| import Mathlib |
| import OAI.Probability.SKRatio.FiniteChain.EntropicCovariance |
|
|
| namespace OAI |
|
|
| section |
| noncomputable section |
| open scoped BigOperators |
| open Real |
| namespace SKRatio.Observation |
| variable {α : Type*} [Fintype α] |
|
|
| def logRemainder (u a : ℝ) : ℝ := u*log (u/a)-u+a |
|
|
| lemma logRemainder_nonneg {u a : ℝ} (hu : 0 < u) (ha : 0 < a) : |
| 0 ≤ logRemainder u a := by |
| have h := mul_le_mul_of_nonneg_left |
| (self_sub_one_le_mul_log (div_nonneg hu.le ha.le)) ha.le |
| have he : a*(u/a*log (u/a))-(a*(u/a-1)) = logRemainder u a := by |
| unfold logRemainder |
| field_simp [ha.ne'] |
| ring |
| rw [← he] |
| linarith only [h] |
|
|
| lemma logRemainder_eq {u a : ℝ} (hu : u ≠ 0) (ha : a ≠ 0) : |
| logRemainder u a = u*log u-u*log a-u+a := by |
| rw [logRemainder,log_div hu ha] |
| ring |
|
|
| lemma ent_nonneg (p : Prior α) {f : α → ℝ} (hf : ∀ x, 0 < f x) : 0 ≤ ent p f := by |
| have hj := convexOn_mul_log.map_sum_le (t := Finset.univ) (w := p.mass) (p := f) |
| (fun x _ => p.nonneg x) p.sum_one (fun x _ => (hf x).le) |
| change avg p f*log (avg p f) ≤ avg p (fun x => f x*log (f x)) at hj |
| unfold ent FiniteLaw.entropy |
| exact sub_nonneg.mpr hj |
|
|
| lemma avg_sum (p : Prior α) {ι : Type*} [Fintype ι] (f : ι → α → ℝ) : |
| avg p (fun x => ∑ i, f i x) = ∑ i, avg p (f i) := by |
| simp only [avg,FiniteLaw.mean,Finset.mul_sum] |
| rw [Finset.sum_comm] |
|
|
| lemma avg_logRemainder (p : Prior α) (f : α → ℝ) (hf : ∀ x, 0 < f x) {a : ℝ} (ha : 0<a) : |
| avg p (fun x => logRemainder (f x) a) = ent p f+logRemainder (avg p f) a := by |
| simp_rw [logRemainder_eq (hf _).ne' ha.ne'] |
| rw [avg_add,avg_sub,avg_sub,avg_mul_const,avg_const, |
| logRemainder_eq (avg_pos p hf).ne' ha.ne'] |
| unfold ent FiniteLaw.entropy |
| change _ = avg p (fun x => f x*log (f x))-avg p f*log (avg p f)+_ |
| ring |
|
|
| lemma ent_minimizes (p : Prior α) (f : α → ℝ) (hf : ∀ x, 0<f x) {a : ℝ} (ha : 0<a) : |
| ent p f ≤ avg p (fun x => logRemainder (f x) a) := by |
| rw [avg_logRemainder p f hf ha] |
| exact le_add_of_nonneg_right (logRemainder_nonneg (avg_pos p hf) ha) |
|
|
| lemma ent_eq_remainder (p : Prior α) (f : α → ℝ) (hf : ∀ x, 0 < f x) : |
| ent p f = avg p (fun x => logRemainder (f x) (avg p f)) := by |
| rw [avg_logRemainder p f hf (avg_pos p hf)] |
| simp only [logRemainder,div_self (avg_pos p hf).ne',log_one,mul_zero,zero_sub,neg_add_cancel,add_zero] |
|
|
| lemma prior_le_one (p : Prior α) (x : α) : p x ≤ 1 := by |
| rw [← p.sum_one] |
| exact Finset.single_le_sum (fun y _ => p.nonneg y) (Finset.mem_univ x) |
|
|
| lemma ent_bound (p : Prior α) (f : α → ℝ) (hf : ∀ x, 0 < f x) : |
| ent p f ≤ ∑ x, logRemainder (f x) 1 := by |
| apply (ent_minimizes p f hf zero_lt_one).trans |
| exact Finset.sum_le_sum (fun x _ => mul_le_of_le_one_left |
| (logRemainder_nonneg (hf x) zero_lt_one) (prior_le_one p x)) |
|
|
| lemma continuous_entropy_masses (f : α → ℝ) : |
| Continuous (fun p : α → ℝ => FiniteLaw.entropy p f) := by |
| have hs : Continuous (fun p : α → ℝ => FiniteLaw.mean p f) := |
| continuous_finsetSum _ (fun x _ => (continuous_apply x : Continuous (fun p : α → ℝ => p x)).mul_const (f x)) |
