1 Introduction
Risk assessment instruments are gaining increasing popularity within the criminal justice system, with versions of such instruments being used or considered for use in pretrial decisionmaking, parole decisions, and in some states even sentencing [1, 2, 3]. In each of these cases, a highrisk classification—particularly a highrisk misclassification—may have a direct adverse impact on a criminal defendant’s outcome. If the use of RPI’s is to become commonplace, it is especially important to ensure that the instruments are free from discriminatory biases that could result in unethical practices and inequitable outcomes for different groups.
In a recent widely popularized investigation conducted by a team at ProPublica, Angwin et al. [4] studied an RPI called COMPAS^{a}^{a}aCOMPAS [5]
is a risk assessment instrument developed by Northpointe Inc.. Of the 22 scales that COMPAS provides, the Recidivism risk and Violent Recidivism risk scales are the most widely used. The empirical results in this paper are based on decile scores coming from the COMPAS Recidivism risk scale.
, concluding that it is biased against black defendants. The authors found that the likelihood of a nonrecidivating black defendant being assessed as high risk is nearly twice that of white defendants. Similarly, the likelihood of a recidivating black defendant being assessed as low risk is nearly half that of white defendants. In technical terms, these findings indicate that the COMPAS instrument has considerably higher false positive rates and lower false negative rates for black defendants than for white defendants.ProPublica’s analysis has met with much criticism from both the academic community and from the Northpointe corporation. Much of the criticism has focussed on the particular choice of fairness criteria selected for the investigation. Flores et al. [6] argue that the correct approach for assessing RPI bias is instead to check for calibration, a fairness criterion that they show COMPAS satisfies. Northpointe in their response[7] argue for a still different approach that checks for a fairness criterion termed predictive parity, which they demonstrate COMPAS also satisfies. We provide precise definitions and a more indepth discussion of these and other fairness criteria in Section 2.1.
In this paper we show that the differences in false positive and false negative rates cited as evidence of racial bias by Angwin et al. [4] are a direct consequence of applying an RPI that that satisfies predictive parity to a population in which recidivism prevalence^{b}^{b}bPrevalence, also termed the base rate, is the proportion of individuals who recidivate in a given population. differs across groups. Our main contribution is twofold. (1) First, we make precise the connection between the predictive parity criterion and error rates in classification. (2) Next, we demonstrate how using an RPI that has different false postive and false negative rates between groups can lead to disparate impact when individuals assessed as high risk receive stricter penalties. Throughout our discussion we use the term disparate impact to refer to settings where a penalty policy has unintended disproportionate adverse impact on a particular group.
It is important to bear in mind that fairness itself—along with the notion of disparate impact—is a social and ethical concept, not a statistical one. A risk prediction instrument that is fair with respect to particular fairness criteria may nevertheless result in disparate impact depending on how and where it is used. In this paper we consider hypothetical use cases in which we are able to directly connect particular fairness properties of an RPI to a measure of disparate impact. We present both theoretical and empirical results to illustrate how disparate impact can arise.
1.1 Outline of paper
We begin in Section 2 by providing some background on several of the different fairness criteria that have appeared in recent literature. We then proceed to demonstrate that an instrument that satisfies predictive parity cannot have equal false positive and negative rates across groups when the recidivism prevalence differs across those groups. In Section 3 we analyse a simple risk assessmentbased sentencing policy and show how differences in false positive and false negative rates can result in disparate impact under this policy. In Section 3.3 we back up our theoretical analysis by presenting some empirical results based on the data made available by the ProPublica investigators. We conclude with a discussion of the issues that biased data presents for the arguments put forth in this paper.
1.2 Data description and setup
The empirical results in this paper are based on the Broward County data made publicly available by ProPublica[8]. This data set contains COMPAS recidivism risk decile scores, 2year recidivism outcomes, and a number of demographic and crimerelated variables on individuals who were scored in 2013 and 2014. We restrict our attention to the subset of defendants whose race is recorded as AfricanAmerican () or Caucasian ().^{c}^{c}cThere are 6 racial groups represented in the data. 85% of individuals are either AfricanAmerican or Caucasian. After applying the same data preprocessing and filtering as reported in the ProPublica analysis, we are left with a data set on individuals, of whom are AfricanAmerican and are Caucasian.
