound μ _ hmeas _ c hb hc) |
| (integrable_exp_of_bound μ _ hmeas _ c' hb hc') (show ∀ ω, Real.exp (c*posteriorSup w ω) ≤ Real.exp (c'*posteriorSup w ω) from by |
| intro ω |
| exact Real.exp_le_exp.mpr (mul_le_mul_of_nonneg_right hle (posteriorSup_bounds w hpos hsum ω).1)) |
| exact hi.trans (posteriorSup_exponential μ ℱ w hw hpos hsum _ (posteriorLevel_dyadic _)) |
|
|
| lemma posteriorSup_exponential_log_ae {Ω ι : Type*} [m : MeasurableSpace Ω] [Fintype ι] [Nonempty ι] |
| (μ : Measure Ω) [IsProbabilityMeasure μ] (ℱ : Filtration ℕ m) |
| (w : ι → ℕ → Ω → ℝ) (hw : ∀ e, Martingale (w e) ℱ μ) |
| (hpos : ∀ e i, ∀ᵐ ω ∂μ, 0 ≤ w e i ω) (hsum : ∀ i, ∀ᵐ ω ∂μ, ∑ e, w e i ω ≤ 1) : |
| (∫ ω, Real.exp (posteriorExponent / Real.log (Fintype.card ι+1) * posteriorSup w ω) ∂μ) ≤ 12 := by |
| have he := posteriorSup_congr_ae μ w (clippedPosterior w) (clippedPosterior_ae_eq μ w hpos hsum) |
| have hi : (∫ ω, Real.exp (posteriorExponent / Real.log (Fintype.card ι+1) * posteriorSup w ω) ∂μ) = |
| ∫ ω, Real.exp (posteriorExponent / Real.log (Fintype.card ι+1) * posteriorSup (clippedPosterior w) ω) ∂μ := by |
| apply integral_congr_ae |
| filter_upwards [he] with ω hω |
| rw [hω] |
| rw [hi] |
| exact posteriorSup_exponential_log μ ℱ (clippedPosterior w) |
| (clippedPosterior_martingale μ ℱ w hw hpos hsum) |
| (clippedPosterior_bounds w).1 (clippedPosterior_bounds w).2 |
|
|
| lemma integrable_posteriorSup_ae {Ω ι : Type*} [m : MeasurableSpace Ω] [Fintype ι] |
| (μ : Measure Ω) [IsFiniteMeasure μ] (ℱ : Filtration ℕ m) |
| (w : ι → ℕ → Ω → ℝ) (hw : ∀ e, Martingale (w e) ℱ μ) |
| (hpos : ∀ e i, ∀ᵐ ω ∂μ, 0 ≤ w e i ω) (hsum : ∀ i, ∀ᵐ ω ∂μ, ∑ e, w e i ω ≤ 1) : |
| Integrable (posteriorSup w) μ := by |
| have he := posteriorSup_congr_ae μ w (clippedPosterior w) (clippedPosterior_ae_eq μ w hpos hsum) |
| apply (integrable_const (Fintype.card ι : ℝ)).mono' (measurable_posteriorSup μ ℱ w hw).aestronglyMeasurable |
| filter_upwards [he] with ω hω |
| rw [hω,Real.norm_eq_abs,abs_of_nonneg (posteriorSup_bounds _ (clippedPosterior_bounds w).1 (clippedPosterior_bounds w).2 ω).1] |
| exact (posteriorSup_bounds _ (clippedPosterior_bounds w).1 (clippedPosterior_bounds w).2 ω).2 |
|
|
| lemma integrable_posteriorSup_exp_ae {Ω ι : Type*} [m : MeasurableSpace Ω] [Fintype ι] |
| (μ : Measure Ω) [IsFiniteMeasure μ] (ℱ : Filtration ℕ m) |
| (w : ι → ℕ → Ω → ℝ) (hw : ∀ e, Martingale (w e) ℱ μ) |
| (hpos : ∀ e i, ∀ᵐ ω ∂μ, 0 ≤ w e i ω) (hsum : ∀ i, ∀ᵐ ω ∂μ, ∑ e, w e i ω ≤ 1) |
| (c : ℝ) (hc : 0 ≤ c) : Integrable (fun ω => Real.exp (c * posteriorSup w ω)) μ := by |
