A posterior probability is the probability of an event after evidence is known. Bayes’ rule combines a prior probability, a likelihood, and the probability of the evidence to obtain P(A|B), the posterior. The same update applies to an uncertain parameter θ: a prior distribution represents beliefs before data, and a posterior distribution represents beliefs after data. section 12.2
Updating beliefs with evidence
Knowing that a randomly chosen person plays professional basketball changes the probability assigned to their being over six feet tall. The conditional probability remains small, but is about 14 times the unconditional probability, illustrating how evidence updates a prior belief. section 12.2
In Black–Litterman, equilibrium expected returns supply the prior and investor views supply uncertain evidence. The model updates them into a posterior distribution of returns, so views influence expected returns without being treated as certain observations. section 12.3.2 section 12.3.3
Under the normality assumptions, the posterior returns distribution is normal and is specified by:
- A posterior mean that combines the prior mean μ with view means b through the view matrix A and uncertainty matrices C and Ω. section 12.3.3
- A posterior covariance formed from the return covariance Σ and the uncertainty remaining after incorporating the views. section 12.3.3
- An optional convention replacing C with τΣ when uncertainty in the equilibrium-return estimate is modeled that way. section 12.3.3
Listing 12.1 Posterior mean and covariance
import numpy as np
import pandas as pd
def compute_posterior_mean(mu, C, A, b, omega):
c_inv = np.linalg.inv(C)
a_t_oinv = A.T @ np.linalg.inv(omega) #1
er = np.linalg.inv(c_inv + a_t_oinv @ A) @ (c_inv @ mu + a_t_oinv @ b)
return pd.Series(er, index=mu.index)
def compute_posterior_cov(sigma, C, A, omega):
return sigma + np.linalg.inv((np.linalg.inv(C) + A.T @ omega @ A))
Use the NumPy implementation to compute posterior means and covariances from μ, C, A, b, Ω, and Σ. The functions are general: they do not require C = τΣ or a diagonal Ω, and their outputs can feed standard asset-allocation optimization. section 12.3.3
The update is useful because it preserves both sources of uncertainty: equilibrium information and the reliability of the views. The derivation of the multivariate formulas is deferred, while the resulting quantities are directly available for portfolio construction. section 12.3.2 section 12.3.3
