| import Mathlib |
| import OAI.Probability.SKRatio.FiniteChain.EntropicCovariance |
|
|
| namespace OAI |
|
|
| section |
| noncomputable section |
| open scoped BigOperators ENNReal NNReal |
| open MeasureTheory ProbabilityTheory Real Set Filter |
| namespace SKRatio.Observation.Channel |
| variable {α : Type*} [Fintype α] {d : ℕ} |
|
|
| def reference (d : ℕ) : Measure (Fin d → ℝ) := |
| Measure.pi (fun _ => gaussianReal 0 1) |
| |
| instance reference_probability (d : ℕ) : IsProbabilityMeasure (reference d) := |
| inferInstanceAs (IsProbabilityMeasure (Measure.pi _)) |
| |
| lemma integrable_exp_linear (a : Fin d → ℝ) : |
| Integrable (fun z => exp (∑ i, a i*z i)) (reference d) := by |
| simp_rw [exp_sum] |
| exact Integrable.fintype_prod (fun i => integrable_exp_mul_gaussianReal (μ := 0) (v := 1) (a i)) |
| |
| lemma integral_exp_linear (a : Fin d → ℝ) : |
| (∫ z, exp (∑ i, a i*z i) ∂reference d) = exp ((∑ i, a i^2)/2) := by |
| simp_rw [exp_sum] |
| rw [reference, integral_fintype_prod_eq_prod (fun i (z : ℝ) => exp (a i*z))] |
| have hi (i : Fin d) : (∫ z : ℝ, exp (a i*z) ∂gaussianReal 0 1) = exp (a i^2/2) := by |
| simpa only [mgf,zero_mul,NNReal.coe_one,one_mul,zero_add,id_eq] using |
| mgf_gaussianReal (μ := 0) (v := 1) (X := id) (p := gaussianReal 0 1) (by simp) (a i) |
| simp_rw [hi] |
| rw [← exp_sum,Finset.sum_div] |
| |
| variable (v : α → Fin d → ℝ) |
| |
| def normSq (x : α) : ℝ := ∑ i, v x i^2 |
| |
| def mean (p : Prior α) (i : Fin d) : ℝ := avg p (fun x => v x i) |
| |
| def secondMoment (p : Prior α) : ℝ := avg p (normSq v) |
| |
| def likelihood (t : ℝ) (x : α) (z : Fin d → ℝ) : ℝ := |
| exp (sqrt t*(∑ i, v x i*z i)-t*normSq v x/2) |
| |
| def density (p : Prior α) (t : ℝ) (z : Fin d → ℝ) : ℝ := |
| ∑ x, p x*likelihood v t x z |
| |
| omit [Fintype α] in |
| lemma likelihood_pos (t : ℝ) (x : α) (z : Fin d → ℝ) : |
| 0 < likelihood v t x z := exp_pos _ |
| |
| omit [Fintype α] in |
| lemma integrable_likelihood (t : ℝ) (x : α) : |
| Integrable (likelihood v t x) (reference d) := by |
| have he : likelihood v t x = fun z => exp (∑ i, (sqrt t*v x i)*z i) * |
| exp (-(t*normSq v x/2)) := by |
| funext z |
| simp only [likelihood,exp_sub,exp_neg,← Finset.mul_sum,mul_assoc,div_eq_mul_inv] |
| rw [he] |
| exact (integrable_exp_linear _).mul_const _ |
| |
| omit [Fintype α] in |
| lemma integral_likelihood {t : ℝ} (ht : 0 ≤ t) (x : α) : |
| (∫ z, likelihood v t x z ∂reference d) = 1 := by |
| have he : likelihood v t x = fun z => exp (∑ i, (sqrt t*v x i)*z i) * |
| exp (-(t*normSq v x/2)) := by |
| funext z |
| simp only [likelihood,exp_sub,exp_neg,← Finset.mul_sum,mul_assoc,div_eq_mul_inv] |
| rw [he,integral_mul_const,integral_exp_linear,← exp_add] |
| simp only [mul_pow,sq_sqrt ht,← Finset.mul_sum,normSq] |
| simp |
| |
| lemma density_jensen (p : Prior α) (t : ℝ) (z : Fin d → ℝ) : |
| exp (sqrt t*(∑ i, mean v p i*z i)-t*secondMoment v p/2) ≤ density v p t z := by |
| have he : (∑ x, p x*(sqrt t*(∑ i, v x i*z i)-t*normSq v x/2)) = |
| sqrt t*(∑ i, mean v p i*z i)-t*secondMoment v p/2 := by |
| simp only [mul_sub,Finset.sum_sub_distrib,mean,secondMoment,avg,FiniteLaw.mean] |
| congr 1 |
| · simp only [Finset.mul_sum,Finset.sum_mul] |
| rw [Finset.sum_comm] |
| exact Finset.sum_congr rfl (fun i _ => Finset.sum_congr rfl (fun x _ => by ring)) |
| · simp only [Finset.mul_sum,Finset.sum_div] |
| exact Finset.sum_congr rfl (fun x _ => by ring) |
| have hj := convexOn_exp.map_sum_le (t := Finset.univ) (w := p.mass) |
