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ADR-0005: Theil mixed-estimation form for Black-Litterman posterior

  • Status: Accepted
  • Date: 2026-05-23

Context

The Black-Litterman posterior mean μ_BL has two algebraically equivalent forms: the precision-weighted form (inverts N×N matrices) and the Theil mixed-estimation form (inverts only K×K, where K = number of views, typically K ≪ N).

Decision

Implement the Theil mixed-estimation form: 

mu_BL = pi + tau·Sigma·Pᵀ · (P·tau·Sigma·Pᵀ + Omega)⁻¹ · (Q − P·pi)
The only inverse is K×K. Posterior covariance Sigma_BL = Sigma + M derived via Woodbury identity to avoid forming (τΣ)⁻¹.

Decision drivers

  • Numerical stability: avoids inverting N×N matrices that may be ill-conditioned.
  • Performance: K is typically 1–5; the K×K solve is trivial.
  • Cholesky-only linear algebra throughout.

Considered options

  • Option A: Theil form. Chosen.
  • Option B: Precision-weighted form. Rejected: forms (τΣ)⁻¹ which is unstable.
  • Option C: Both, user-selectable. Rejected: API bloat with no user-visible benefit.

Consequences

  • The K=0 (no views) case is a fast path returning μ_BL = π and M = 0.
  • All linear solves use scipy.linalg.cho_factor / cho_solve.
  • The He-Litterman 1999 reproduction test validates against published numbers regardless of which form was used.
  • Theil, H. (1971). Principles of Econometrics.
  • He, G. & Litterman, R. (1999). "The Intuition Behind Black-Litterman Model Portfolios."