iccv2019论文全集8601-pc-fairness-a-unified-framework-for-measuring-causality-based-fairness

1、PC-Fairness: A Unifi ed Framework for Measuring Causality-based Fairness Yongkai Wu University of Arkansas Lu Zhang University of Arkansas Xintao Wu University of Arkansas Hanghang Tong University of Illinois at Urbana-Champaign Abstract

2、 A recent trend of fair machine learning is to defi ne fairness as causality-based notions which concern the causal connection between protected attributes and decisions. However, one common challenge of all causality-based fairness notions is identifi ability, i.e., whether they can be uniquely mea

3、sured from observational data, which is a critical barrier to applying these notions to real-world situations. In this paper, we develop a framework for measuring different causality-based fair- ness. We propose a unifi ed defi nition that covers most of previous causality-based fairness notions, na

4、mely the path-specifi c counterfactual fairness (PC fairness). Based on that, we propose a general method in the form of a constrained opti- mization problem for bounding the path-specifi c counterfactual fairness under all unidentifi able situations. Experiments on synthetic and real-world datasets

5、 show the correctness and effectiveness of our method. 1Introduction Fair machine learning is now an important research fi eld which studies how to develop predictive machine learning models such that decisions made with their assistance fairly treat all groups of people irrespective of their protec

6、ted attributes such as gender, race, etc. A recent trend in this fi eld is to defi ne fairness as causality-based notions which concern the causal connection between protected attributes and decisions. Based on Pearls structural causal models 8, a number of causality-based fairness notions have been

7、 proposed for capturing fairness in different situations, including total effect 19, 16, 20, direct/indirect discrimination 19, 16, 7, 20, and counterfactual fairness 5, 14, 15, 9. One common challenge of all causality-based fairness notions is identifi ability, i.e., whether they can be uniquely me

8、asured from observational data. As causality-based fairness notions are defi ned based on different types of causal effects, such as total effect on interventions, direct/indirect discrimination on path-specifi c effects, and counterfactual fairness on counterfactual effects, their identifi ability

9、depends on the identifi ability of these causal effects. Unfortunately, in many situations these causal effects are in general unidentifi able, referred to as unidentifi able situations 12 . Identifi ability is a critical barrier for the causality-based fairness to be applied to real applications. I

10、n previous works, simplifying assumptions are proposed to evade this problem 5,19,4 . However, these simplifi cations may severely damage the performance of predictive models. In 20 the authors propose a method to bound indirect discrimination as the path-specifi c effect in unidentifi able situatio

11、ns, and in 14 a method is proposed to bound counterfactual fairness. Nevertheless, the tightness of these methods is not analyzed. In addition, it is not clear whether these methods can be applied to other unidentifi able situations, and more importantly, a combination of multiple unidentifi able si

12、tuations. 33rd Conference on Neural Information Processing Systems (NeurIPS 2019), Vancouver, Canada. In this paper, we propose a framework for handling different causality-based fairness notions. We fi rst propose a general representation of all types of causal effects, i.e., the path-specifi c cou

13、nterfactual effect, based on which we defi ne a unifi ed fairness notion that covers most previous causality-based fairness notions, namely the path-specifi c counterfactual fairness (PC fairness). We summarize all unidentifi able situations that are discovered in the causal inference literature. Th

14、en, we develop a constrained optimization problem for bounding the PC fairness, which is motivated by the method proposed in 2 for bounding confounded causal effects. The key idea is to parameterize the causal model using so-called response-function variables, whose distribution captures all randomn

15、ess encoded in the causal model, so that we can explicitly traverse all possible causal models to fi nd the tightest possible bounds. In the experiments, we evaluate the proposed method and compare it with previous bounding methods using both synthetic and real-world datasets. The results show that

16、our method is capable of bounding causal effects under any unidentifi able situation or combinations. When only path-specifi c effect or counterfactual effect is considered, our method provides tighter bounds than methods in 20 or 14. The proposed framework settles a general theoretical foundation f

