| import OAI.Probability.FreeIsing.Cavity.LogOdds |
| import OAI.Probability.FreeIsing.Gibbs.Leaf |
|
|
| namespace OAI |
|
|
| noncomputable section |
|
|
| namespace Problem367.GibbsLeaf |
|
|
| def eliminationConstant (beta z : ℝ) : ℝ := |
| 2 * Real.exp ((Real.log (Real.cosh (z + beta)) + Real.log (Real.cosh (z - beta))) / 2) |
|
|
| theorem eliminationConstant_pos (beta z : ℝ) : 0 < eliminationConstant beta z := by |
| unfold eliminationConstant |
| positivity |
|
|
| theorem cosh_eq_eliminationConstant_mul_message (beta z : ℝ) (s : IsingSpin) : |
| 2 * Real.cosh (z + beta * spinSign s) = |
| eliminationConstant beta z * Real.exp |
| (cavityMessage (Real.tanh beta) z * spinSign s) := by |
| unfold eliminationConstant |
| rw [mul_assoc, ← Real.exp_add, cavityMessage_eq_log_cosh_sub] |
| cases s |
| · simp only [spinSign_neg, mul_neg_one] |
| rw [show (Real.log (Real.cosh (z + beta)) + Real.log (Real.cosh (z - beta))) / 2 + |
| -((Real.log (Real.cosh (z + beta)) - Real.log (Real.cosh (z - beta))) / 2) = |
| Real.log (Real.cosh (z - beta)) by ring] |
| rw [Real.exp_log (Real.cosh_pos _)] |
| congr 2 |
| · simp only [spinSign_pos, mul_one] |
| rw [show (Real.log (Real.cosh (z + beta)) + Real.log (Real.cosh (z - beta))) / 2 + |
| (Real.log (Real.cosh (z + beta)) - Real.log (Real.cosh (z - beta))) / 2 = |
| Real.log (Real.cosh (z + beta)) by ring] |
| rw [Real.exp_log (Real.cosh_pos _)] |
|
|
| theorem energy_update_field {d : ℕ} (beta : ℝ) (h : TreeVertex d → ℝ) |
| (D : Finset (TreeVertex d)) {p : TreeVertex d} (hp : p ∈ D) |
| (z : ℝ) (eta : TreeVertex d → IsingSpin) : |
| energy beta (Function.update h p (h p + z)) D eta = |
| energy beta h D eta + z * spinSign (eta p) := by |
| classical |
| have hsum : (∑ x ∈ D.erase p, Function.update h p (h p + z) x * spinSign (eta x)) = |
| ∑ x ∈ D.erase p, h x * spinSign (eta x) := by |
| apply Finset.sum_congr rfl |
| intro x hx |
| rw [Function.update_of_ne (Finset.mem_erase.mp hx).1] |
| unfold energy |
| rw [← Finset.sum_erase_add D |
| (fun x => Function.update h p (h p + z) x * spinSign (eta x)) hp, |
| ← Finset.sum_erase_add D (fun x => h x * spinSign (eta x)) hp, hsum] |
| simp only [Function.update_self] |
| ring |
|
|
| theorem sum_weight_update_eq_cavity |
| {d : ℕ} {D : Finset (TreeVertex d)} {v p : TreeVertex d} |
| (beta : ℝ) (h : TreeVertex d → ℝ) (hv : v ∈ D) (hp : p ∈ D.erase v) |
| (hedges : inducedEdges D = insert (p, v) (inducedEdges (D.erase v))) |
| (eta : TreeVertex d → IsingSpin) : |
| (∑ s : IsingSpin, weight beta h D (Function.update eta v s)) = |
| eliminationConstant beta (h v) * |
| weight beta (Function.update h p (h p + cavityMessage (Real.tanh beta) (h v))) |
| (D.erase v) eta := by |
| rw [sum_weight_update_of_edges beta h hv (Finset.mem_erase.mp hp).1 hedges, |
| cosh_eq_eliminationConstant_mul_message] |
| simp only [weight, energy_update_field beta h (D.erase v) hp, Real.exp_add] |
| ring |
|
|
| theorem partition_eq_cavity_partition |
| {d : ℕ} {D : Finset (TreeVertex d)} {v p : TreeVertex d} |
| (beta : ℝ) (h : TreeVertex d → ℝ) (hv : v ∈ D) (hp : p ∈ D.erase v) |
| (hedges : inducedEdges D = insert (p, v) (inducedEdges (D.erase v))) : |
| (∑ eta : D → IsingSpin, weight beta h D (extendFiniteSpin D eta)) = |
| eliminationConstant beta (h v) * |
| ∑ eta : D.erase v → IsingSpin, |
| weight beta (Function.update h p (h p + cavityMessage (Real.tanh beta) (h v))) |
| (D.erase v) (extendFiniteSpin (D.erase v) eta) := by |
| classical |
| rw [sum_weight_eq_erase_sum beta h hv (Finset.mem_erase.mp hp).1 hedges, Finset.mul_sum] |
| apply Finset.sum_congr rfl |
| intro eta _ |
| rw [← sum_weight_update_of_edges beta h hv (Finset.mem_erase.mp hp).1 hedges] |
| exact sum_weight_update_eq_cavity beta h hv hp hedges _ |
|
|
| def partition {d : ℕ} (beta : ℝ) (h : TreeVertex d → ℝ) |
| (D : Finset (TreeVertex d)) : ℝ := |
| ∑ eta : D → IsingSpin, weight beta h D (extendFiniteSpin D eta) |
|
|
| def mass {d : ℕ} (beta : ℝ) (h : TreeVertex d → ℝ) |
| (D : Finset (TreeVertex d)) (eta : TreeVertex d → IsingSpin) : ℝ := |
| weight beta h D eta / partition beta h D |
|
|
| theorem partition_pos {d : ℕ} (beta : ℝ) (h : TreeVertex d → ℝ) |
| (D : Finset (TreeVertex d)) : 0 < partition beta h D := by |
| classical |
| let : Nonempty (D → IsingSpin) := ⟨fun _ => .neg⟩ |
| exact Finset.sum_pos (fun eta _ => Real.exp_pos _) Finset.univ_nonempty |
|
|
| theorem sum_mass_update_eq_cavity |
| {d : ℕ} {D : Finset (TreeVertex d)} {v p : TreeVertex d} |
| (beta : ℝ) (h : TreeVertex d → ℝ) (hv : v ∈ D) (hp : p ∈ D.erase v) |
| (hedges : inducedEdges D = insert (p, v) (inducedEdges (D.erase v))) |
| (eta : TreeVertex d → IsingSpin) : |
| (∑ s : IsingSpin, mass beta h D (Function.update eta v s)) = |
| mass beta (Function.update h p (h p + cavityMessage (Real.tanh beta) (h v))) |
| (D.erase v) eta := by |
| classical |
| simp only [mass, ← Finset.sum_div] |
| rw [sum_weight_update_eq_cavity beta h hv hp hedges] |
| unfold partition |
| rw [partition_eq_cavity_partition beta h hv hp hedges] |
| exact mul_div_mul_left _ _ (ne_of_gt (eliminationConstant_pos beta (h v))) |
|
|
| end Problem367.GibbsLeaf |
|
|
| end |
|
|
| end OAI |
|
|