| (p := fun x => sqrt t*(∑ i, v x i*z i)-t*normSq v x/2) |
| (fun x _ => p.nonneg x) p.sum_one (fun _ _ => Set.mem_univ _) |
| simpa only [smul_eq_mul,he,density,likelihood] using hj |
| |
| lemma density_pos (p : Prior α) (t : ℝ) (z : Fin d → ℝ) : |
| 0 < density v p t z := (exp_pos _).trans_le (density_jensen v p t z) |
| |
| lemma integrable_density (p : Prior α) (t : ℝ) : |
| Integrable (density v p t) (reference d) := |
| integrable_finsetSum _ (fun x _ => (integrable_likelihood v t x).const_mul (p x)) |
| |
| lemma integral_density (p : Prior α) {t : ℝ} (ht : 0 ≤ t) : |
| (∫ z, density v p t z ∂reference d) = 1 := by |
| unfold density |
| rw [integral_finsetSum _ (fun x _ => (integrable_likelihood v t x).const_mul (p x))] |
| simp_rw [integral_const_mul,integral_likelihood v ht,mul_one] |
| exact p.sum_one |
| |
| def interaction (p : Prior α) (x y : α) : ℝ := |
| ((∑ i, (v x i+v y i-mean v p i)^2)-normSq v x-normSq v y+secondMoment v p)/2 |
| |
| lemma interaction_sum (p : Prior α) (x y : α) : |
| interaction v p x y = ∑ i, (v x i*v y i-v x i*mean v p i-v y i*mean v p i+ |
| (mean v p i)^2/2+avg p (fun z => v z i^2)/2) := by |
| have he : secondMoment v p = ∑ i, avg p (fun z => v z i^2) := by |
| simp only [secondMoment,avg,FiniteLaw.mean,normSq,Finset.mul_sum] |
| exact Finset.sum_comm |
| rw [interaction,normSq,normSq,he,← Finset.sum_sub_distrib,← Finset.sum_sub_distrib, |
| ← Finset.sum_add_distrib,Finset.sum_div] |
| exact Finset.sum_congr rfl (fun i _ => by ring) |
| |
| omit v in |
| lemma signed_sum_left (a u : α → ℝ) (ha : ∑ x, a x = 0) : |
| (∑ x, ∑ y, a x*a y*u x) = 0 := by |
| apply Finset.sum_eq_zero |
| intro x _ |
| calc |
| _ = (a x*u x)*∑ y, a y := by |
| rw [Finset.mul_sum] |
| exact Finset.sum_congr rfl (fun y _ => by ring) |
| _ = 0 := by rw [ha,mul_zero] |
| |
| omit v in |
| lemma signed_sum_right (a u : α → ℝ) (ha : ∑ x, a x = 0) : |
| (∑ x, ∑ y, a x*a y*u y) = 0 := by |
| rw [Finset.sum_comm] |
| convert signed_sum_left a u ha using 1 |
| exact Finset.sum_congr rfl (fun x _ => Finset.sum_congr rfl (fun y _ => by ring)) |
| |
| omit v in |
| lemma signed_sum_product (a u : α → ℝ) : |
| (∑ x, ∑ y, a x*a y*(u x*u y)) = (∑ x, a x*u x)^2 := by |
| rw [pow_two,Finset.sum_mul_sum] |
| exact Finset.sum_congr rfl (fun x _ => Finset.sum_congr rfl (fun y _ => by ring)) |
| |
| lemma signed_interaction (p : Prior α) (a : α → ℝ) (ha : ∑ x, a x = 0) : |
| (∑ x, ∑ y, a x*a y*interaction v p x y) = ∑ i, (∑ x, a x*v x i)^2 := by |
| simp_rw [interaction_sum,Finset.mul_sum] |
| calc |
| _ = ∑ x, ∑ i, ∑ y, a x*a y*(v x i*v y i-v x i*mean v p i-v y i*mean v p i+ |
| (mean v p i)^2/2+avg p (fun z => v z i^2)/2) := by |
| exact Finset.sum_congr rfl (fun x _ => Finset.sum_comm) |
| _ = ∑ i, ∑ x, ∑ y, a x*a y*(v x i*v y i-v x i*mean v p i-v y i*mean v p i+ |
| (mean v p i)^2/2+avg p (fun z => v z i^2)/2) := Finset.sum_comm |
| _ = _ := by |
| apply Finset.sum_congr rfl |
| intro i _ |
| simp only [mul_add,mul_sub,Finset.sum_add_distrib,Finset.sum_sub_distrib] |
| rw [signed_sum_product a (fun x => v x i), |
| signed_sum_left a (fun x => v x i*mean v p i) ha, |
| signed_sum_right a (fun x => v x i*mean v p i) ha, |
| signed_sum_left a (fun _ => (mean v p i)^2/2) ha, |
| signed_sum_left a (fun _ => avg p (fun z => v z i^2)/2) ha] |
| ring |
| |
| lemma interaction_remainder_bound (p : Prior α) {K t : ℝ} (hK : 0 ≤ K) (ht : 0 ≤ t) |
| (hbound : ∀ x y, |interaction v p x y| ≤ K) (hKt : K*t ≤ 1) (x y : α) : |