17、or causality-based fairness. We make no assumption about the hidden confounders so that hidden confounders are allowed to exist in the causal model. We also make no assumption about the data generating process and whether the observation data is generated by linear or non-linear functions would not

18、introduce bias into our results. We only assume that the causal graph is given, which is a common assumption in structural causal models. Relationship to other work.In 3 , the author introduces the term “path-specifi c counterfactual fairness”, which states that a decision is fair toward an individu

19、al if it coincides with the one that would have been taken in a counterfactual world in which the sensitive attribute along the unfair pathways were different. They develop a correction method called PSCF for eliminating the individual-level unfair information contained in the observations while ret

20、aining fair information. Compared to 3 , we formally defi ne a general fairness notion which, besides the individual-level fairness, is also applied to fairness in any sub-group of the population. In addition, we further consider the identifi ability issue in causal inference that is inevitably brou

21、ght by conditioning on the individual level. Unidentifi able situation means that there exist two causal models which exactly agree with the same observational distribution (hence cannot be distinguished using statistic methods such as maximum likelihood), but lead to very different causal effects.

22、In our paper, we address various unidentifi able situations by developing a general bounding method. The authors in 6 study the conditional path-specifi c effect and develop a complete identifi cation algorithm with the application to the problem of algorithmic fairness. Similar to our proposed noti

23、on, their notion is also quantifi ed via conditional distributions over the interventional variant. However, the conditional path-specifi c effect generalizes the conditional causal effect, where the factual condition is assumed to be “non-contradictory” (such as age in measuring the effect of smoki

24、ng on lung cancer) 12. The path-specifi c counterfactual effect, on the other hand, generalizes the counterfactual effect, where the factual condition can be contradictory to the observation. Formally, in the conditional path- specifi c effect, the condition is performed on the pre-intervention dist

25、ribution, but in the path-specifi c counterfactual effect, the condition is performed on the post-intervention distribution. 2Preliminaries In our notations, an uppercase denotes a variable, e.g.,X; a bold uppercase denotes a set of variables, e.g., X; and a lowercase denotes a value or a set of val

26、ues of the variables, e.g., x and x. 2.1Causal Model and Causal Graph Defi nition 1(Structural Causal Model 8).A structural causal modelMis represented by a quadriple hU,V,F,P(U)i where 1. U is a set of exogenous variables that are determined by factors outside the model. 2. P(U) is a joint probabil

27、ity distribution defi ned over U. 3. V is a set of endogenous variables that are determined by variables in U V. 4. Fis a set of structural equations fromU VtoV . Specifi cally, for eachV V, there is a functionfV Fmapping fromU (VV )toV, i.e.,v = fV(paV,uV), wherepaVis a realization of a set of endo

28、genous variablesPAV V Vthat directly determinesV, and uVis a realization of a set of exogenous variables that directly determines V . 2 XY UXUY correlated XY XY UXUY independent XY A Markovian modelA semi-Markovian model Figure 1: Causal graphs of a Markovian model and a semi-Markovian models In gen

29、eral,fV()can be an equation of any type. In some cases, people may assume thatfV()is of a specifi c type, e.g., the nonlinear additive function ifv = fV(paV) + uV. On the other hand, if all exogenous variables inUare assumed to be mutually independent, then the causal model is called a Markovian mod

30、el; otherwise, it is called a semi-Markovian model. In this paper, we dont make assumptions about the type of equations and independence relationships among exogenous variables. The causal modelMis associated with a causal graphG = hV,EiwhereVis a set of nodes andEis a set of edges. Each node ofVcor

31、responds to a variable ofVinM. Each edge inE, denoted by a directed arrow, points from a nodeX U Vto a different nodeY ViffYuses values of Xas input. A causal path fromXtoYis a directed path which traces arrows directed fromXto Y . The causal graph is usually simplifi ed by removing all exogenous va