2 Assessing fairness
2.1 Background
We begin by with some notation. Let denote the risk score based on covariates , with higher values of corresponding to higher levels of assessed risk. We will interchangeably refer to as a score or an instrument. For simplicity, our discussion of fairness criteria will focus on a setting where there exist just two groups. We let denote the group to which an individual belongs, and do not preclude from being one of the elements of . We denote the outcome indicator by , with indicating that the given individual goes on to recidivate. Lastly, we introduce the quantity , which denotes the highrisk score threshold. Defendants whose score exceeds will be referred to as highrisk, while the remaining defendants will be referred to as lowrisk.
With this notation in hand, we now proceed to define and discuss several fairness criteria that commonly appear in the literature, beginning with those mentioned in the introduction. We indicate cases where a given criterion is known to us to also commonly appear under some other name. All of the criteria presented below can also be assessed conditionally by further conditioning on some covariates in . We discuss this point in greater detail in Section 3.1.
Definition 1 (Calibration).
A score is said to be wellcalibrated if it reflects the same likelihood of recidivism irrespective of the individuals’ group membership. That is, if for all values of ,
(2.1) 
Within the educational and psychological testing and assessment literature, the notion of calibration features among the widely accepted and adopted standards for empirical fairness assessment. In this literature, an instrument that is wellcalibrated is referred to as being free from predictive bias. This criterion has recently been applied to the PCRA^{d}^{d}dThe Post Conviction Risk Assessment (PCRA) tool was developed by the Administrative Office of the United States Courts for the purpose of improving “the effectiveness and efficiency of postconviction supervision”[9] instrument, with initial findings suggesting that calibration is satisfied with respect race[10, 11], but not with respect to gender[12]. In their response to the ProPublica investigation, Flores et al. [6]
verify that COMPAS is wellcalibrated using logistic regression modeling.
Definition 2 (Predictive parity).
A score satisfies predictive parity at a threshold if the likelihood of recidivism among highrisk offenders is the same regardless of group membership. That is, if,
(2.2) 
Predictive parity at a given threshold amounts to requiring that the positive predictive value
(PPV) of the classifier
be the same across groups. While predictive parity and calibration look like very similar criteria, wellcalibrated scores can fail to satisfy predictive parity at a given threshold. This is because the relationship between (2.2) and (2.1) depends on the conditional distribution of , which can differ across groups in ways that result in PPV imbalance. In the simple case where itself is binary, a score that is wellcalibrated will also satisfy predictive parity. Northpointe’s refutation[7] of the ProPublica analysis shows that COMPAS satisfies predictive parity for threshold choices of interest.Definition 3 (Error rate balance).
A score satisfies error rate balance at a threshold if the false positive and false negative error rates are equal across groups. That is, if,
(2.3)  
(2.4) 
where the expressions in the first line are the groupspecific false positive rates, and those in the second line are the groupspecific false negative rates.
ProPublica’s analysis considered a threshold of , which they showed leads to considerable imbalance in both false positive and false negative rates. While this choice of cutoff met with some criticism, we will see later in this section that error rate imbalance persists—indeed, must persist—for any choice of cutoff at which the score satisfies the predictive parity criterion. Error rate balance is also closely connected to the notions of
equalized odds
and equal opportunity as introduced in the recent work of Hardt et al. [13].Definition 4 (Statistical parity).
A score satisfies statistical parity at a threshold if the proportion of individuals classified as highrisk is the same for each group. That is, if,
(2.5) 
Statistical parity also goes by the name of equal acceptance rates[14] or group fairness[15], though it should be noted that these terms are in many cases not used synonymously. While our discussion focusses primarily on first three fairness criteria, statistical parity is widely used within the machine learning community and may be the criterion with which many readers are most familiar[16, 17]. Statistical parity is wellsuited to contexts such as employment or admissions, where it may be desirable or required by law or regulation to employ or admit individuals in equal proportion across racial, gender, or geographical groups. It is, however, a difficult criterion to motivate in the recidivism prediction setting, and thus will not be further considered in this work.
