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IAI Actuarial Core Principles · Actuarial Statistics · Bayesian inference: priors, posteriors, loss functions and credible intervals

A posterior distribution for a claim rate λ is Exponential with mean 4 (rate 0.25). Under absolute error loss L(λ,a) = |λ − a|, the Bayes estimate is the posterior median. What is it, to two decimal places?

The Bayes estimate is 2.77. Absolute error loss gives the posterior median. For an exponential distribution with rate 0.25, the median is ln2 divided by 0.25, which equals 4 × 0.6931 = 2.77. The mean of 4 is the quadratic loss answer.

  1. A4.00
  2. B2.77Correct
  3. C2.00
  4. D1.39
  5. 5.55

Explanation

The median m solves 1 − exp(−0.25m) = 0.5, so m = ln2/0.25 = 4 ln2 = 2.77. The value 4 is the mean, which is optimal under quadratic loss, not absolute loss.

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