The Error Equation

Figure 8: Distribution of errors with bias reduced by half vs. noise reduced by half

The illustration concerns forecasts at the fictional company GoodSell. Amy Simkin wants to reduce noise; her boss wants to reduce bias. Here, bias is average error and noise is the standard deviation of forecasts. Their initial magnitudes are assumed equal.

In panel A, we assume that Amy’s boss decided to do things his way: he found out what the bias was, then somehow managed to reduce it by half (perhaps by providing feedback to the overoptimistic forecasters). Nothing was done about noise. The improvement is visible: the whole distribution of forecasts has shifted closer to the true value.

In panel B, we show what would have happened if Amy had won the argument. Bias is unchanged, but noise is reduced by half. The paradox here is that noise reduction seems to have made things worse. The forecasts are now more concentrated (less noisy) but not more accurate (not less biased). Whereas 84% of forecasts were on one side of the true value, almost all (98%) now err in the direction of overshooting the true value. Noise reduction seems to have made the forecasts more precisely wrong—hardly the sort of improvement for which Amy hoped!

Despite appearances, however, overall error has been reduced just as much in panel B as in panel A. The illusion of deterioration in panel B arises from an erroneous intuition about bias. The relevant measure of bias is not the imbalance of positive and negative errors. It is average error, which is the distance between the peak of the bell curve and the true value. In panel B, this average error has not changed from the original situation—it is still high, at 10%, but not worse. True, the presence of bias is now more striking, because it accounts for a larger proportion of overall error (80% rather than 50%). But that is because noise has been reduced. Conversely, in panel A, bias has been reduced, but noise has not. The net result is that MSE is the same in both panels: because the starting magnitudes are equal, halving either noise or bias produces the same reduction in MSE.

As this example illustrates, MSE conflicts with common intuitions about the scoring of predictive judgments. To minimize MSE, you must concentrate on avoiding large errors. If you measure length, for example, the effect of reducing an error from 11cm to 10cm is 21 times as large as the effect of going from an error of 1cm to a perfect hit. Unfortunately, people’s intuitions in this regard are almost the mirror image of what they should be: people are very keen to get perfect hits and highly sensitive to small errors, but they hardly care at all about the difference between two large errors. Even if you sincerely believe that your goal is to make accurate judgments, your emotional reaction to results may conflict with accuracy as measured by squared error.

Of course, the best solution here would be to reduce both noise and bias. Bias and noise contribute separate terms to the equation, so both deserve attention. In practice, a change intended to reduce one may affect the other. In that regard, if GoodSell decides to reduce noise, the fact that noise reduction makes bias more visible—indeed, impossible to miss—may turn out to be a blessing. Making the remaining bias more visible may help the company address it next.

Admittedly, reducing noise would be less of a priority if bias were much larger than noise. But the GoodSell example offers another lesson worth highlighting. In this simplified model, we have assumed that noise and bias are equal. Given the form of the error equation, their contributions to total error are equal, too: bias accounts for 50% of overall error, and so does noise. Yet, as we have noted, 84% of the forecasters err in the same direction. It takes a bias this large (six out of seven people making mistakes in the same direction!) to have as much effect as noise has. We should not be surprised, therefore, to find situations in which there is more noise than bias.

We illustrated the application of the error equation to a single case, one particular region of GoodSell’s territory. Of course, it is always desirable to carry out a noise audit on multiple cases at once. Nothing changes. The error equation is applied to the separate cases; and an overall equation is obtained by taking the averages of MSE, bias squared and noise squared over the cases. It would have been better for Amy Simkin to obtain multiple forecasts for several regions, either from the same or from different forecasters. Averaging results would give her a more accurate picture of bias and noise in the forecasting system of GoodSell.

The Cost of Noise

The error equation explains why reducing system noise deserves attention alongside reducing statistical bias. (We should emphasize that statistical bias is not a synonym for social discrimination; it is simply the average error in a set of judgments.)

The error equation and the conclusions we have drawn from it depend on the use of MSE as the measure of overall error. MSE measures how closely numerical forecasts and estimates approach the true value, with costs that rise with squared distance. It is useful for this purpose, but it need not represent every practical cost of a prediction error.

Applying the equation to evaluative judgments requires specifying a reference value and accepting squared error as the relevant loss. Those choices are often difficult or disputed. Furthermore, even if errors could be specified, their costs would rarely be symmetrical and would be unlikely to be precisely proportional to their square.

