OAI/Probability/InvariantIsing/Cavity/CavityGaussianSpectral.lean

GitHub source file

11 declarations, 0 contain sorry, 0 declared as axioms

Square completion for an arbitrary positive definite real precision matrix, expressed against the standard finite-dimensional Gaussian.

  1. lemma OAI.InvariantIsing.cavity_eigenbasis_action line 14
  2. lemma OAI.InvariantIsing.cavity_eigenbasis_coordinate line 21
  3. lemma OAI.InvariantIsing.cavity_eigenbasis_quadratic line 29
  4. lemma OAI.InvariantIsing.cavity_eigenbasis_linear line 40
  5. lemma OAI.InvariantIsing.cavity_eigenbasis_exponent line 49
  6. lemma OAI.InvariantIsing.cavity_precision_inverse_cancel line 63
  7. lemma OAI.InvariantIsing.cavity_eigenbasis_inverse_coordinate line 71
  8. lemma OAI.InvariantIsing.cavity_eigenbasis_inverse_quadratic line 81
  9. lemma OAI.InvariantIsing.cavity_precision_normalizer line 92
  10. theorem OAI.InvariantIsing.cavity_gaussian_precision_integrable line 100
  11. theorem OAI.InvariantIsing.cavity_gaussian_precision_integral line 114

    Source documentation

    Standard-Gaussian square completion with an arbitrary positive definite precision matrix. No simultaneous diagonalization with the original covariance is assumed.

    theorem cavity_gaussian_precision_integral {d : ℕ}
    (Q : Matrix (Fin d) (Fin d) ℝ) (hQ : Q.PosDef)
    (l : EuclideanSpace ℝ (Fin d)) :
    (∫ x : EuclideanSpace ℝ (Fin d),
      Real.exp ((‖x‖ ^ 2 - ⟪x, Q.toEuclideanLin x⟫) / 2 + ⟪l, x⟫)
      ∂stdGaussian (EuclideanSpace ℝ (Fin d))) =
      (Real.sqrt Q.det)⁻¹ * Real.exp (⟪l, Q⁻¹.toEuclideanLin l⟫ / 2) := by
…