| |exp (t*interaction v p x y)-1-t*interaction v p x y| ≤ K^2*t^2 := by |
| have htb : |t*interaction v p x y| ≤ K*t := by |
| rw [abs_mul,abs_of_nonneg ht,mul_comm K] |
| exact mul_le_mul_of_nonneg_left (hbound x y) ht |
| apply (abs_exp_sub_one_sub_id_le (htb.trans hKt)).trans |
| have hs := (sq_le_sq₀ (abs_nonneg (t*interaction v p x y)) (mul_nonneg hK ht)).mpr htb |
| simpa only [sq_abs,mul_pow] using hs |
| |
| def pairShift (p : Prior α) (x y : α) : ℝ := |
| (normSq v x+normSq v y-secondMoment v p)/2 |
| |
| def pairKernel (p : Prior α) (t : ℝ) (x y : α) (z : Fin d → ℝ) : ℝ := |
| exp ((∑ i, (sqrt t*(v x i+v y i-mean v p i))*z i)-t*pairShift v p x y) |
| |
| lemma integrable_pairKernel (p : Prior α) (t : ℝ) (x y : α) : |
| Integrable (pairKernel v p t x y) (reference d) := by |
| have he : pairKernel v p t x y = fun z => |
| exp (∑ i, (sqrt t*(v x i+v y i-mean v p i))*z i)*exp (-(t*pairShift v p x y)) := by |
| funext z; simp [pairKernel,exp_sub,exp_neg,div_eq_mul_inv] |
| rw [he] |
| exact (integrable_exp_linear _).mul_const _ |
| |
| lemma integral_pairKernel (p : Prior α) {t : ℝ} (ht : 0 ≤ t) (x y : α) : |
| (∫ z, pairKernel v p t x y z ∂reference d) = exp (t*interaction v p x y) := by |
| have he : pairKernel v p t x y = fun z => |
| exp (∑ i, (sqrt t*(v x i+v y i-mean v p i))*z i)*exp (-(t*pairShift v p x y)) := by |
| funext z; simp [pairKernel,exp_sub,exp_neg,div_eq_mul_inv] |
| rw [he,integral_mul_const,integral_exp_linear,← exp_add] |
| congr 1 |
| simp only [mul_pow,sq_sqrt ht,← Finset.mul_sum,interaction,pairShift] |
| ring |
| |
| lemma likelihood_pair (p : Prior α) (t : ℝ) (x y : α) (z : Fin d → ℝ) : |
| likelihood v t x z * likelihood v t y z * |
| exp (-(sqrt t * (∑ i, mean v p i * z i) - t*secondMoment v p/2)) = pairKernel v p t x y z := by |
| unfold likelihood pairKernel pairShift |
| rw [← exp_add, ← exp_add] |
| congr 1 |
| simp only [add_mul, sub_mul, mul_add, mul_sub, Finset.mul_sum, Finset.sum_add_distrib, |
| Finset.sum_sub_distrib, mul_assoc] |
| ring |
| |
| def majorant (p : Prior α) (a : α → ℝ) (t : ℝ) (z : Fin d → ℝ) : ℝ := |
| (∑ x, a x*likelihood v t x z)^2 * exp (-(sqrt t * (∑ i, mean v p i * z i) - t*secondMoment v p/2)) |
| |
| lemma majorant_eq_sum (p : Prior α) (a : α → ℝ) (t : ℝ) (z : Fin d → ℝ) : |
| majorant v p a t z = ∑ x, ∑ y, (a x*a y)*pairKernel v p t x y z := by |
| unfold majorant |
| rw [pow_two, Finset.sum_mul_sum] |
| simp only [Finset.sum_mul] |
| apply Finset.sum_congr rfl |
| intro x _ |
| apply Finset.sum_congr rfl |
| intro y _ |
| calc |
| _ = (a x*a y)*(likelihood v t x z*likelihood v t y z* |
| exp (-(sqrt t * (∑ i, mean v p i * z i) - t*secondMoment v p/2))) := by ring |
| _ = _ := by rw [likelihood_pair] |
| |
| lemma majorant_nonneg (p : Prior α) (a : α → ℝ) (t : ℝ) (z : Fin d → ℝ) : |
| 0 ≤ majorant v p a t z := mul_nonneg (sq_nonneg _) (exp_pos _).le |
| |
| lemma integrable_majorant (p : Prior α) (a : α → ℝ) (t : ℝ) : |
| Integrable (majorant v p a t) (reference d) := by |
| simp_rw [show majorant v p a t = fun z => ∑ x, ∑ y, (a x*a y)*pairKernel v p t x y z from |
| funext (majorant_eq_sum v p a t)] |
| exact integrable_finsetSum _ (fun x _ => integrable_finsetSum _ (fun y _ => |
| (integrable_pairKernel v p t x y).const_mul _)) |
| |
| lemma integral_majorant (p : Prior α) (a : α → ℝ) {t : ℝ} (ht : 0 ≤ t) : |
| (∫ z, majorant v p a t z ∂reference d) = ∑ x, ∑ y, (a x*a y)*exp (t*interaction v p x y) := by |