2.2 Further related work
Though the study of discrimination in decision making and predictive modeling is rapidly evolving, it also has a long and rich multidisciplinary history. Romei and Ruggieri [18] provide an excellent overview of some of the work in this broad subject area. The recent work of Barocas and Selbst [19] offers a broad examination of algorithmic fairness framed within the context of antidiscrimination laws governing employment practices. HannahMoffat [20], Skeem [21], and Monahan and Skeem [22] examine legal and ethical issues relating specifically to the use of risk assessment instruments in sentencing, citing the potential for race and gender discrimination as a major concern.
In work concurrent with our own, several other researchers have also investigated the compatibility of different notions of fairness. Kleinberg et al. [23] show that calibration cannot be satisfied simultaneously with the fairness criteria of balance for the negative class and balance for the positive class. Translated into the present context, the latter criteria require that the average score assigned to nonrecidivists (the negative class) should be the same for both groups, and that the same should hold among recidivists (the positive class). The work of CorbettDavies et al. [24] closely parallels the results that we present in Section 2.3, reaching the same conclusion regarding the incompatibility of predictive parity and error rate balance in the setting of unequal prevalence.
2.3 Predictive parity, false positive rates, and false negative rates
. Error bars represent 95% confidence intervals. These Figures confirm that COMPAS is (approximately) wellcalibrated, satisfies predictive parity for highrisk cutoff values of 4 or higher, but fails to have error rate balance.
In this section we present our first main result, which establishes that predictive parity is incompatible with error rate balance when prevalence differs across groups. To better motivate the discussion, we begin by presenting an empirical fairness assessment of the COMPAS RPI. Figure 1 shows plots of the observed recidivism rates and error rates corresponding to the fairness notions of calibration, predictive parity, and error rate balance. We see that the COMPAS RPI is (approximately) wellcalibrated, and also satisfies predictive parity provided that the highrisk cutoff is or greater. However, COMPAS fails on both false positive and false negative error rate balance across the range of highrisk cutoffs.
Angwin et al. [4] focussed on a highrisk cutoff of for their analysis, which some critics have argued is too low, suggesting that is more suitable. As can be seen from Figures 0(c) and 0(d), significant error rate imbalance persists at this cutoff as well. Moreover, the error rates achieved at so high a cutoff are at odds with evidence suggesting that the use of RPI’s is of interest in settings where false negatives have a higher cost than false positives, with relative cost estimates ranging from 2.6 to upwards of 15. [25, 26]
As we now proceed to show, the error rate imbalance exhibited by COMPAS is not a coincidence, nor can it be remedied in the present context. When the recidivism prevalence–i.e., the base rate —differs across groups, any instrument that satisfies predictive parity at a given threshold must have imbalanced false positive or false negative errors rates at that threshold. To understand why predictive parity and error rate balance are mutually exclusive in the setting of unequal recidivism prevalence, it is instructive to think of how these quantities are all related.
Given a particular choice of
, we can summarize an instrument’s performance in terms of a confusion matrix, as shown in Table
1 below.LowRisk  HighRisk  

TN  FP  
FN  TP 
All of the fairness metrics presented in Section 2.1 can be thought of as imposing constraints on the values (or the distribution of values) in this table. Another constraint—one that we have no direct control over—is imposed by the recidivism prevalence within groups. It is not difficult to show that the prevalence (), positive predictive value (), and false positive and negative error rates (, ) are related via the equation
(2.6) 
From this simple expression we can see that if an instrument satisfies predictive parity—that is, if the PPV is the same across groups—but the prevalence differs between groups, the instrument cannot achieve equal false positive and false negative rates across those groups.
This observation enables us to better understand why we observe such large discrepancies in FPR and FNR between black and white defendants in Figure 1. The recidivism rate among black defendants in the data is 51%, compared to 39% for White defendants. Thus at any threshold where the COMPAS RPI satisfies predictive parity, equation (2.6) tells us that some level of imbalance in the error rates must exist. Since not all of the fairness criteria can be satisfied at the same time, it becomes important to understand the potential impact of failing to satisfy particular criteria. This question is explored in the context of a hypothetical riskbased sentencing framework in the next section.
