Credibility Theory Cheat Sheet

Classical (Limited Fluctuation) Credibility

• Full credibility standard: n_0 = (z_(alpha/2) / k)^2 * (sigma^2 / mu^2) for full credibility at level k
• Partial credibility: Z = sqrt(n / n_0)
• Credibility premium: P = Z * X-bar + (1 - Z) * M

Buhlmann Credibility

• Credibility factor: Z = n / (n + k) where k = v / a
• v (expected value of process variance): v = E[Var(X|Theta)]
• a (variance of hypothetical means): a = Var(E[X|Theta])
• Buhlmann premium: P = Z * X-bar + (1-Z) * mu
• mu (overall mean): mu = E[E[X|Theta]] = E[X]

Buhlmann-Straub

• Z_i = m_i / (m_i + k) where m_i is the exposure for risk i
• k = v / a (same as Buhlmann, but exposure-weighted)
• Credibility-weighted overall mean: X-bar-bar = sum(Z_i * X-bar_i) / sum(Z_i)

Conjugate Prior Families

Likelihood	Prior	Posterior
Poisson(lambda)	Gamma(alpha, beta)	Gamma(alpha + sum x_i, beta + n)
Binomial(n, p)	Beta(a, b)	Beta(a + sum x_i, b + sum(n_i - x_i))
Normal(mu, sigma^2)	Normal(mu_0, sigma_0^2)	Normal(weighted mean, combined precision)
Exponential(theta)	Gamma(alpha, beta)	Gamma(alpha + n, beta + sum x_i)