Developing classification algorithms that are fair with respect to sensitive attributes of the data is an important problem due to the increased deployment of classification algorithms in societ al contexts. Several recent works have focused on studying classification with respect to specific fairness metrics, modeled the corresponding fair classification problem as constrained optimization problems, and developed tailored algorithms to solve them. Despite this, there still remain important metrics for which there are no fair classifiers with theoretical guarantees; primarily because the resulting optimization problem is non-convex. The main contribution of this paper is a meta-algorithm for classification that can take as input a general class of fairness constraints with respect to multiple non-disjoint and multi-valued sensitive attributes, and which comes with provable guarantees. In particular, our algorithm can handle non-convex "linear fractional" constraints (which includes fairness constraints such as predictive parity) for which no prior algorithm was known. Key to our results is an algorithm for a family of classification problems with convex constraints along with a reduction from classification problems with linear fractional constraints to this family. Empirically, we observe that our algorithm is fast, can achieve near-perfect fairness with respect to various fairness metrics, and the loss in accuracy due to the imposed fairness constraints is often small.
The most relevant works to ours in technique are @cite_36 @cite_17 @cite_11 , all of which considered the Bayesian classification model for statistical parity or equalized odds. They either reduce their problem to unconstrained optimization problem by the Lagrangian principle or can be alternately expressed in that form. Our work uses similar techniques but in comparison can handle a wider class of fairness metrics.
@cite_36 @cite_11 @cite_17
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In this paper we present an average-case analysis of the Bayesian classifier, a simple induction algorithm that fares remarkably well on many learning tasks. Our analysis assumes a monotone conjunctive target concept, and independent, noise-free Boolean attributes. We calculate the probability that the algorithm will induce an arbitrary pair of concept descriptions and then use this to compute the probability of correct classification over the instance space. The analysis takes into account the number of training instances, the number of attributes, the distribution of these attributes, and the level of class noise. We also explore the behavioral implications of the analysis by presenting predicted learning curves for artificial domains, and give experimental results on these domains as a check on our reasoning. Sequential techniques can enhance the efficiency of the approximate Bayesian computation algorithm, as in 's (2007) partial rejection control version. While this method is based upon the theoretical works of Del (2006), the application to approximate Bayesian computation results in a bias in the approximation to the posterior. An alternative version based on genuine importance sampling arguments bypasses this difficulty, in connection with the population Monte Carlo method of (2004), and it includes an automatic scaling of the forward kernel. When applied to a population genetics example, it compares favourably with two other versions of the approximate algorithm. Copyright 2009, Oxford University Press. Recent work in text classification has used two different first-order probabilistic models for classification, both of which make the naive Bayes assumption. Some use a multi-variate Bernoulli model, that is, a Bayesian Network with no dependencies between words and binary word features (e.g. Larkey and Croft 1996; Koller and Sahami 1997). Others use a multinomial model, that is, a uni-gram language model with integer word counts (e.g. Lewis and Gale 1994; Mitchell 1997). This paper aims to clarify the confusion by describing the differences and details of these two models, and by empirically comparing their classification performance on five text corpora. We find that the multi-variate Bernoulli performs well with small vocabulary sizes, but that the multinomial performs usually performs even better at larger vocabulary sizes--providing on average a 27 reduction in error over the multi-variate Bernoulli model at any vocabulary size. These lecture notes highlight the mathematical and computational structure relating to the formulation of, and development of algorithms for, the Bayesian approach to inverse problems in differential equations. This approach is fundamental in the quantification of uncertainty within applications in volving the blending of mathematical models with data. The finite dimensional situation is described first, along with some motivational examples. Then the development of probability measures on separable Banach space is undertaken, using a random series over an infinite set of functions to construct draws; these probability measures are used as priors in the Bayesian approach to inverse problems. Regularity of draws from the priors is studied in the natural Sobolev or Besov spaces implied by