For a company that makes elevators, for example, the consequences of errors in estimating the maximum load of an elevator are obviously asymmetrical: underestimation is costly, but overestimation could be catastrophic. Squared error is similarly irrelevant to the decision of when to leave home to catch a train. For that decision, the consequences of being either one minute late or five minutes late are the same. And when the insurance company of chapter 2 prices policies or estimates the value of claims, errors in both directions are costly, but there is no reason to assume that their costs are equivalent.

These examples highlight the need to specify the roles of predictive and evaluative judgments in decisions. A widely accepted maxim of good decision making is that you should not mix your values and your facts. Good decision making must be based on objective and accurate predictive judgments that are completely unaffected by hopes and fears, or by preferences and values. For the elevator company, the first step would be a neutral calculation of the maximum technical load of the elevator under different engineering solutions. Safety becomes a dominant consideration only in the second step, when an evaluative judgment determines the choice of an acceptable safety margin to set the maximum capacity. (To be sure, that choice will also greatly depend on factual judgments involving, for example, the costs and benefits of that safety margin.) Similarly, the first step in deciding when to leave for the station should be an objective determination of the probabilities of different travel times. The respective costs of missing your train and of wasting time at the station become relevant only in your choice of the risk you are willing to accept.

The same logic applies to much more consequential decisions. A military commander must weigh many considerations when deciding whether to launch an offensive, but much of the intelligence on which the leader relies is a matter of predictive judgment. A government responding to a health crisis, such as a pandemic, must weigh the pros and cons of various options, but no evaluation is possible without accurate predictions about the likely consequences of each option (including the decision to do nothing).

In all these examples, the final decisions require evaluative judgments. The decision makers must consider multiple options and apply their values to make the optimal choice. But the decisions depend on underlying predictions, which should be value-neutral. Their goal is accuracy; MSE is the measure used in this analysis. Predictive judgments will be improved by procedures that reduce noise, as long as they do not increase bias to a larger extent.

Speaking of the Error Equation

The following maxims summarize the authors’ argument. The equal-reduction claim refers to the illustration’s equal starting magnitudes, and improvements are assessed using MSE.

“Oddly, reducing bias and noise by the same amount has the same effect on accuracy.”

“Reducing noise in predictive judgment is always useful, regardless of what you know about bias.”

“When judgments are split 84 to 16 between those that are above and below the true value, there is a large bias—that’s when bias and noise are equal.”

“Predictive judgments are involved in every decision, and accuracy should be their only goal. Keep your values and your facts separate.”

CHAPTER 6

The Analysis of Noise

The illustration above concerns variability in judgments of a single case. When we focus on a single case, all variability of judgment is error, and the two constituents of error are bias and noise. Of course, the judgment systems we are examining, including those involving courts and insurance companies, are designed to deal with different cases and to discriminate among them. Federal judges and claims adjusters would be of little use if they returned the same judgment for all the cases that come their way. Much of the variability in judgments of different cases is intentional.

However, variability in judgments of the same case is still undesirable—it is system noise. As we will show, a noise audit in which the same people make judgments about several cases permits a more detailed analysis of system noise.

A Noise Audit of Sentencing

To illustrate the analysis of noise with multiple cases, we turn to an exceptionally detailed noise audit of sentencing by federal judges. The analysis was published in 1981 as part of the movement toward sentencing reform that we described in chapter 1. The study narrowly focused on sentencing decisions, but the lessons it offers are general and bear on other professional judgments. The goal of the noise audit was to go beyond the vivid but anecdotal evidence of noise assembled by Judge Frankel and others and to “determine the extent of sentencing disparity” more systematically.

The study’s authors developed sixteen hypothetical cases in which the defendant had been found guilty and was to be sentenced. The vignettes depicted either robberies or cases of fraud and differed on six other dimensions, including whether the defendant was a principal or an accomplice in the crime, whether he had a criminal record, whether (for the robbery cases) a weapon had been used, and so on.

The researchers organized carefully structured interviews with a national sample of 208 active federal judges. In the course of ninety minutes, the judges were presented with all sixteen cases and asked to set a sentence.

To appreciate what can be learned from this study, you will find an exercise in visualization helpful. Picture a large table with sixteen columns for the crimes, labeled from A to P, and 208 rows for the judges, labeled 1 to 208. Each cell, from A1 to P208, shows the prison term set for a particular case by a particular judge. Figure 9 illustrates what this 3,328-cell table would look like. To study noise, we will want to focus on the sixteen columns, each of which is a separate noise audit.