3 Assessing impact
In this section we show how differences in false positive and false negative rates can result in disparate impact under policies where a highrisk assessment results in a stricter penalty for the defendant. Such situations may arise when risk assessments are used to inform bail, parole, or sentencing decisions. In Pennsylvania and Virginia, for instance, statutes permit the use of RPI’s in sentencing, provided that the sentence ultimately falls within accepted guidelines[1]. We use the term “penalty” somewhat loosely in this discussion to refer to outcomes both in the pretrial and postconviction phase of legal proceedings. For instance, even though pretrial outcomes such as the amount at which bail is set are not punitive in a legal sense, we nevertheless refer to bail amount as a “penalty” for the purpose of our discussion.
There are notable cases where RPI’s are used for the express purpose of informing risk reduction efforts. In such settings, individuals assessed as high risk receive what may be viewed as a benefit rather than a penalty. The PCRA score, for instance, is intended to support precisely this type of decisionmaking at the federal courts level [11]. Our analysis in this section specifically addresses use cases where highrisk individuals receive stricter penalties.
To begin, consider a setting in which guidelines indicate that a defendant is to receive a penalty . A very simple riskbased approach, which we will refer to as the MinMax^{e}^{e}eThe term MinMax as used throughout this paper has no intended connection the decisiontheoretic notion of minimax decision rules. Min and Max in this context refer to the minimum and maximum allowable sentences as stipulated by sentencing guidelines. policy, would be to assign penalties as follows:
(3.1) 
In this simple setting, we can precisely characterize the extent of disparate impact in terms of recognizable quantities. Our analysis will focus on the quantity
which is the expected difference in sentence duration between defendants in different groups, with potentially different outcomes . is taken to serve as our the measure of disparate impact.
Proposition 3.1.
The expected difference in penalty under the MinMax policy is given by
A proof can be found in Appendix A. We will discuss two immediate Corollaries of this result.
Corollary 3.1 (NonRecidivists).
Among individuals who do not recidivate, the difference in average penalty under the MinMax policy is
(3.2) 
where denotes the false positive rate among individuals in group .
Corollary 3.2 (Recidivists).
Among individuals who recidivate, the difference in average penalty under the MinMax policy is
(3.3) 
where denotes the false negative rate among individuals in group .
When using an RPI that satisfies predictive parity in populations where recidivism prevalence differs across groups, it will generally be the case that the higher recidivism prevalence group will have a higher FPR and lower FNR. From equations (3.2) and (3.3), we can see that this would on average result in greater penalties for defendants in the higher prevalence group, both among recidivists and nonrecidivists.
An interesting special case to consider is one where . This could arise in sentencing decisions for offenders convicted of lowseverity crimes who have good prior records. In such cases, socalled restorative sanctions may be imposed as an alternative to a period of incarceration. If we further take , then
, which can be interpreted as the probability that a defendant receives a sentence imposing some period of incarceration.
It is easy to see that in such settings a nonrecidivist in group is times more likely to be incarcerated compared to a nonrecidivist in group .^{f}^{f}fWe are overloading notation in this expression: Here, , similarly for . This naturally raises the question of whether overall differences in error rates are observed to persist across more granular subpopulations, such as the subset of individuals eligible for restorative sanctions. We explore this question in the section below.
3.1 Conditioning on other covariates
One might expect that differences in false positive rates are largely attributable to the subset of defendants who are charged with more serious offenses and who have a larger number of prior arrests/convictions. While it is true that the false positive rates within both racial groups are higher for defendants with worse criminal histories, considerable betweengroup differences in these error rates persist across low prior count subgroups. Figure 2 shows plots of false positive rates across different ranges of prior count for all defendants and also for the subset charged with a misdemeanor offense, which is the lowest severity criminal offense category. As one can see, differences in false positive rates between Black defendants and White defendants persist across prior record subgroups.
In general, all of the theoretical results presented in this section extend to the setting where we further condition on the covariates . The main difference is that all classification metrics would need to be evaluated conditional on . For instance, assuming that and are constant on a set Corollary 3.1 would say that the difference in average penalty under the MinMax policy among nonrecidivists for whom is given by
(3.4)  
(3.5) 
The false positive rates shown in Figure 2(a) correspond precisely to the quantities for choices of given by different prior record count bins. The leftmost bars correspond to taking . Similarly the leftmost bars in Figure 2(a) correspond to taking . In Appendix B
we present a logistic regression analysis showing that significant differences in false positive rates persist even after adjusting for a number of other recidivismrelated covariates.
