Bayesian Live Risk

We propose Bayesian Live Risk (BLR), a framework in which the spectral representation of portfolio loss is treated as a posterior state updated in real time. The key claim is architectural: a risk engine should output not only a current risk estimate, but also quantified uncertainty about that estimate and a persistent memory state carrying historically learned stress structure. This yields a self-aware risk engine that can update, doubt, and remember.

The central object is therefore not a single VaR number but a richer risk state \[ \mathcal{K}_t = (\Pi_t, U_t, M_t), \] where \(\Pi_t\) is the current posterior law of portfolio loss, \(U_t\) quantifies uncertainty about that law, and \(M_t\) records whether current conditions resemble historically learned stress episodes. Classical tail measures are recovered as projections of this state, in particular \(\mathrm{VaR}_\alpha(t) = q_\alpha(\Pi_t)\) and \(\mathrm{ES}_\alpha(t) = e_\alpha(\Pi_t)\).

BLR combines posterior risk aggregation, particle-filter dynamics, uncertainty diagnostics, and adaptive crisis recall on a spectral representation of the loss distribution. The formal spine is machine-verified in Lean 4: posterior averaging preserves coherence, posterior width and model uncertainty are non-negative, valid posterior mixtures remain valid distributions, posterior particles admit a spectral equilibrium interpretation, and the online update obeys an \(O(1/T)\) regret bound. These results establish that self-aware posterior risk is coherent, convergent, and structurally compatible with the repo's broader spectral theory.

The present paper is a theory-and-architecture paper with simulation evidence and first local historical benchmarks, not yet a full real-data validation paper. On simulated GFC-like paths, posterior width acts as an early warning signal before the posterior VaR level fully adjusts. On local historical windows built from a one-factor SPX control path and a simple SPX+VIX stress basket, the same architecture yields early warning signals and action separation under a simple memory-aware capital rule, while also exposing an honest calibration gap on the harder one-factor proxy. A first explicit 1-day ES backtest layer and a first frozen 10-day extension now sharpen that asymmetry rather than removing it: the stress basket still carries the clearest action-layer evidence, the thin SPX control path remains a real calibration stress test, and only one narrow forecast slice on the control path becomes roughly competitive after selector hardening. Broader cross-asset and multi-horizon validation remains future work and is stated as a limitation rather than a solved claim. The main contribution is therefore not a finalized backtest result but a new risk architecture: risk that knows what it does not know, and remembers what past crises taught it.