Estimate the parameters for severity, frequency, and aggregate distributions using Bayesian Estimation.

Bayesian estimation treats the parameter as random, not the data. You start with a prior belief, see the data, and update to a posterior. That single sentence is the entire exam.

The Bayesian machinery. Let $\theta$ be the parameter (claim frequency rate, severity scale, aggregate mean) and $\mathbf{x}=(x_1,\dots ,x_n)$ the observed data. You need three objects.

• Prior $\pi (\theta )$: your belief before seeing data.
• Likelihood $f(\mathbf{x}\mid \theta )={\prod }_{i}f(x_i\mid \theta )$: the model density evaluated at the data.
• Posterior $\pi (\theta \mid \mathbf{x})$: updated belief.

$$\pi (\theta \mid \mathbf{x})=\frac{\pi (\theta )f(\mathbf{x}\mid \theta )}{\int \pi (\theta )f(\mathbf{x}\mid \theta )d\theta }$$

KEY: Drop the denominator while you work. Match the numerator to a known density up to a constant of proportionality and read off the parameters.

Point estimates from the posterior. The choice of loss function picks the estimator.

• Squared-error loss: ${\hat{\theta }}_{\text{Bayes}}=E[\theta \mid \mathbf{x}]$, the posterior mean.
• Absolute-error loss: posterior median.
• Zero-one loss: posterior mode (the maximum a posteriori, or MAP).

The exam defaults to squared-error unless it says otherwise. The mean is what you compute.

Conjugate priors for actuarial models. A prior is conjugate when the posterior lies in the same family. Three pairings cover most exam problems.

Common mistakes

• Multiplying instead of normalizing. The posterior is proportional to prior times likelihood, but you still need the constant if asked for a probability, not just a mean.
• Confusing prior and posterior in the predictive integral. The predictive uses the posterior, not the prior.
• Using the Gamma posterior scale as $\theta$ instead of $\theta /(1+n\theta )$. The denominator is essential.

Bottom line

• Posterior ∝ Prior × Likelihood. The denominator is a normalizing constant that makes the posterior integrate to 1.
• Bayes estimate depends on the loss: squared-error gives the posterior mean, absolute-error gives the posterior median, 0-1 loss gives the posterior mode.
• Gamma prior + Poisson data gives a Gamma posterior with shape $\alpha +\sum n_i$ and scale $\theta /(1+n\theta )$; the predictive count distribution is Negative Binomial.
• Beta prior + Bernoulli data gives a Beta$(a+\sum x_i,b+n-\sum x_i)$ posterior.

Exam shortcut

Identify the conjugate pair first. If prior and likelihood are conjugate, write the posterior parameters directly without computing any integral. For Gamma-Poisson, compute $Z=n\theta /(1+n\theta )$ and form the credibility-weighted mean. Skip the explicit Gamma update. When asked for predictive expectation of next-period claims under Gamma-Poisson, it equals the posterior mean of $\lambda$. The Negative Binomial mean reduces to it.

The full lesson (about 1,466 words, 10 min read) adds 2 worked examples, all 6 common mistakes, a self-check, free in the app.

Learning objectives

• 4d

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