the choice of functions in the random series construction, and the Kolmogorov continuity theorem is used to extend regularity considerations to the space of Holder continuous functions. Bayes’ theorem is de rived in this prior setting, and here interpreted as finding conditions under which the posterior is absolutely continuous with respect to the prior, and determining a formula for the Radon-Nikodym derivative in terms of the likelihood of the data. Having established the form of the posterior, we then describe various properties common to it in the infinite dimensional setting. These properties include well-posedness, approximation theory, and the existence of maximum a posteriori estimators. We then describe measure-preserving dynamics, again on the infinite dimensional space, including Markov chain-Monte C arlo and sequential Monte Carlo methods, and measure-preserving reversible stochastic differential equations. By formulating the theory and algorithms on the underlying infinite dimensional space, we obtain a framework suitable for rigorous analysis of the accuracy of reconstructions, of computational complexity, as well as naturally constructing algorithms which perform well under mesh refinement, since they are inherently well-defined in infinite dimensions. We investigate the class of computable probability distributions and explore the fundamental limitations of using this class to describe and compute conditional distributions. In addition to proving the existence of noncomputable conditional distributions, and thus ruling out the possibility of generic probabilistic inference algorithms (even inefficient ones), we highlight some positive results showing that posterior inference is possible in the presence of additional structure like exchangeability and noise, both of which are common in Bayesian hierarchical modeling. This theoretical work bears on the development of probabilistic programming languages (which enable the specification of complex probabilistic models) and their implementations (which can be used to perform Bayesian reasoning). The probabilistic programming approach is particularly well suited for defining infinite-dimensional, recursively-defined stochastic processes of the sort used in nonparametric Bayesian statistics. We present a new construction of the Mondrian process as a partition-valued Markov process in continuous time, which can be viewed as placing a distribution on an infinite kd-tree data structure. (Copies available exclusively from MIT Libraries, Rm. 14-0551, Cambridge, MA 02139-4307. Ph. 617-253-5668; Fax 617-253-1690.) We describe algorithms for learning Bayesian networks from a combination of user knowledge and statistical data. The algorithms have two components: a scoring metric and a search procedure. The scoring metric takes a network structure, statistical data, and a user's prior knowledge, and returns a score proportional to the posterior probability of the network structure given the data. The search procedure generates networks for evaluation by the scoring metric. Our contributions are threefold. First, we identify two important properties of metrics, which we call event equivalence and parameter modularity. These properties have been mostly ignored, but when combined, greatly simplify the encoding of a user's prior knowledge. In particular, a user can express her knowledge-for the most part-as a single prior Bayesian network for the domain. Second, we describe local search and annealing algorithms to be used in conjunction with scoring metrics. In the special case where each node has at most one parent, we show that heuristic search can be replaced with a polynomial algorithm to identify the networks with the highest score. Third, we describe a methodology for evaluating Bayesian-network learning algorithms. We apply this approach to a comparison of metrics and search procedures. Three Bayesian related approaches, namely, variational Bayesian (VB), minimum message length (MML) and Bayesian Ying-Yang (BYY) harmony learning, have been applied to automatically determining an appropriate number of components during learning Gaussian mixture model (GMM). This paper aims to provide a comparative investigation on these approaches with not only a Jeffreys prior but also a conjugate Dirichlet-Normal-Wishart (DNW) prior on GMM. In addition to adopting the existing algorithms either directly or with some modifications, the algorithm for VB with Jeffreys prior and the algorithm for BYY with DNW prior are developed in this paper to fill the missing gap. The performances of automatic model selection are evaluated through extensive experiments, with several empirical findings: 1) Considering priors merely on the mixing weights, each of three approaches makes biased mistakes, while considering priors on all the parameters of GMM makes each approach reduce its bias and also improve its performance. 2) As Jeffreys prior is replaced by the DNW prior, all the three approaches improve their performances. Moreover, Jeffreys prior makes MML slightly better than VB, while the DNW prior makes VB better than MML. 3) As the hyperparameters of DNW prior are further optimized by each of its own learning principle, BYY improves its performances while VB and MML deteriorate their performances when there are too many free hyper-parameters. Actually, VB and MML lack a good guide for optimizing the hyper-parameters of DNW prior. 