32、riables from the graph. In a Markovian model, exogenous variables can be directly removed without loss of information. In a semi-Markovian model, after removing exogenous variables we also need to add dashed bi-directed edges between the children of correlated exogenous variables to indicate the exi

33、stence of unobserved common cause factors, i.e., hidden confounders. Examples are demonstrated in Figure 1. 2.2Causal Effects Quantitatively measuring causal effects in the causal model is facilitated with thedo-operator 8 which forces some variableXto take certain valuex, formally denoted bydo(X =

34、x)ordo(x). In a causal modelM, the interventiondo(x) is defi ned as the substitution of structural equation X = fX(PAX,UX)withX = x. For an observed variableY(Y 6= X) which is affected by the intervention, its interventional variant is denoted byYx. The distribution ofYx, also referred to as the pos

35、t-intervention distribution of Y under do(x), is denoted by P(Yx= y) or simply P(yx). By using the do-operator, the total causal effect is defi ned as follows. Defi nition 2(Total Causal Effect 8).The total causal effect of the value change ofXfromx0tox1 on Y = y is given by TCE(x1,x0) = P(yx1) P(yx

36、0). The total causal effect is defi ned as the effect ofXonYwhere the intervention is transferred along all causal paths fromXtoY. If we force the intervention to be transferred only along a subset of all causal paths fromXtoY , the causal effect is then called the path-specifi c effect, defi ned as

37、 follows. Defi nition 3 (Path-specifi c Effect 1).Given a causal path set, the -specifi c effect of the value change of X from x0to x1on Y = y through (with reference x0) is given by PE(x1,x0) = P(yx1|,x0| ) P(yx0), whereP(Yx1|,x0| )represents the post-intervention distribution ofYwhere the effect o

38、f intervention do(x1)is transmitted only alongwhile the effect of reference interventiondo(x0)is transmitted along the other paths. Defi nition 2 and 3 consider the average causal effect over the entire population without any prior observations. If we have certain observations about a subset of attr

39、ibutesO = oand use them as con- ditions when inferring the causal effect, then the causal inference problem becomes a counterfactual inference problem meaning that the causal inference is performed on the sub-population specifi ed byO = oonly. Symbolically, the distribution ofYxconditioning on factu

40、al observationO = ois denoted by P(yx |o). The counterfactual effect is defi ned as follows. Defi nition 4(Counterfactual Effect 12).Given a factual conditionO = o, the counterfactual effect of the value change of X from x0to x1on Y = y is given by CE(x1,x0|o) = P(yx1|o) P(yx0|o). 3 Table 1: Connect

41、ion between previous fairness notions and PC fairness DescriptionReferencesRelating to PC fairness Total effect19, 16O = and = (System) Direct discrimination19, 7, 16O = or S and = d= S Y (System) Indirect discrimination19, 7, 16O = or S and = i Individual direct discrimination17O = S,X and = d= S Y

42、 Group direct discrimination18O = Q = PAYS and = d= S Y Counterfactual fairness5, 9, 14O = S,X and = Counterfactual error rate15O = S,Y and = dor i 3 Path-specifi c Counterfactual Fairness In this section, we defi ne a unifi ed fairness notion for representing different causality-based fairness noti

43、ons. The key component of our notion is a general representation of causal effects. Consider an intervention onXwhich is transmitted along a subset of causal pathstoY, conditioning on observation O = o. Based on that, we defi ne path-specifi c counterfactual effect as follows. Defi nition 5 (Path-sp

44、ecifi c Counterfactual Effect).Given a factual conditionO = oand a causal path set , the path-specifi c counterfactual effect of the value change ofXfromx0tox1onY = y through (with reference x0) is given by PCE(x1,x0|o) = P(yx1|,x0| |o) P(yx0|o). In the context of fair machine learning, we useS s+,s

