Quant Toolbox > 23. Bayesian statistics > Classical equivalent and estimation uncertainty

Classical-equivalent

Given a posterior distribution fpos(·) (23.6), we can always single out oneestimate θcl_eq as follows

• posterior expectation

θ̂mean ≡∫θfpos(θ)dθ (23.8)

• maximum a posteriori (MAP) probability estimate

θ̂MAP ≡ argmaxθ{fpos(θ)} (23.9)

ARPM - Advanced Risk and Portfolio Management - arpm.co This update: Feb-21-2017 - Last update

Quant Toolbox > 23. Bayesian statistics > Classical equivalent and estimation uncertainty

Estimation uncertainty

We define the estimation uncertainty as the dispersion of the posteriordistribution fpos(·) (23.6)

• Covariance

s2θ ≡∫

(θ − θ̂mean)(θ − θ̂mean)′ fpos(θ)dθ (23.10)

• Modal square-dispersion (31.115)

s2θ ≡ (−∇2θ,θ ln fpos |θ=θ̂MAP

)−1 (23.11)

The classical-equivalent θ̂ and the uncertainty s2θ determine alocation-dispersion ellipsoid (31.101)

E(θ̂, r2s2θ) ≡ {θ : (θ − θ̂)′(s2θ)−1(θ − θ̂)=r2} (23.12)

ARPM - Advanced Risk and Portfolio Management - arpm.co This update: Feb-21-2017 - Last update