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b317718 | 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 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 | import OAI.Probability.FreeIsing.Cavity.BinaryMixture
import OAI.Probability.FreeIsing.Cavity.Homeomorph
import OAI.Probability.FreeIsing.Estimates.ChainMarginals
import OAI.Probability.FreeIsing.Trees.CoordinateResponse
namespace OAI
noncomputable section
open scoped BigOperators
namespace Problem367.FinitePathGibbs
theorem mean_eq_posterior_mean (beta root : ℝ) (h : ℕ → ℝ) (n : ℕ)
(f : (Fin (n + 1) → IsingSpin) → ℝ) :
mean beta root h n f =
FinitePosterior.mean (energy beta root h n) (fun _ => 0) f 0 := by
unfold mean FinitePosterior.mean FinitePosterior.partition
simp only [FinitePosterior.weight, zero_mul, add_zero]
simp_rw [mass_eq_exp_energy, partition_eq_sum_exp, div_mul_eq_mul_div]
rw [← Finset.sum_div]
congr 1
apply Finset.sum_congr rfl
intro x _
ring
theorem coordinate_mean_mem_Ioo (beta root : ℝ) (h : ℕ → ℝ) (n : ℕ)
(i : Fin (n + 1)) :
mean beta root h n (fun x => spinSign (x i)) ∈ Set.Ioo (-1 : ℝ) 1 := by
rw [mean_eq_posterior_mean]
let : Nonempty (Fin (n + 1) → IsingSpin) := ⟨fun _ => .pos⟩
apply FinitePosterior.spin_mean_mem_Ioo
· intro x
cases x i <;> simp [spinSign]
· exact ⟨fun _ => .neg, rfl⟩
· exact ⟨fun _ => .pos, rfl⟩
theorem coordinate_mean_affine (beta root : ℝ) (h : ℕ → ℝ) (n : ℕ)
(i : Fin n) :
mean beta root h n (fun x => spinSign (x i.succ)) =
PosteriorChain.intercept beta (backwardField beta h n i) +
PosteriorChain.slope beta (backwardField beta h n i) *
mean beta root h n (fun x => spinSign (x i.castSucc)) := by
have hh := InhomogeneousChain.coordinateMean_affine
(fun j => PosteriorChain.transition beta (backwardField beta h n j))
(rootLaw beta root h n) spinSign
(fun j => PosteriorChain.intercept beta (backwardField beta h n j))
(fun j => PosteriorChain.slope beta (backwardField beta h n j))
(sum_rootLaw beta root h n)
(fun j => PosteriorChain.transition_sum beta (backwardField beta h n j))
(fun j => PosteriorChain.transition_spinMean_affine beta (backwardField beta h n j)) n i
simpa only [InhomogeneousChain.coordinateMean, ← mass_eq_jointWeight, mean] using hh
theorem backwardField_eq_cavityFromSites {beta : ℝ} (hbeta : 0 ≤ beta)
(root : ℝ) (h : ℕ → ℝ) (n : ℕ) (i : Fin n) :
backwardField beta h n i =
(cavityFromSites (tanh_nonneg_of_beta_nonneg hbeta) (Real.tanh_lt_one beta)
(Real.artanh (mean beta root h n (fun x => spinSign (x i.castSucc))),
Real.artanh (mean beta root h n (fun x => spinSign (x i.succ))))).2 := by
have hh := PosteriorChain.cavityFromSites_affine_mean_snd hbeta
(backwardField beta h n i)
(Real.artanh (mean beta root h n (fun x => spinSign (x i.castSucc))))
rw [Real.tanh_artanh (coordinate_mean_mem_Ioo beta root h n i.castSucc),
← coordinate_mean_affine beta root h n i] at hh
exact hh.symm
end Problem367.FinitePathGibbs
namespace Problem367.TreePathResponse
theorem measurable_mean {d : ℕ} (beta : ℝ)
(D : Finset (TreeVertex d)) (u : TreeVertex d) :
Measurable (fun h : TreeVertex d → ℝ => mean beta h D u) := by
unfold mean LeafGibbs.expectation GibbsLeaf.energy
fun_prop
theorem measurable_field {d : ℕ} (beta : ℝ)
(D : Finset (TreeVertex d)) (u : TreeVertex d) :