| have he := posteriorSup_congr_ae μ w (clippedPosterior w) (clippedPosterior_ae_eq μ w hpos hsum) |
| have hm := measurable_posteriorSup μ ℱ w hw |
| apply (integrable_const (Real.exp (c*Fintype.card ι))).mono' (hm.const_mul c).exp.aestronglyMeasurable |
| filter_upwards [he] with ω hω |
| rw [Real.norm_eq_abs,abs_of_pos (Real.exp_pos _),hω] |
| exact Real.exp_le_exp.mpr (mul_le_mul_of_nonneg_left |
| (posteriorSup_bounds _ (clippedPosterior_bounds w).1 (clippedPosterior_bounds w).2 ω).2 hc) |
|
|
| lemma posteriorSup_joint_event_weight {Ω Ξ ι : Type*} [m : MeasurableSpace Ω] [MeasurableSpace Ξ] |
| [Fintype ι] [Nonempty ι] (μ : Measure Ω) [IsProbabilityMeasure μ] |
| (ν : Measure Ξ) [IsProbabilityMeasure ν] (π : Ξ → Ω) (hπ : MeasurePreserving π ν μ) |
| (ℱ : Filtration ℕ m) (w : ι → ℕ → Ω → ℝ) (hw : ∀ e, Martingale (w e) ℱ μ) |
| (hpos : ∀ e i, ∀ᵐ ω ∂μ, 0 ≤ w e i ω) (hsum : ∀ i, ∀ᵐ ω ∂μ, ∑ e, w e i ω ≤ 1) |
| (V : Set Ξ) : |
| (∫ ξ in V, posteriorSup w (π ξ) ∂ν) ≤ |
| (12/posteriorExponent) * Real.log (Fintype.card ι+1) * ν.real V * (1-Real.log (ν.real V)) := by |
| let c := posteriorExponent / Real.log (Fintype.card ι+1) |
| have hm : 1 ≤ Fintype.card ι := Fintype.card_pos_iff.mpr ‹Nonempty ι› |
| have hB : 0 < Real.log (Fintype.card ι+1) := Real.log_pos (by exact_mod_cast (show 1 < Fintype.card ι+1 by omega)) |
| have hc : 0 < c := div_pos posteriorExponent_pos hB |
| have hXe := integrable_posteriorSup_exp_ae μ ℱ w hw hpos hsum c hc.le |
| have hXI := integrable_posteriorSup_ae μ ℱ w hw hpos hsum |
| have hExp : (∫ ξ, Real.exp (c * posteriorSup w (π ξ)) ∂ν) ≤ 12 := by |
| have hi : (∫ ξ, Real.exp (c * posteriorSup w (π ξ)) ∂ν) = |
| ∫ ω, Real.exp (c * posteriorSup w ω) ∂μ := by |
| rw [← hπ.map_eq] |
| exact (integral_map_of_stronglyMeasurable hπ.measurable |
| (((measurable_posteriorSup μ ℱ w hw).const_mul c).exp.stronglyMeasurable)).symm |
| rw [hi] |
| exact posteriorSup_exponential_log_ae μ ℱ w hw hpos hsum |
| have hi := setIntegral_entropy_of_exp ν (fun ξ => posteriorSup w (π ξ)) |
| (hπ.integrable_comp_of_integrable hXI) c hc (hπ.integrable_comp_of_integrable hXe) hExp V |
| have hscale : 12/c = (12/posteriorExponent) * Real.log (Fintype.card ι+1) := by |
| dsimp [c] |
| field_simp |
| simpa only [hscale] using hi |
|
|
| lemma ae_atom_pos {α : Type*} [Countable α] [MeasurableSpace α] [MeasurableSingletonClass α] |
| (μ : Measure α) : ∀ᵐ x ∂μ, μ {x} ≠ 0 := by |
| simp only [ae_iff] |
| have h : {x | ¬ μ {x} ≠ 0} = ⋃ x : {x // μ {x} = 0}, ({x.val} : Set α) := by |
| ext x |
| simp |
| rw [h] |
| exact measure_iUnion_null fun x => x.property |
|
|
| end DirectionalZeroOne |
|
|
| end OAI |
|
|