45、to denote the protected attribute, Y y+,y+to denote the decision, andXto denote a set of non-protected attributes. The underlying mechanism of the population over the spaceS X Yis represented by a causal model M, which is associated with a causal graphG. A historical datasetDis drawn from the popula

46、tion, which is used to construct a predictorh : X,S Y. The causal model for the population over space S X Ycan be considered the same asMexcept that functionfYis replaced with a predictorh. We use to denote all causal paths from S to Y in the causal graph. Then, we defi ne the path-specifi c counter

47、factual fairness based on Defi nition 5. Defi nition 6 (Path-specifi c Counterfactual Fairness (PC Fairness).Given a factual condition O = owhereO S,X,Y and a causal path set, predictor Yachieves the PC fairness ifPCE(s1,s0|o) = 0wheres1,s0 s+,s. We also say that Yachieves the-PC fairness if ? ?PCE(

48、s1,s0|o)? . We show that previous causality-based fairness notions can be expressed as special cases of the PC fairness. Their connections are summarised in Table 1, wheredcontains the direct edge fromSto Y, andiis a path set that contains all causal paths passing through any redlining attributes (i

49、.e., a set of attributes inX that cannot be legally justifi ed if used in decision-making). Based on whetherO equalsor not, the previous notions can be categorized into the ones that deal with the system level (O = ) and the ones that have certain conditions (O 6= ). Based on whetherequalsor not, th

50、e previous notions can be categorized into the ones that deal with the total causal effect ( = ), the ones that consider the direct discrimination ( = d), and the ones that consider the indirect discrimination ( = i). In addition to unifying the existing notions, the notion of PC fairness also resol

51、ves new types of fairness that the previous notions cannot do. One example is individual indirect discrimination, which means discrimination along the indirect paths for a particular individual. Individual indirect discrimination has not been studied yet in the literature, probably due to the diffi

52、culty in defi nition and identifi cation. However, it can be directly defi ned and analyzed using PC fairness by letting O = S,X and = i. 4 Measuring Path-specifi c Counterfactual Fairness In this section, we develop a general method for bounding the path-specifi c counterfactual effect in any unide

53、ntifi able situation. In the causal inference fi eld, researchers have studied the reasons 4 XY Figure 2: The “bow graph”. XWY = X W Z Y Z Figure 3: The “kite graph”. X YYx x Figure 4: The “w graph”. for unidentifi ability under different cases. WhenO = and , the reason for unidentifi ability can be

54、 the existence of the “kite graph” (see Figure 3) in the causal graph 1. WhenO 6= and = , the reason for unidentifi ability can be the existence of the “w graph” (see Figure 4) 11. In any situation, as long as there exists a “hedge graph” (where the simplest case is the “bow graph” as shown in Figur

55、e 2), then the causal effect is unidentifi able 12 . Obviously, all above unidentifi able situations can exist in the path-specifi c counterfactual effect. Our method is motivated by 2 which formulates the bounding problem as a constrained optimization problem. The general idea is to parameterize th

56、e causal model and use the observational distribution P(V) to impose constraints on the parameters. Then, the path-specifi c counterfactual effect of interest is formulated as an objective function of maximization or minimization for estimating its upper or lower bound. The bounds are guaranteed to

57、be tight as we traverse all possible causal models when solving the optimization problem. Thus, a byproduct of the method is a unique estimation of the path-specifi c counterfactual effect in the identifi able situation. For presenting our method, we fi rst introduce a key concept called the respons

58、e-function variable. 4.1Response-function Variable Response-function variables are proposed in 2 for parameterizing the causal model. Consider an arbitrary endogenous variable denoted byV V, its endogenous parents denoted byPAV, its exogenous parents denoted byUV, and its associated structural funct

59、ion in the causal model denoted byv = fV(paV,uV). In general,UVcan be a variable of any type with any domain size, andfVcan be any function, making the causal model very diffi cult to be handled. However, we can note that, for each particular valueuVofUV, the functional mapping fromPAVtoVis a particular determinist