3.2 Connections to measures of differences in distribution
In their analysis of the PCRA instrument, Skeem and Lowenkamp [11] remark that some applications of the risk score could create disparate impact due to differences in the score distributions between black and white offenders. To summarize the distributional difference in scores between the two groups, the authors report a Cohen’s of , with a corresponding nonoverlap of 13.5%. A natural question to ask is whether the level of disparity in sentence duration, , is in some sense closely related to such measures of distributional difference. With a small generalization of the % nonoverlap measure, we can answer this question in the affirmative.
The % nonoverlap of two distributions is generally calculated assuming both distributions are normal, and thus has a onetoone correspondence to Cohen’s [27].^{g}^{g}g, where
is a pooled estimate of standard deviation.
However, as we can see from Figure 3, the COMPAS decile score is far from being normally distributed in either group. A more reasonable way to calculate % nonoverlap in such cases is to note that in the Gaussian case % nonoverlap is equivalent to the total variation distance. Letting
denote the score distribution among individuals in group with recidivism outcome , one can establish the following sharp bound on .Proposition 3.2 (Percent overlap bound).
Under the MinMax policy,
This result is simple to understand. When there is some nonoverlap between the score distributions for two groups, the worst case scenario is that the nonoverlap is entirely due to mass shifting from scores below to those above . In such cases, the inequality becomes an equality.
3.3 Empirical results
In this section we present some empirical results based on two hypothetical sentencing rules: the MinMax rule introduced in the previous section, and the Interpolation rule, which we will introduce below. Though the offenders in our data set come from Broward County, Florida, our empirical analysis is modelled on the sentencing guidelines of the State of Pennsylvania.
The penalty ranges and are selected by approximately matching each offender’s charge degree (M2  F1) to a sentence range in Pennsylvania’s Basic Sentencing Matrix (PA Code §303.16). This matrix provides sentence ranges based on the charge degree for the current offense and the defendant’s prior record score (0  5+). We do not have enough information in the Broward County data to reliably assign a prior record score for each individual. Our results are based on using the sentencing range corresponding to a prior record score of for all defendants in the data.
Figure 4 shows the expected sentences for black and white defendants broken down by observed recidivism outcome. The axis in these figures is taken to be the offense gravity score, which for the purpose of this analysis is mapped to charge degree as indicated in Table 2.
Offense gravity score  2  3  5  7  8 

Charge Degree  (M2)  (M1)  (F3)  (F2)  (F1) 
Results are shown for both the MinMax policy introduced earlier in this section, and the Interpolation policy, which is given by
(3.6) 
Unlike the MinMax policy, which is based on the coarsened score, the Interpolation policy assigns sentences by linearly interpolating between and based on the assigned decile score. We see that under both policies there are consistent trends in the expected sentences. Black defendants are observed to receive higher sentences than white defendants both within the nonrecidivating subgroup and the recidivating subgroup (except in the F1 charge degree category, where sample sizes are small and results are nonsignificant). Since white defendants have higher false negative rates and lower false positive rates than black defendants, the empirical results are consistent with the theoretical results presented earlier in this section.
4 Revisiting predictive parity
In this final section we revisit the notion of predictive parity and further discuss its implications for general classifiers. We know from equation (2.6) that when the positive predictive values are constrained to be equal but the prevalences differ across groups, the false positive and false negative rates cannot both be equal across those groups. While we have no direct control over recidivism prevalence, we do have some control over the PPV and error rates of our classifiers. At least in principle, we are free to tune our classifiers in any of the following ways:

Allow unequal false negative rates to retain equal PPV’s and achieve equal false positive rates

Allow unequal false positive rates to retain equal PPV’s and achieve equal false negative rates

Allow unequal PPV’s to achieve equal false positive and false negative rates
Figure 5 helps to put these tradeoffs into perspective. From (2.6), we can see that FPR is a linear function of FNR under constraints on PPV and . This means that, if PPV is fixed at a given value, tuning strategy (i) may require a very large increase in in order to balance FPR. The black line shows feasible combinations of when is forced to equal the observed value . We can see that to get to match , we would need to increase to around , which would be a substantial drop in accuracy. In view of Corollaries 3.1 and 3.2 Strategies (i) and (ii) may generally be undesirable because while they reduce disparate impact for one subgroup (e.g., among nonrecidivists), they may increase it in the other.
