Predictive Distribution

Study guides: Exam MAS-II, Exam 7

The predictive distribution is the distribution of the next observation given the ones already seen, with the unknown risk parameter integrated out against its posterior. It is what a Bayesian Credibility forecast is really a mean of.

f(x_{n+1} \mid \mathbf{x}) = \int f(x_{n+1} \mid \theta)\, \pi(\theta \mid \mathbf{x})\, d\theta

• The prior predictive (or marginal) distribution uses \pi(\theta) instead of the posterior — it is the denominator of Bayes Theorem and describes a risk you know nothing about
• The Bayesian credibility estimate is its mean: E[X_{n+1} \mid \mathbf{x}] = E_\theta\!\left[\mu(\theta) \mid \mathbf{x}\right]
• It is wider than the conditional distribution f(x \mid \hat\theta) at a plugged-in point estimate, because it carries parameter uncertainty as well as process variance
• Gamma-Poisson gives a Negative Binomial predictive; Beta-Bernoulli gives a Beta-Binomial — mixing always produces a heavier tail than the component
• Only the mean is needed for a credibility estimate, but the full distribution is what prices a limit or a reinsurance layer

Negative Binomial from a Gamma-Poisson Mixture Claim counts are Poisson(\Lambda) with \Lambda \sim Gamma(\alpha = 2, \theta = 0.5) and no experience yet. Find E[N] and \mathrm{Var}(N) under the prior predictive distribution.

Answer Mixing a Poisson over a Gamma gives a Negative Binomial with r = \alpha = 2, \beta = \theta = 0.5: E[N] = E[\Lambda] = \alpha\theta = 1.0 \mathrm{Var}(N) = E[\Lambda] + \mathrm{Var}(\Lambda) = \alpha\theta + \alpha\theta^{2} = 1.0 + 0.5 = 1.5 The variance exceeds the mean — the parameter uncertainty adds \alpha\theta^2 on top of Poisson process variance, which is why mixed claim counts are overdispersed.