A loss function satisfies the definition of a random variable so we can establish a cumulative distribution function and an expected value. However, more commonly, the loss function is expressed as a function of some other random variable. For example, the time that a light bulb operates before failure is a random variable and we can specify the loss, arising from having to cope in the dark and/or replace the bulb, as a function of failure time. For a continuous random variable X with probability density function f and loss function λ, the expected loss (som ...
One of the consequences of Bayesian inference is that in addition to experimental data, the loss function does not in itself wholly determine a decision. What is important is the relationship between the loss function and the prior probability. So it is possible to have two different loss functions which lead to the same decision when the prior probability distributions associated with each compensate for the details of each loss function.
Combining the three elements of the prior probability, the data, and the loss function then allo ...