| simp_rw [majorant_eq_sum] |
| rw [integral_finsetSum _ (fun x _ => integrable_finsetSum _ (fun y _ => |
| (integrable_pairKernel v p t x y).const_mul _))] |
| apply Finset.sum_congr rfl |
| intro x _ |
| rw [integral_finsetSum _ (fun y _ => (integrable_pairKernel v p t x y).const_mul _)] |
| simp_rw [integral_const_mul, integral_pairKernel v p ht] |
| |
| lemma interaction_exp_sum_bound (p : Prior α) (a : α → ℝ) |
| (ha : ∑ x, a x = 0) {K t : ℝ} (ht : 0 ≤ t) (hK : 0 ≤ K) (hbound : ∀ x y, |interaction v p x y| ≤ K) (hKt : K*t ≤ 1) : |
| (∑ x, ∑ y, a x*a y*exp (t*interaction v p x y)) ≤ |
| t*(∑ i, (∑ x, a x*v x i)^2) + |
| K^2*t^2*(∑ x, |a x|)^2 := by |
| calc |
| _ ≤ ∑ x, ∑ y, ((a x*a y + t*(a x*a y*interaction v p x y)) + |
| (K^2*t^2)*(|a x| * |a y|)) := by |
| apply Finset.sum_le_sum |
| intro x _ |
| apply Finset.sum_le_sum |
| intro y _ |
| have hr := interaction_remainder_bound v p hK ht hbound hKt x y |
| have hb : a x*a y*(exp (t*interaction v p x y)-1-t*interaction v p x y) ≤ |
| |a x| * |a y| * (K^2*t^2) := by |
| calc |
| _ ≤ |a x*a y*(exp (t*interaction v p x y)-1-t*interaction v p x y)| := le_abs_self _ |
| _ = |a x| * |a y| * |exp (t*interaction v p x y)-1-t*interaction v p x y| := by simp [abs_mul] |
| _ ≤ _ := mul_le_mul_of_nonneg_left hr (mul_nonneg (abs_nonneg _) (abs_nonneg _)) |
| nlinarith only [hb] |
| _ = _ := by |
| simp only [Finset.sum_add_distrib, ← Finset.mul_sum] |
| rw [ha, signed_interaction v p a ha] |
| simp only [mul_zero, Finset.sum_const_zero, zero_add, ← Finset.sum_mul] |
| ring |
| |
| def quadraticIntegrand (p : Prior α) (a : α → ℝ) (t : ℝ) (z : Fin d → ℝ) : ℝ := |
| (∑ x, a x*likelihood v t x z)^2 / density v p t z |
| |
| lemma quadraticIntegrand_nonneg (p : Prior α) (a : α → ℝ) (t : ℝ) (z : Fin d → ℝ) : |
| 0 ≤ quadraticIntegrand v p a t z := div_nonneg (sq_nonneg _) (density_pos v p t z).le |
| |
| lemma quadraticIntegrand_le_majorant (p : Prior α) (a : α → ℝ) (t : ℝ) (z : Fin d → ℝ) : |
| quadraticIntegrand v p a t z ≤ majorant v p a t z := by |
| unfold quadraticIntegrand |
| calc |
| _ ≤ (∑ x, a x*likelihood v t x z)^2 / |
| exp (sqrt t * (∑ i, mean v p i*z i) - t*secondMoment v p/2) := |
| div_le_div_of_nonneg_left (sq_nonneg _) (exp_pos _) (density_jensen v p t z) |
| _ = _ := by simp only [majorant, div_eq_mul_inv, exp_neg] |
| |
| omit [Fintype α] in |
| lemma continuous_likelihood (t : ℝ) (x : α) : Continuous (likelihood v t x) := by |
| unfold likelihood |
| fun_prop |
| |
| lemma continuous_density (p : Prior α) (t : ℝ) : Continuous (density v p t) := by |
| unfold density |
| exact continuous_finsetSum _ (fun x _ => (continuous_likelihood v t x).const_mul _) |
| |
| lemma continuous_quadraticIntegrand (p : Prior α) (a : α → ℝ) (t : ℝ) : |
| Continuous (quadraticIntegrand v p a t) := by |
| apply Continuous.div |
| ((continuous_finsetSum _ (fun x _ => (continuous_likelihood v t x).const_mul (a x))).pow 2) |
| (continuous_density v p t) |
| intro z |
| exact (density_pos v p t z).ne' |
| |
| lemma integrable_quadraticIntegrand (p : Prior α) (a : α → ℝ) (t : ℝ) : |
| Integrable (quadraticIntegrand v p a t) (reference d) := by |
| apply (integrable_majorant v p a t).mono' (continuous_quadraticIntegrand v p a t).aestronglyMeasurable |
| filter_upwards [] with z |
| rw [Real.norm_eq_abs, abs_of_nonneg (quadraticIntegrand_nonneg v p a t z)] |
| exact quadraticIntegrand_le_majorant v p a t z |
| |
| theorem quadratic_bound (p : Prior α) (a : α → ℝ) (ha : ∑ x, a x = 0) |
