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The Black-Litterman model combines equilibrium-implied expected returns with additional views whose uncertainty is stated explicitly. Fischer Black and Robert Litterman introduced it to make portfolio optimization less dependent on unstable unconstrained return estimates.^[ref]

In a standard Gaussian formulation, let the unknown return-mean vector have prior $\mu \sim N(\pi ,C)$, often $C=\tau \Sigma$ for asset-return covariance matrix $\Sigma$. Views take the form $q=P\mu +\epsilon$, with independent $\epsilon \sim N(0,\Omega )$. The Gaussian conditional expectation formula gives posterior meanThe matrix $P$ describes absolute or relative views, while $\Omega$ describes their uncertainty.

For one mean with prior $4\%$, prior variance $0.0004$, and a view of $8\%$ with equal variance, the posterior mean is $6\%$. Increasing the view variance pulls the result toward $4\%$; making it very small pulls toward $8\%$. The uncertainty $C$ about a mean is distinct from the covariance matrix $\Sigma$ of fluctuating returns, and must not be substituted for it without specifying the portfolio model.

Robert Litterman developed the Black-Litterman model with Fischer Black. Their 1992 paper makes equilibrium expected returns a neutral starting point for portfolio modeling, then incorporates views that can differ in their confidence. The contribution addresses how uncertain inputs enter modern portfolio theory, rather than treating every proposed expected return as equally reliable.^[ref]

An elementary Bayesian inference analogy illustrates the role of uncertainty. For an unknown quantity $\mu$ with normal distribution prior $N(m_0,s_0^2)$ and an independent observation error giving $y\mid \mu \sim N(\mu ,s_y^2)$, Gaussian conditional expectation gives the posterior meanThe more uncertain input receives less weight. This scalar example explains a modeling principle; the full Black-Litterman model treats related expected returns jointly, with a covariance matrix describing their uncertainty.