Formula sheets

Bühlmann credibility $Z=\frac{n}{n+k},\qquad k=\frac{EPV}{VHM}=\frac{E[\operatorname{Var}(X\mid\Theta)]}{\operatorname{Var}(E[X\mid\Theta])}$

Bühlmann–Straub with exposures $m_i$: $Z_i=\dfrac{m_i}{m_i+k}$.

Limited fluctuation full credibility (frequency, Poisson): $n_F=\left(\dfrac{z_{(1+p)/2}}{r}\right)^2$; partial credibility $Z=\sqrt{n/n_F}$.

Gamma–Poisson conjugacy: prior $\Gamma(\alpha,\theta)$, observe $\sum x$ claims over $n$ periods $\Rightarrow$ posterior $\Gamma\!\left(\alpha+\sum x,\;\dfrac{\theta}{1+n\theta}\right)$.

Metropolis–Hastings acceptance $\;\alpha=\min\!\left(1,\dfrac{\pi(y)q(x\mid y)}{\pi(x)q(y\mid x)}\right)$.

Ridge / lasso $\;\min_\beta \|y-X\beta\|^2+\lambda\|\beta\|_2^2$ (ridge), $+\lambda\|\beta\|_1$ (lasso).