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f61e547 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 | 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
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