| {K t : ℝ} (ht : 0 ≤ t) (hK : 0 ≤ K) (hbound : ∀ x y, |interaction v p x y| ≤ K) (hKt : K*t ≤ 1) : |
| (∫ z, quadraticIntegrand v p a t z ∂reference d) ≤ |
| t*(∑ i, (∑ x, a x*v x i)^2) + |
| K^2*t^2*(∑ x, |a x|)^2 := by |
| calc |
| _ ≤ ∫ z, majorant v p a t z ∂reference d := |
| integral_mono (integrable_quadraticIntegrand v p a t) (integrable_majorant v p a t) |
| (quadraticIntegrand_le_majorant v p a t) |
| _ = _ := integral_majorant v p a ht |
| _ ≤ _ := interaction_exp_sum_bound v p a ha ht hK hbound hKt |
| |
| def posterior (p : Prior α) (t : ℝ) (z : Fin d → ℝ) : Prior α where |
| mass x := p x * likelihood v t x z / density v p t z |
| nonneg x := div_nonneg (mul_nonneg (p.nonneg x) (likelihood_pos v t x z).le) |
| (density_pos v p t z).le |
| sum_one := by |
| rw [← Finset.sum_div] |
| exact div_self (density_pos v p t z).ne' |
|
|
| lemma posterior_avg_weight (p : Prior α) (t : ℝ) (z : Fin d → ℝ) (f : α → ℝ) : |
| density v p t z * avg (posterior v p t z) f = ∑ x, (p x*f x)*likelihood v t x z := by |
| unfold avg FiniteLaw.mean posterior |
| rw [Finset.mul_sum] |
| apply Finset.sum_congr rfl |
| intro x _ |
| dsimp |
| field_simp [(density_pos v p t z).ne'] |
|
|
| lemma integrable_posterior_avg_weight (p : Prior α) (t : ℝ) (f : α → ℝ) : |
| Integrable (fun z => density v p t z * avg (posterior v p t z) f) (reference d) := by |
| simp_rw [posterior_avg_weight] |
| exact integrable_finsetSum _ (fun x _ => (integrable_likelihood v t x).const_mul _) |
|
|
| lemma integral_posterior_avg_weight (p : Prior α) {t : ℝ} (ht : 0 ≤ t) (f : α → ℝ) : |
| (∫ z, density v p t z * avg (posterior v p t z) f ∂reference d) = avg p f := by |
| simp_rw [posterior_avg_weight] |
| rw [integral_finsetSum _ (fun x _ => (integrable_likelihood v t x).const_mul _)] |
| simp_rw [integral_const_mul, integral_likelihood v ht, mul_one] |
| rfl |
|
|
| lemma scalar_entropy_nonneg {r : ℝ} (hr : 0 < r) : 0 ≤ r*log r-r+1 := by |
| have h := mul_le_mul_of_nonneg_left (one_sub_inv_le_log_of_pos hr) hr.le |
| have hi : r*r⁻¹ = 1 := mul_inv_cancel₀ hr.ne' |
| nlinarith |
|
|
| lemma scalar_entropy_le_square {r : ℝ} (hr : 0 < r) : r*log r-r+1 ≤ (r-1)^2 := by |
| have h := mul_le_mul_of_nonneg_left (log_le_sub_one_of_pos hr) hr.le |
| nlinarith |
|
|
| def divergenceIntegrand (p q : Prior α) (t : ℝ) (z : Fin d → ℝ) : ℝ := |
| density v q t z * log (density v q t z/density v p t z) - density v q t z + density v p t z |
|
|
| lemma divergenceIntegrand_eq (p q : Prior α) (t : ℝ) (z : Fin d → ℝ) : |
| divergenceIntegrand v p q t z = density v p t z * |
| ((density v q t z/density v p t z)*log (density v q t z/density v p t z)- |
| density v q t z/density v p t z+1) := by |
| unfold divergenceIntegrand |
| field_simp [(density_pos v p t z).ne'] |
|
|
| lemma divergenceIntegrand_nonneg (p q : Prior α) (t : ℝ) (z : Fin d → ℝ) : |
| 0 ≤ divergenceIntegrand v p q t z := by |
| rw [divergenceIntegrand_eq] |
| exact mul_nonneg (density_pos v p t z).le |
| (scalar_entropy_nonneg (div_pos (density_pos v q t z) (density_pos v p t z))) |
|
|
| lemma density_sub (p q : Prior α) (t : ℝ) (z : Fin d → ℝ) : |
| (∑ x, (q x-p x)*likelihood v t x z) = density v q t z-density v p t z := by |
| simp only [sub_mul, Finset.sum_sub_distrib, density] |
|
|
| lemma divergenceIntegrand_le_quadratic (p q : Prior α) (t : ℝ) (z : Fin d → ℝ) : |
| divergenceIntegrand v p q t z ≤ quadraticIntegrand v p (fun x => q x-p x) t z := by |
