Bayesian Credibility Premium: Formula, Derivation and Poisson-Gamma Example

Understand Bayesian Credibility Premium

Key rules to remember

Bayesian premium (predictive mean)

E[X_{n+1} | x] = ∫ μ(θ) f(θ | x) dθ

Use a sum for a discrete θ. μ(θ) = E[X | θ]. X_{n+1} is independent of the past given θ.

Posterior density

f(θ | x) ∝ f(θ) × Π f(xᵢ | θ)

Drop every factor not involving θ, then recognise the distribution from its kernel.

Credibility form

P = Z × x̄ + (1 − Z) × E[μ(θ)]

Valid exactly only when the posterior mean of μ(θ) is linear in the data, as in conjugate cases above.

Poisson-gamma model

X | λ ~ Poisson(λ), λ ~ Gamma(α, β) (mean α ÷ β). Posterior: Gamma(α + Σxᵢ, β + n)

Here β is the rate parameter. Check which parametrisation the question uses.

Poisson-gamma premium and Z

P = (α + Σxᵢ) ÷ (β + n) = Z x̄ + (1 − Z)(α ÷ β), with Z = n ÷ (n + β)

So k = β. Z is found by matching the coefficient of x̄.

Normal-normal (known variance)

X | θ ~ N(θ, σ²), θ ~ N(μ, s²). Z = n ÷ (n + σ² ÷ s²)

Posterior mean = Z x̄ + (1 − Z) μ.

Binomial-beta (m trials each)

X | q ~ Bin(m, q), q ~ Beta(a, b). Posterior mean of q = (a + Σxᵢ) ÷ (a + b + nm)

Premium per trial is this value, so Z = nm ÷ (a + b + nm).

How to solve Bayesian Credibility Premium questions

1. 1Write the model: the distribution of X given θ, and the prior of θ. Note the parametrisation.
2. 2Write μ(θ) = E[X | θ], the quantity you want to predict.
3. 3Find the posterior: multiply likelihood by prior, keep only θ terms, and identify the distribution and its parameters.
4. 4Compute the posterior mean of μ(θ). This is the Bayesian premium.
5. 5Rewrite it as Z × x̄ + (1 − Z) × prior mean. Read Z from the coefficient of x̄.
6. 6Check that Z is between 0 and 1 and that the prior mean equals E[μ(θ)] from the original prior.
7. 7Substitute the data and state the premium with units. If asked, comment on how Z changes with n.

Quickest way: Conjugate shortcut

Common mistakes in Bayesian Credibility Premium

• Using the posterior mean of θ instead of the posterior mean of μ(θ).

  In Poisson-gamma μ(λ) = λ, so the two agree, and students assume that is always true.

  Fix: Always write μ(θ) = E[X | θ] first. For exponential with mean 1 ÷ θ you need E[1 ÷ θ | x], not E[θ | x].

• Mixing up the gamma rate and scale parameters.

  Texts and questions use different conventions.

  Fix: Check the stated mean. If the prior mean is α ÷ β, β is a rate and k = β. If it is αθ, convert.

• Taking Z = n ÷ (n + α).

  Students remember a shape parameter and not the correct one.

  Fix: Derive it: (α + nx̄) ÷ (β + n) gives Z = n ÷ (n + β).

• Claiming every Bayesian premium is a credibility premium.

  The textbook examples are all conjugate and linear.

  Fix: State that the form is exact only when the posterior mean is linear in the data. Otherwise it is only approximated by the Bühlmann premium.

• Using the sum of claims where the mean is needed in Z × x̄.

  The posterior formula contains Σxᵢ, so the sum gets plugged into the credibility form.

  Fix: Use x̄ = Σxᵢ ÷ n in the credibility form. Check Z is below 1.

• Forgetting the prior mean, or using the posterior mean, in the (1 − Z) term.

  Confusion between prior and posterior.

  Fix: The complement weight always multiplies the prior mean of μ(θ), here α ÷ β.

Worked examples