| exact (continuous_finsetSum _ (fun x _ => (continuous_apply x : Continuous (fun p : α → ℝ => p x)).mul_const (f x*log (f x)))).sub |
| (continuous_mul_log.comp hs) |
|
|
| lemma avg_ent_eq_remainder (p : Prior α) (f : α → ℝ) (hf : ∀ x, 0<f x) : |
| ent p f = avg p (fun x => f x*log (f x/avg p f)) := by |
| have he (x : α) : f x*log (f x/avg p f) = f x*log (f x)-f x*log (avg p f) := by |
| rw [log_div (hf x).ne' (avg_pos p hf).ne',mul_sub] |
| simp_rw [he] |
| rw [avg_sub,avg_mul_const] |
| rfl |
|
|
| lemma avg_log_le_log_avg (p : Prior α) (f : α → ℝ) (hf : ∀ x, 0<f x) : |
| avg p (fun x => log (f x)) ≤ log (avg p f) := by |
| have hj := strictConcaveOn_log_Ioi.concaveOn.le_map_sum (t := Finset.univ) |
| (w := p.mass) (p := f) (fun x _ => p.nonneg x) p.sum_one (fun x _ => hf x) |
| exact hj |
|
|
| lemma ent_le_cov_log (p : Prior α) (f : α → ℝ) (hf : ∀ x, 0 < f x) : |
| ent p f ≤ avg p (fun x => f x*log (f x))-avg p f*avg p (fun x => log (f x)) := by |
| have h := mul_le_mul_of_nonneg_left (avg_log_le_log_avg p f hf) (avg_pos p hf).le |
| unfold ent FiniteLaw.entropy |
| change avg p (fun x => f x*log (f x))-avg p f*log (avg p f) ≤ _ |
| linarith only [h] |
|
|
| lemma logSum (p : Prior α) (a b : α → ℝ) (ha : ∀ x,0<a x) (hb : ∀ x,0<b x) : |
| avg p a*log (avg p a/avg p b) ≤ avg p (fun x => a x*log (a x/b x)) := by |
| let q := reweight p b hb |
| have he : avg q (fun x => a x/b x) = avg p a/avg p b := by |
| rw [avg_reweight] |
| congr 1 |
| congr 1 |
| funext x |
| field_simp [(hb x).ne'] |
| have hj := convexOn_mul_log.map_sum_le (t := Finset.univ) (w := q.mass) |
| (p := fun x => a x/b x) (fun x _ => q.nonneg x) q.sum_one |
| (fun x _ => (div_pos (ha x) (hb x)).le) |
| change avg q (fun x => a x/b x)*log (avg q (fun x => a x/b x)) ≤ |
| avg q (fun x => (a x/b x)*log (a x/b x)) at hj |
| rw [he,avg_reweight] at hj |
| have he₂ : (fun x => b x*(a x/b x*log (a x/b x))) = |
| fun x => a x*log (a x/b x) := by |
| funext x |
| field_simp [(hb x).ne'] |
| rw [he₂] at hj |
| have h := mul_le_mul_of_nonneg_right hj (avg_pos p hb).le |
| convert! h using 1 <;> field_simp [(avg_pos p hb).ne'] |
|
|
| theorem entropy_average_le {ι : Type*} [Fintype ι] (p : Prior α) (q : Prior ι) |
| (f : ι → α → ℝ) (hf : ∀ i x, 0<f i x) : |
| ent p (fun x => avg q (fun i => f i x)) ≤ avg q (fun i => ent p (f i)) := by |
| have hswap (F : ι → α → ℝ) : avg p (fun x => avg q (fun i => F i x)) = |
| avg q (fun i => avg p (F i)) := by |
| simp only [avg,FiniteLaw.mean,Finset.mul_sum] |
| rw [Finset.sum_comm] |
| apply Finset.sum_congr rfl |
| intro i _ |
| apply Finset.sum_congr rfl |
| intro x _ |
| ring |
| rw [avg_ent_eq_remainder p _ (fun x => avg_pos q (fun i => hf i x))] |
| rw [hswap f] |
| calc |
| _ ≤ avg p (fun x => avg q (fun i => f i x*log (f i x/avg p (f i)))) := by |
| apply avg_mono p |
| intro x |
| exact logSum q (fun i => f i x) (fun i => avg p (f i)) (fun i => hf i x) |
| (fun i => avg_pos p (hf i)) |
| _ = avg q (fun i => ent p (f i)) := by |
| rw [hswap] |
| congr 1 |
| funext i |
| exact (avg_ent_eq_remainder p (f i) (hf i)).symm |
|
|
| end SKRatio.Observation |
|
|
| end |
| end |
|
|
| end OAI |
|
|
