03-decision-02-loss-functions.Rmd

## Loss Functions

Quantifying the loss can be tricky, and Table \@ref(tab:loss-functions) summarizes three different examples with three different loss functions.

If you're declaring the average payoff for an insurance claim, and if you are **linear** in how you value money, that is, twice as much money is exactly twice as good, then one can prove that the optimal one-number estimate is the **median** of the posterior distribution. But in different situations, other measures of loss may apply.

If you are advising a patient on his/her life expectancy, it is easy to imagine that large errors are far more problematic than small ones. And perhaps the loss increases as the **square** of how far off your single number estimate is from the truth. For example, if she is told that her average life expectancy is two years, and it is actually ten, then her estate planning will be catastrophically bad, and she will die in poverty. In the case when the loss is proportional to the **quadratic** error, one can show that the optimal one-number estimate is the **mean** of the posterior distribution.

Finally, in some cases, the penalty is 0 if you are exactly correct, but constant if you're at all wrong. This is the case with the old saying that close only counts with horseshoes and hand grenades; i.e., coming close but not succeeding is not good enough. And it would apply if you want a prize for correctly guessing the number of jelly beans in a jar. Here, of course, instead of minimizing expected losses, we want to **maximize the expected gain**. If a Bayesian is in such a situation, then his/her best one-number estimate is the **mode** of his/her posterior distribution, which is the most likely value.

There is a large literature on decision theory, and it is directly linked to risk analysis, which arises in many fields. Although it is possible for frequentists to employ a certain kind of decision theory, it is much more natural for Bayesians.

```{r loss-functions, echo=FALSE}
temp <- matrix(c("Linear","Median",
"Quadratic","Mean",
"0/1","Mode"), nrow=3, byrow=TRUE)

colnames(temp) <- c("Loss","Best Estimate")

knitr::kable(
x = temp, booktabs = TRUE,
caption = "Loss Functions", align = 'c'
)
```

When making point estimates of unknown parameters, we should make the choices that minimize the loss. Nevertheless, the best estimate depends on the kind of loss function we are using, and we will discuss in more depth how these best estimates are determined in the next section.