| rw [divergenceIntegrand_eq, quadraticIntegrand, density_sub] |
| calc |
| _ ≤ density v p t z*(density v q t z/density v p t z-1)^2 := |
| mul_le_mul_of_nonneg_left |
| (scalar_entropy_le_square (div_pos (density_pos v q t z) (density_pos v p t z))) |
| (density_pos v p t z).le |
| _ = _ := by field_simp [(density_pos v p t z).ne'] |
|
|
| lemma continuous_divergenceIntegrand (p q : Prior α) (t : ℝ) : |
| Continuous (divergenceIntegrand v p q t) := by |
| have hd := (continuous_density v q t).div (continuous_density v p t) |
| (fun z => (density_pos v p t z).ne') |
| exact (((continuous_density v q t).mul (hd.log (fun z => |
| (div_pos (density_pos v q t z) (density_pos v p t z)).ne'))).sub |
| (continuous_density v q t)).add (continuous_density v p t) |
|
|
| lemma integrable_divergenceIntegrand (p q : Prior α) (t : ℝ) : |
| Integrable (divergenceIntegrand v p q t) (reference d) := by |
| apply (integrable_quadraticIntegrand v p (fun x => q x-p x) t).mono' |
| (continuous_divergenceIntegrand v p q t).aestronglyMeasurable |
| filter_upwards [] with z |
| rw [Real.norm_eq_abs, abs_of_nonneg (divergenceIntegrand_nonneg v p q t z)] |
| exact divergenceIntegrand_le_quadratic v p q t z |
|
|
| lemma prior_sub_l1 (p q : Prior α) : (∑ x, |q x-p x|) ≤ 2 := by |
| calc |
| _ ≤ ∑ x, (q x+p x) := Finset.sum_le_sum (fun x _ => by |
| simpa only [abs_of_nonneg (q.nonneg x), abs_of_nonneg (p.nonneg x)] using abs_sub (q x) (p x)) |
| _ = 2 := by rw [Finset.sum_add_distrib, q.sum_one, p.sum_one]; norm_num |
|
|
| lemma prior_sub_sum (p q : Prior α) : (∑ x, (q x-p x)) = 0 := by |
| rw [Finset.sum_sub_distrib, q.sum_one, p.sum_one, sub_self] |
|
|
| theorem entropy_step_bound (p q : Prior α) {K t : ℝ} (ht : 0 ≤ t) |
| (hK : 0 ≤ K) (hbound : ∀ x y, |interaction v p x y| ≤ K) (hKt : K*t ≤ 1) : |
| (∫ z, divergenceIntegrand v p q t z ∂reference d) ≤ |
| t*(∑ i, (mean v q i-mean v p i)^2)+4*K^2*t^2 := by |
| have hmean : (∑ i, (∑ x, (q x-p x)*v x i)^2) = |
| ∑ i, (mean v q i-mean v p i)^2 := by |
| simp only [sub_mul, Finset.sum_sub_distrib, mean, avg, FiniteLaw.mean] |
| have he : (∑ x, |q x-p x|)^2 ≤ 4 := by |
| have hl : 0 ≤ ∑ x, |q x-p x| := Finset.sum_nonneg (fun _ _ => abs_nonneg _) |
| nlinarith [prior_sub_l1 p q] |
| calc |
| _ ≤ ∫ z, quadraticIntegrand v p (fun x => q x-p x) t z ∂reference d := |
| integral_mono (integrable_divergenceIntegrand v p q t) (integrable_quadraticIntegrand v p _ t) |
| (divergenceIntegrand_le_quadratic v p q t) |
| _ ≤ t*(∑ i, (mean v q i-mean v p i)^2)+K^2*t^2*(∑ x, |q x-p x|)^2 := by |
| simpa only [hmean] using quadratic_bound v p (fun x => q x-p x) (prior_sub_sum p q) ht hK hbound hKt |
| _ ≤ _ := by nlinarith [mul_le_mul_of_nonneg_left he (show 0 ≤ K^2*t^2 by positivity)] |
|
|
| lemma integrable_relativeEntropy (p q : Prior α) (t : ℝ) : |
| Integrable (fun z => density v q t z*log (density v q t z/density v p t z)) (reference d) := by |
| have he : (fun z => density v q t z*log (density v q t z/density v p t z)) = |
| fun z => divergenceIntegrand v p q t z+density v q t z-density v p t z := by |
| funext z |
| unfold divergenceIntegrand |
| ring |
| rw [he] |
| exact ((integrable_divergenceIntegrand v p q t).add (integrable_density v q t)).sub |
| (integrable_density v p t) |
|
|
| lemma integral_relativeEntropy (p q : Prior α) {t : ℝ} (ht : 0 ≤ t) : |
| (∫ z, density v q t z*log (density v q t z/density v p t z) ∂reference d) = |
| ∫ z, divergenceIntegrand v p q t z ∂reference d := by |
| have hi := integral_add |
