Posterior Expected Loss

A Posterior Expected Loss is an Expected Value or a Loss Function that is based on a posterior distribution.

• AKA: Bayesian Expected Loss.
  • …
• Counter-Example(s):
  • Posterior Risk,
  • Squared Error Loss,
  • Absolute Loss.
• See: Bayesian Estimator, Bayes Theorem, Bayesian Inference, Prior Distribution, Bayes Envelope Function.

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References

2019

• (Vidakovic, 2019) ⇒ Brani Vidakovic (2019). Handbok 4: https://www2.isye.gatech.edu/~brani/isyebayes/bank/handout4.pdf Retrieved:2019-06-08.
  • QUOTE: Definition 1. Bayesian expected loss is the expectation of the loss function with respect to posterior measure, i.e.,
    [math]\displaystyle{ \rho(a, \pi) = E ^{\theta |X}L(a, \theta) = \int_{\theta} L(\theta, a)\pi(\theta|x)d\theta }[/math]
    The Expected Loss Principle. In comparing two actions [math]\displaystyle{ a_1 = \delta_1(X) }[/math] and [math]\displaystyle{ a_2 = \delta_2(X) }[/math], after data [math]\displaystyle{ X }[/math] had been observed, preferred action is the one for which the posterior expected loss is smaller. An action [math]\displaystyle{ a^∗ }[/math] that minimizes the posterior expected loss is called Bayes action.

2011

• (Wikipedia, 2019) ⇒ http://en.wikipedia.org/wiki/Loss_function#Bayesian_expected_loss
  • In a Bayesian approach, the expectation is calculated using the posterior distribution π^* of the parameter θ: [math]\displaystyle{ \rho(\pi^*,a) = \int_\Theta L(\theta, a) \, \operatorname{d} \pi^* (\theta) }[/math]. One then should choose the action a^* which minimises the expected loss. Although this will result in choosing the same action as would be chosen using the Bayes risk, the emphasis of the Bayesian approach is that one is only interested in choosing the optimal action under the actual observed data, whereas choosing the actual Bayes optimal decision rule, which is a function of all possible observations, is a much more difficult problem.

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