We need to figure out good aways of modeling/representing uncertainty.

unifying principle

• data: samples / distribution
• representation: simplify / keep what matters
• control: convex / gradient / MPC

Different things requires different techniques but broadly the structure.

contributions

• factor models of covariance
• 1d projections of distributions and distribution shaping
• informational representation: conditioning + shrinking horizon reoptimization

factor model

Suppose someone hands you a model for returns:

\begin{equation} r \approx F_{\text{base}} s + \varepsilon, s \sim \mathcal{N}\qty(0, \Omega_{\text{base}}), \epsilon \sim \mathcal{N}\qty(0, D_{\text{base}}) \end{equation}

call \(F_{\text{base}}\) factor exposures (e.g., input features), and \(D_{\text{base}}\) is the idiosyncratic risk (diagonal.)

Problems!

• you don’t update a lot
• can’t capture trans client factor; or small regime shifts
• missing statistical factors that explali nasset returns
• statistical factors seems to be capturing “theme"s (groups of stocks) rather than individual stock