| ((integrable_relativeEntropy v p q t).sub (integrable_density v q t)) (integrable_density v p t) |
| have hj := integral_sub (integrable_relativeEntropy v p q t) (integrable_density v q t) |
| simp only [Pi.sub_apply] at hi hj |
| rw [hj, integral_density v q ht, integral_density v p ht] at hi |
| change (∫ z, divergenceIntegrand v p q t z ∂reference d) = _ at hi |
| linarith |
|
|
| lemma mean_abs_le (p : Prior α) {B : ℝ} (hB : ∀ x i, |v x i| ≤ B) (i : Fin d) : |
| |mean v p i| ≤ B := by |
| calc |
| _ ≤ ∑ x, |p x*v x i| := Finset.abs_sum_le_sum_abs _ _ |
| _ ≤ ∑ x, p x*B := Finset.sum_le_sum (fun x _ => by |
| rw [abs_mul,abs_of_nonneg (p.nonneg x)] |
| exact mul_le_mul_of_nonneg_left (hB x i) (p.nonneg x)) |
| _ = B := by rw [← Finset.sum_mul,p.sum_one,one_mul] |
|
|
| omit [Fintype α] in |
| lemma normSq_nonneg (x : α) : 0 ≤ normSq v x := Finset.sum_nonneg (fun _ _ => sq_nonneg _) |
|
|
| omit [Fintype α] in |
| lemma normSq_le {B : ℝ} (hB0 : 0 ≤ B) (hB : ∀ x i, |v x i| ≤ B) (x : α) : |
| normSq v x ≤ d*B^2 := by |
| calc |
| _ ≤ ∑ _i : Fin d, B^2 := Finset.sum_le_sum (fun i _ => by |
| simpa only [sq_abs] using (sq_le_sq₀ (abs_nonneg _) hB0).mpr (hB x i)) |
| _ = _ := by simp |
|
|
| lemma interaction_abs_le (p : Prior α) {B : ℝ} (hB0 : 0 ≤ B) |
| (hB : ∀ x i, |v x i| ≤ B) (x y : α) : |
| |interaction v p x y| ≤ 5*d*B^2 := by |
| have hmean (i : Fin d) := mean_abs_le v p hB i |
| have hsec : 0 ≤ secondMoment v p := avg_nonneg p (normSq_nonneg v) |
| have hsecb : secondMoment v p ≤ d*B^2 := by |
| exact (avg_mono p (normSq_le v hB0 hB)).trans_eq (avg_const p _) |
| have hs0 : 0 ≤ ∑ i, (v x i+v y i-mean v p i)^2 := |
| Finset.sum_nonneg (fun _ _ => sq_nonneg _) |
| have hsum : (∑ i, (v x i+v y i-mean v p i)^2) ≤ 9*d*B^2 := by |
| calc |
| _ ≤ ∑ _i : Fin d, (3*B)^2 := by |
| apply Finset.sum_le_sum |
| intro i _ |
| have hh : |v x i+v y i-mean v p i| ≤ 3*B := by |
| have hh := (abs_sub (v x i+v y i) (mean v p i)).trans |
| (add_le_add (abs_add_le (v x i) (v y i)) le_rfl) |
| linarith [hB x i,hB y i,hmean i] |
| simpa only [sq_abs] using (sq_le_sq₀ (abs_nonneg _) (by positivity)).mpr hh |
| _ = _ := by simp;ring |
| rw [abs_le] |
| unfold interaction |
| constructor <;> nlinarith [normSq_nonneg v x,normSq_nonneg v y, |
| normSq_le v hB0 hB x,normSq_le v hB0 hB y, |
| mul_nonneg (Nat.cast_nonneg (α := ℝ) d) (sq_nonneg B)] |
|
|
| theorem exists_interaction_bound : ∃ K : ℝ, 0 ≤ K ∧ |
| ∀ p : Prior α, ∀ x y, |interaction v p x y| ≤ K := by |
| let B := ∑ x, ∑ i, |v x i| |
| have hB0 : 0 ≤ B := Finset.sum_nonneg (fun _ _ => Finset.sum_nonneg (fun _ _ => abs_nonneg _)) |
| have hB (x : α) (i : Fin d) : |v x i| ≤ B := by |
| exact (Finset.single_le_sum (f := fun j => |v x j|) |
| (fun _ _ => abs_nonneg _) (Finset.mem_univ i)).trans |
| (Finset.single_le_sum (f := fun y => ∑ j, |v y j|) |
| (fun _ _ => Finset.sum_nonneg (fun _ _ => abs_nonneg _)) (Finset.mem_univ x)) |
| exact ⟨5*d*B^2,by positivity,fun p x y => interaction_abs_le v p hB0 hB x y⟩ |
|
|
| lemma reweight_density (p : Prior α) (f : α → ℝ) (hf : ∀ x, 0 < f x) |
| (t : ℝ) (z : Fin d → ℝ) : |
| density v (reweight p f hf) t z = |
| density v p t z*avg (posterior v p t z) f/avg p f := by |
| rw [posterior_avg_weight] |
| unfold density reweight |
| dsimp |
| rw [Finset.sum_div] |
| exact Finset.sum_congr rfl (fun x _ => by ring) |
|
|
| lemma posterior_avg_eq_density_ratio (p : Prior α) (f : α → ℝ) (hf : ∀ x, 0 < f x) |
| (t : ℝ) (z : Fin d → ℝ) : |
