| import OAI.Probability.FreeIsing.Cavity.LogOdds | |
| namespace OAI | |
| noncomputable section | |
| namespace Problem367 | |
| theorem binary_mean_eq_tanh_half_log {a b : ℝ} (ha : 0 < a) (hb : 0 < b) : | |
| (a - b) / (a + b) = Real.tanh ((Real.log a - Real.log b) / 2) := by | |
| have hab : 0 < a + b := add_pos ha hb | |
| have hmem : (a - b) / (a + b) ∈ Set.Ioo (-1 : ℝ) 1 := by | |
| constructor | |
| · apply (lt_div_iff₀ hab).2 | |
| linarith | |
| · apply (div_lt_iff₀ hab).2 | |
| linarith | |
| have hratio : (1 + (a - b) / (a + b)) / (1 - (a - b) / (a + b)) = a / b := by | |
| apply (div_eq_div_iff (by linarith [hmem.2]) hb.ne').2 | |
| field_simp | |
| ring | |
| have hfield : Real.artanh ((a - b) / (a + b)) = (Real.log a - Real.log b) / 2 := by | |
| rw [Real.artanh_eq_half_log ⟨hmem.1.le, hmem.2.le⟩, hratio, | |
| Real.log_div ha.ne' hb.ne'] | |
| ring | |
| rw [← hfield, Real.tanh_artanh hmem] | |
| def finiteBranchWeight {I : Type*} (D : Finset I) (beta x : ℝ) (z : I → ℝ) (s : ℝ) : ℝ := | |
| Real.exp (x * s) * ∏ i ∈ D, Real.cosh (z i + beta * s) | |
| theorem finiteBranchWeight_pos {I : Type*} (D : Finset I) (beta x : ℝ) | |
| (z : I → ℝ) (s : ℝ) : 0 < finiteBranchWeight D beta x z s := by | |
| exact mul_pos (Real.exp_pos _) (Finset.prod_pos fun i _ => Real.cosh_pos _) | |
| theorem log_finiteBranchWeight {I : Type*} (D : Finset I) (beta x : ℝ) | |
| (z : I → ℝ) (s : ℝ) : | |
| Real.log (finiteBranchWeight D beta x z s) = | |
| x * s + ∑ i ∈ D, Real.log (Real.cosh (z i + beta * s)) := by | |
| rw [finiteBranchWeight, Real.log_mul (Real.exp_pos _).ne' | |
| (Finset.prod_ne_zero_iff.mpr fun i _ => (Real.cosh_pos _).ne'), Real.log_exp, | |
| Real.log_prod (fun i _ => (Real.cosh_pos _).ne')] | |
| theorem finiteBranchWeight_half_log_odds {I : Type*} (D : Finset I) (beta x : ℝ) | |
| (z : I → ℝ) : | |
| (Real.log (finiteBranchWeight D beta x z 1) - | |
| Real.log (finiteBranchWeight D beta x z (-1))) / 2 = | |
| x + ∑ i ∈ D, cavityMessage (Real.tanh beta) (z i) := by | |
| rw [log_finiteBranchWeight, log_finiteBranchWeight] | |
| simp only [mul_one, mul_neg_one, ← sub_eq_add_neg] | |
| have hsum : (∑ i ∈ D, Real.log (Real.cosh (z i + beta))) - | |
| (∑ i ∈ D, Real.log (Real.cosh (z i - beta))) = | |
| 2 * ∑ i ∈ D, cavityMessage (Real.tanh beta) (z i) := by | |
| rw [← Finset.sum_sub_distrib, Finset.mul_sum] | |
| apply Finset.sum_congr rfl | |
| intro i hi | |
| rw [cavityMessage_eq_log_cosh_sub beta (z i)] | |
| ring | |
| linarith | |
| theorem finite_cavity_mean {I : Type*} (D : Finset I) (beta x : ℝ) (z : I → ℝ) : | |
| (finiteBranchWeight D beta x z 1 - finiteBranchWeight D beta x z (-1)) / | |
| (finiteBranchWeight D beta x z 1 + finiteBranchWeight D beta x z (-1)) = | |
| Real.tanh (x + ∑ i ∈ D, cavityMessage (Real.tanh beta) (z i)) := by | |
| rw [binary_mean_eq_tanh_half_log (finiteBranchWeight_pos D beta x z 1) | |
| (finiteBranchWeight_pos D beta x z (-1)), finiteBranchWeight_half_log_odds] | |
| theorem finite_cavity_field {I : Type*} (D : Finset I) (beta x : ℝ) (z : I → ℝ) : | |
| Real.artanh ((finiteBranchWeight D beta x z 1 - finiteBranchWeight D beta x z (-1)) / | |
| (finiteBranchWeight D beta x z 1 + finiteBranchWeight D beta x z (-1))) = | |
| x + ∑ i ∈ D, cavityMessage (Real.tanh beta) (z i) := by | |
| rw [finite_cavity_mean, Real.artanh_tanh] | |
| theorem finite_cavity_field_difference_bound {I : Type*} (D : Finset I) | |
| {beta : ℝ} (hbeta : 0 ≤ beta) (x : ℝ) (z : I → ℝ) : | |
| |Real.artanh ((finiteBranchWeight D beta x z 1 - finiteBranchWeight D beta x z (-1)) / | |
| (finiteBranchWeight D beta x z 1 + finiteBranchWeight D beta x z (-1))) - x| ≤ | |
| (D.card : ℝ) * beta := by | |
| rw [finite_cavity_field, add_sub_cancel_left] | |
| calc | |
| |∑ i ∈ D, cavityMessage (Real.tanh beta) (z i)| ≤ | |
| ∑ i ∈ D, |cavityMessage (Real.tanh beta) (z i)| := Finset.abs_sum_le_sum_abs _ _ | |
| _ ≤ ∑ _i ∈ D, beta := Finset.sum_le_sum fun i _ => abs_cavityMessage_tanh_le hbeta (z i) | |
| _ = (D.card : ℝ) * beta := by simp | |
| end Problem367 | |
| end | |
| end OAI | |