File size: 3,423 Bytes
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 | import OAI.Probability.FreeIsing.Cavity.Homeomorph
import OAI.Probability.FreeIsing.Estimates.InhomogeneousSpinChain
namespace OAI
noncomputable section
namespace Problem367.PosteriorChain
private theorem tanh_add_ratio (x y : ℝ) :
Real.tanh (x + y) =
(Real.tanh x + Real.tanh y) / (1 + Real.tanh x * Real.tanh y) := by
have hm := cavityMessage_argument_mem (Real.tanh_sq_lt_one x).le y
have hd : 1 + Real.tanh x * Real.tanh y ≠ 0 := by linarith [hm.1]
apply (eq_div_iff hd).2
simp only [Real.tanh_eq_sinh_div_cosh]
field_simp
rw [Real.sinh_add, Real.cosh_add]
ring
private theorem tanh_sub_ratio (x y : ℝ) :
Real.tanh (x - y) =
(Real.tanh x - Real.tanh y) / (1 - Real.tanh x * Real.tanh y) := by
simpa only [sub_eq_add_neg, Real.tanh_neg, mul_neg] using tanh_add_ratio x (-y)
theorem affine_mean_eq_cavity_mean (beta z H : ℝ) :
intercept beta z + slope beta z * Real.tanh H =
Real.tanh (z + cavityMessage (Real.tanh beta)
(H - cavityMessage (Real.tanh beta) z)) := by
have htheta : Real.tanh beta ^ 2 ≤ 1 := (Real.tanh_sq_lt_one beta).le
have hinner : Real.tanh (cavityMessage (Real.tanh beta) z) =
Real.tanh beta * Real.tanh z :=
Real.tanh_artanh (cavityMessage_argument_mem htheta z)
have houter : Real.tanh (cavityMessage (Real.tanh beta)
(H - cavityMessage (Real.tanh beta) z)) =
Real.tanh beta * Real.tanh (H - cavityMessage (Real.tanh beta) z) :=
Real.tanh_artanh (cavityMessage_argument_mem htheta _)
rw [tanh_add_ratio, houter, tanh_sub_ratio H, hinner]
unfold intercept slope
rw [tanh_add_ratio, tanh_sub_ratio]
have hz := cavityMessage_argument_mem (Real.tanh_sq_lt_one z).le beta
have hh := cavityMessage_argument_mem (Real.tanh_sq_lt_one H).le
(cavityMessage (Real.tanh beta) z)
rw [hinner] at hh
have hdplus : 1 + Real.tanh z * Real.tanh beta ≠ 0 := by linarith [hz.1]
have hdminus : 1 - Real.tanh z * Real.tanh beta ≠ 0 := by linarith [hz.2]
have hdinner : 1 - Real.tanh H * (Real.tanh beta * Real.tanh z) ≠ 0 := by
linarith [hh.2]
have hout := cavityMessage_argument_mem (Real.tanh_sq_lt_one z).le
(cavityMessage (Real.tanh beta) (H - cavityMessage (Real.tanh beta) z))
rw [houter, tanh_sub_ratio H, hinner] at hout
have hdouter : 1 + Real.tanh z * (Real.tanh beta *
((Real.tanh H - Real.tanh beta * Real.tanh z) /
(1 - Real.tanh H * (Real.tanh beta * Real.tanh z)))) ≠ 0 := by
linarith [hout.1]
apply (eq_div_iff hdouter).2
have hdinner' : 1 - Real.tanh z * Real.tanh beta * Real.tanh H ≠ 0 := by
convert hdinner using 1
ring
field_simp [hdplus, hdminus, hdinner, hdinner']
ring
theorem cavityFromSites_affine_mean {beta : ℝ} (hbeta : 0 ≤ beta) (z H : ℝ) :
cavityFromSites (tanh_nonneg_of_beta_nonneg hbeta) (Real.tanh_lt_one beta)
(H, Real.artanh (intercept beta z + slope beta z * Real.tanh H)) =
(H - cavityMessage (Real.tanh beta) z, z) := by
apply cavityFromSites_eq
constructor
· ring
· simp only [affine_mean_eq_cavity_mean, Real.artanh_tanh]
theorem cavityFromSites_affine_mean_snd {beta : ℝ} (hbeta : 0 ≤ beta) (z H : ℝ) :
(cavityFromSites (tanh_nonneg_of_beta_nonneg hbeta) (Real.tanh_lt_one beta)
(H, Real.artanh (intercept beta z + slope beta z * Real.tanh H))).2 = z := by
rw [cavityFromSites_affine_mean hbeta]
end Problem367.PosteriorChain
end
end OAI
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