The preferred approach, at least in some cases, may be to pursue strategy (iii). This amounts to using a score that does not satisfy predictive parity in the first place, but can also be achieved by allowing the highrisk cutoff to differ across groups. The shaded regions in Figure 5 show feasible values of when we allow to be within some of the observed value of . We can see that even at small values of the feasible region is quite large.
5 Discussion
The primary contribution of this paper was to show how disparate impact can result from the use of a recidivism prediction instrument that is known to satisfy the fairness criterion of predictive parity. Our analysis focussed on the simple setting where a binary risk assessment was used to inform a binary penalty policy. While all of the formulas have natural analogs in the nonbinary score and penalty setting, we find that many of the salient features are already present in the analysis of the simpler binarybinary problem.
A key limitation of our analysis stems from potential biases in the observed data that may affect our ability to draw valid inferences concerning the fairness of an RPI. Throughout this paper we have implicitly operated under the assumption that the observed recidivism outcome is a suitable outcome measure for the purpose of assessing the fairness properties of a recidivism prediction instrument. However, the true outcome of interest in this context is reoffense, which is not what we observe. In the latest statistics released by the Federal Bureau of Investigation[28], it is reported that 46% of violent crimes and 19.4% of property crimes were successfully cleared by law enforcement agencies. Many criminal offenders are simply never identified. It is therefore possible that a nonnegligible fraction of the individuals in our data for whom we observed did in truth reoffend. If this is indeed the case, and if there are group differences in the rates at which offenders are caught, the findings of empirical fairness assessments may be misleading. Understanding how such forms of data bias affect the ability to assess instruments with respect to different fairness criteria is a subject of our ongoing research efforts.
6 Conclusion
In closing, we would like to note that there is a large body of literature showing that datadriven risk assessment instruments tend to be more accurate than professional human judgements [29, 30], and investigating whether humandriven decisions are themselves prone to exhibiting racial bias [31, 32]. We should not abandon the datadriven approach on the basis of negative headlines. Rather, we need to work to ensure that the instruments we use are demonstrably free from the kinds of biases that could lead to disparate impact in the specific contexts in which they are to be applied.
Appendix A Proofs
Proof of Proposition 3.1.
To simplify notation, we let denote the event .
∎
Appendix B Covariateadjusted false positive rates
In this section we present the results of a logistic regression analysis that we conducted in order to assess whether the observed differences in false positive rates between black and white defendants can be entirely accounted for by other covariates. We find that adjusting for covariates decreases the gap, but it nevertheless remains large and statistically significant.
For the purpose of this analysis we consider only the subset of defendants who do not recidivate. The outcome variable for the logistic regression is taken to be
where denotes the COMPAS decile score. In this setup, denotes a True Negative and denotes a False Positive. Statistically significant positive coefficient estimates correspond to variables associated with increased likelihood of false positives.
Table 3 shows the results of regressing on race alone. The coefficient of race in this model is large, positive, and statistically significant. Without adjusting for other covariates, the odds that a nonrecidivating Black defendant receives a highrisk assessment are times higher than those of a White defendant.
Table 4 shows the results of regressing on race, age, gender, number of priors, and charge degree. The coefficient of race is smaller than it was in the unadjusted model, but it is nevertheless large and statistically significant. Even after adjusting for these other factors, the odds that a nonrecidivating Black defendant receives a highrisk assessment are times higher than those of a White defendant.
Estimate  Std. Error  z value  Pr(z)  

(Intercept)  1.183  0.061  19.33  0.0000 
raceBlack  0.976  0.077  12.60  0.0000 
Estimate  Std. Error  z value  Pr(z)  

(Intercept)  1.397  0.176  7.92  0.0000 
raceBlack  0.547  0.087  6.30  0.0000 
Age  0.079  0.005  17.48  0.0000 
sexMale  0.291  0.098  2.97  0.0030 
Number of Priors  0.283  0.016  17.78  0.0000 
chargeMisdemeanor  0.109  0.088  1.25  0.2123 
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