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

Which loss function leads to the posterior median as the Bayes estimate of a parameter?

Absolute error loss yields the posterior median as the Bayes estimate. Quadratic loss gives the posterior mean and all-or-nothing loss gives the posterior mode, so only absolute error corresponds to the median.

  1. AQuadratic loss (θ̂ − θ)²
  2. BAbsolute error loss |θ̂ − θ|Correct
  3. CAll-or-nothing (0–1) loss
  4. DCubic loss |θ̂ − θ|³
  5. Loss proportional to 1/θ

Explanation

Minimising expected absolute error gives the median of the posterior. Quadratic loss gives the mean and all-or-nothing loss gives the posterior mode.

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