| avg (posterior v p t z) f = avg p f*(density v (reweight p f hf) t z/density v p t z) := by |
| rw [reweight_density] |
| field_simp [(avg_pos p hf).ne',(density_pos v p t z).ne'] |
|
|
| lemma posterior_entropy_identity (p : Prior α) (f : α → ℝ) (hf : ∀ x, 0 < f x) |
| (t : ℝ) (z : Fin d → ℝ) : |
| density v p t z*ent (posterior v p t z) f = |
| density v p t z*avg (posterior v p t z) (fun x => f x*log (f x)) - |
| avg p f*(density v (reweight p f hf) t z*log (density v (reweight p f hf) t z/density v p t z)) - |
| log (avg p f)*(density v p t z*avg (posterior v p t z) f) := by |
| change density v p t z*(avg (posterior v p t z) (fun x => f x*log (f x))- |
| avg (posterior v p t z) f*log (avg (posterior v p t z) f)) = _ |
| rw [posterior_avg_eq_density_ratio v p f hf,log_mul (avg_pos p hf).ne' |
| (div_pos (density_pos v (reweight p f hf) t z) (density_pos v p t z)).ne'] |
| field_simp [(density_pos v p t z).ne'] |
| ring |
|
|
| lemma integrable_posterior_entropy_weight (p : Prior α) (f : α → ℝ) (hf : ∀ x, 0 < f x) (t : ℝ) : |
| Integrable (fun z => density v p t z*ent (posterior v p t z) f) (reference d) := by |
| simp_rw [posterior_entropy_identity v p f hf] |
| exact ((integrable_posterior_avg_weight v p t _).sub |
| ((integrable_relativeEntropy v p (reweight p f hf) t).const_mul _)).sub |
| ((integrable_posterior_avg_weight v p t f).const_mul _) |
|
|
| lemma integral_posterior_entropy (p : Prior α) (f : α → ℝ) (hf : ∀ x, 0 < f x) |
| {t : ℝ} (ht : 0 ≤ t) : |
| (∫ z, density v p t z*ent (posterior v p t z) f ∂reference d) = |
| ent p f-avg p f*(∫ z, divergenceIntegrand v p (reweight p f hf) t z ∂reference d) := by |
| simp_rw [posterior_entropy_identity v p f hf] |
| have h₁ := integrable_posterior_avg_weight v p t (fun x => f x*log (f x)) |
| have h₂ := (integrable_relativeEntropy v p (reweight p f hf) t).const_mul (avg p f) |
| have h₃ := (integrable_posterior_avg_weight v p t f).const_mul (log (avg p f)) |
| have h₄ := integral_sub (h₁.sub h₂) h₃ |
| have h₅ := integral_sub h₁ h₂ |
| simp only [Pi.sub_apply] at h₄ h₅ |
| rw [h₄,h₅,integral_const_mul,integral_const_mul, |
| integral_posterior_avg_weight v p ht,integral_posterior_avg_weight v p ht, |
| integral_relativeEntropy v p (reweight p f hf) ht] |
| unfold ent FiniteLaw.entropy avg |
| ring |
|
|
| theorem entropy_retention (p : Prior α) (f : α → ℝ) (hf : ∀ x, 0 < f x) |
| (hp : ∀ x, 0 < p x) {C K t : ℝ} (hC : 0 < C) (ht : 0 ≤ t) |
| (hK : 0 ≤ K) (hbound : ∀ x y, |interaction v p x y| ≤ K) (hKt : K*t ≤ 1) |
| (hvar : ∀ a : Fin d → ℝ, ∀ s : ℝ, |
| var (tilt p (linear v a) s) (linear v a) ≤ C*(∑ i, a i^2)) : |
| (1-2*C*t)*ent p f-4*K^2*t^2*avg p f ≤ |
| ∫ z, density v p t z*ent (posterior v p t z) f ∂reference d := by |
| have h := entropy_step_bound v p (reweight p f hf) ht hK hbound hKt |
| have hd := vector_mean_displacement_sq p (reweight p f hf) v hp hC hvar |
| rw [relativeEntropy_reweight p f hf hp] at hd |
| change (∑ i, (mean v (reweight p f hf) i-mean v p i)^2) ≤ _ at hd |
| have he := mul_le_mul_of_nonneg_left hd ht |
| have hh := mul_le_mul_of_nonneg_left |
| (h.trans (add_le_add he le_rfl)) (avg_pos p hf).le |
| rw [integral_posterior_entropy v p f hf ht] |
| have heq : avg p f*(t*(2*C*(ent p f/avg p f))+4*K^2*t^2) = |
| 2*C*t*ent p f+4*K^2*t^2*avg p f := by |
| field_simp [(avg_pos p hf).ne'] |
| rw [heq] at hh |
| nlinarith only [hh] |
|
|
| end SKRatio.Observation.Channel |
|
|
| end |
| end |
|
|
| end OAI |
|
|
