Loss Models: From Data to Decisions, 4th Edition

15.2 Inference and prediction

In one sense the analysis is complete. We begin with a distribution that quantifies our knowledge about the parameter and/or the next observation, and we end with a revised distribution. But we suspect that your boss may not be satisfied if you produce a distribution in response to his or her request. No doubt a specific number, perhaps with a margin for error, is what is desired. The usual Bayesian solution is to pose a loss function.

Definition 15.9 A loss function describes the penalty paid by the investigator when _j is the estimate and θ_j is the true value of the jth parameter.

It is be possible to have a multidimensional loss function l(,θ) that allowed the loss to depend simultaneously on the errors in the various parameter estimates.

Definition 15.10 The Bayes estimate for a given loss function is the one that minimizes the expected loss given the posterior distribution of the parameter in question.

The three most commonly used loss functions are defined as follows.

Definition 15.11 For squared-error loss, the loss function is (all subscripts are dropped for convenience) . For absolute loss, it is . For zero–one loss it is and is 1 otherwise.