Measurable (fun h : TreeVertex d → ℝ => field beta h D u) := by
have ha : Measurable Real.artanh := by
unfold Real.artanh
fun_prop
exact ha.comp (measurable_mean beta D u)
def orientedCavity {d : ℕ} {beta : ℝ} (hbeta : 0 ≤ beta)
(h : TreeVertex d → ℝ) (D : Finset (TreeVertex d))
(u v : TreeVertex d) : ℝ :=
(cavityFromSites (tanh_nonneg_of_beta_nonneg hbeta) (Real.tanh_lt_one beta)
(field beta h D u, field beta h D v)).2
theorem measurable_orientedCavity {d : ℕ} {beta : ℝ} (hbeta : 0 ≤ beta)
(D : Finset (TreeVertex d)) (u v : TreeVertex d) :
Measurable (fun h : TreeVertex d → ℝ => orientedCavity hbeta h D u v) := by
exact (measurable_cavityFromSites (tanh_nonneg_of_beta_nonneg hbeta)
(Real.tanh_lt_one beta)).snd.comp
((measurable_field beta D u).prodMk (measurable_field beta D v))
theorem root_response_eq_canonical_product {d : ℕ} {beta : ℝ}
(hbeta : 0 ≤ beta) (h : TreeVertex d → ℝ)
{D : Finset (TreeVertex d)} (hD : ConnectedFinset D)
(hr : ConnectedHull.root d ∈ D) {v : TreeVertex d} (hv : v ∈ D) :
response beta h D (ConnectedHull.root d) v =
Real.tanh beta ^ v.1.length *
∏ i ∈ Finset.range v.1.length, responseAttenuation (Real.tanh beta)
(orientedCavity hbeta h D (ConnectedHull.takeVertex v i)
(ConnectedHull.takeVertex v (i + 1))) := by
obtain ⟨y, hy⟩ := exists_path_fields beta h hD hr hv
have hmean (i : Fin (v.1.length + 1)) :
mean beta h D (ConnectedHull.path v i) =
FinitePathGibbs.mean beta (y 0) (PathPosterior.childFields y) v.1.length
(fun x => spinSign (x i)) :=
hy (fun x => spinSign (x i))
have hback (i : ℕ) (hi : i < v.1.length) :
FinitePathGibbs.backwardField beta (PathPosterior.childFields y) v.1.length i =
orientedCavity hbeta h D (ConnectedHull.takeVertex v i)
(ConnectedHull.takeVertex v (i + 1)) := by
have hh := FinitePathGibbs.backwardField_eq_cavityFromSites hbeta
(y 0) (PathPosterior.childFields y) v.1.length ⟨i, hi⟩
rw [← hmean (Fin.castSucc ⟨i, hi⟩), ← hmean (Fin.succ ⟨i, hi⟩)] at hh
exact hh
have hprod :
(∏ i ∈ Finset.range v.1.length, responseAttenuation (Real.tanh beta)
(FinitePathGibbs.backwardField beta (PathPosterior.childFields y) v.1.length i)) =
∏ i ∈ Finset.range v.1.length, responseAttenuation (Real.tanh beta)
(orientedCavity hbeta h D (ConnectedHull.takeVertex v i)
(ConnectedHull.takeVertex v (i + 1))) := by
apply Finset.prod_congr rfl
intro i hi
rw [hback i (Finset.mem_range.mp hi)]
have hcov : covariance beta h D (ConnectedHull.root d) v =
(1 - mean beta h D (ConnectedHull.root d) ^ 2) * Real.tanh beta ^ v.1.length *
∏ i ∈ Finset.range v.1.length, responseAttenuation (Real.tanh beta)
(orientedCavity hbeta h D (ConnectedHull.takeVertex v i)
(ConnectedHull.takeVertex v (i + 1))) := by
rw [← hprod]
have hm := hy (fun x => spinSign (x 0))
have hn := hy (fun x => spinSign (x (Fin.last v.1.length)))
have hp := hy (fun x => spinSign (x 0) * spinSign (x (Fin.last v.1.length)))
simp only [ConnectedHull.path_zero, ConnectedHull.path_last] at hm hn hp
unfold covariance LeafGibbs.covariance mean
rw [hp, hm, hn]
exact FinitePathGibbs.endpointCovariance_eq beta (y 0)
(PathPosterior.childFields y) v.1.length
rw [response, hcov]
have hp := (variance_pos beta h hr).ne'
field_simp
end Problem367.TreePathResponse
end
end OAI
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