4) BYY considerably outperforms both VB and MML for any type of priors and whether hyper-parameters are optimized. Being different from VB and MML that rely on appropriate priors to perform model selection, BYY does not highly depend on the type of priors. It has model selection ability even without priors and performs already very well with Jeffreys prior, and incrementally improves as Jeffreys prior is replaced by the DNW prior. Finally, all algorithms are applied on the Berkeley segmentation database of real world images. Again, BYY considerably outperforms both VB and MML, especially in detecting the objects of interest from a confusing background. This article surveys Bayesian methods for categorical data analysis, with primary emphasis on contingency table analysis. Early innovations were proposed by Good (1953, 1956, 1965) for smoothing proportions in contingency tables and by Lindley (1964) for inference about odds ratios. These approaches primarily used conjugate beta and Dirichlet priors. Altham (1969, 1971) presented Bayesian analogs of small-sample frequentist tests for 2 x 2 tables using such priors. An alternative approach using normal priors for logits received considerable attention in the 1970s by Leonard and others (e.g., Leonard 1972). Adopted usually in a hierarchical form, the logit-normal approach allows greater flexibility and scope for generalization. The 1970s also saw considerable interest in loglinear modeling. The advent of modern computational methods since the mid-1980s has led to a growing literature on fully Bayesian analyses with models for categorical data, with main emphasis on generalized linear models such as logistic regression for binary and multi-category response variables. Motivated by settings in which predictive models may be required to be non-discriminatory with respect to certain attributes (such as race), but even collecting the sensitive attribute may be forbidden or restricted, we initiate the study of fair learning under the constraint of differential privacy. We design two learning algorithms that simultaneously promise differential privacy and equalized odds, a 'fairness' condition that corresponds to equalizing false positive and negative rates across protected groups. Our first algorithm is a private implementation of the equalized odds post-processing approach of [, 2016]. This algorithm is appealingly simple, but must be able to use protected group membership explicitly at test time, which can be viewed as a form of 'disparate treatment'. Our second algorithm is a differentially private version of the oracle-efficient in-processing approach of [, 2018] that can be used to find the optimal fair classifier, given access to a subroutine that can solve the original (not necessarily fair) learning problem. This algorithm is more complex but need not have access to protected group membership at test time. We identify new tradeoffs between fairness, accuracy, and privacy that emerge only when requiring all three properties, and show that these tradeoffs can be milder if group membership may be used at test time. We conclude with a brief experimental evaluation. The most prevalent notions of fairness in machine learning are statistical definitions: they fix a small collection of pre-defined groups, and then ask for parity of some statistic of the classifier across these groups. Constraints of this form are susceptible to intentional or inadvertent "fairness gerrymandering", in which a classifier appears to be fair on each individual group, but badly violates the fairness constraint on one or more structured subgroups defined over the protected attributes. We propose instead to demand statistical notions of fairness across exponentially (or infinitely) many subgroups, defined by a structured class of functions over the protected attributes. This interpolates between statistical definitions of fairness and recently proposed individual notions of fairness, but raises several computational challenges. It is no longer clear how to audit a fixed classifier to see if it satisfies such a strong definition of fairness. We prove that the computational problem of auditing subgroup fairness for both equality of false positive rates and statistical parity is equivalent to the problem of weak agnostic learning, which means it is computationally hard in the worst case, even for simple structured subclasses. We then derive two algorithms that provably converge to the best fair classifier, given access to oracles which can solve the agnostic learning problem. The algorithms are based on a formulation of subgroup fairness as a two-player zero-sum game between a Learner and an Auditor. Our first algorithm provably converges in a polynomial number of steps. Our second algorithm enjoys only provably asymptotic convergence, but has the merit of simplicity and faster per-step computation. We implement the simpler algorithm using linear regression as a heuristic oracle, and show that we can effectively both audit and learn fair classifiers on real datasets. We consider the problem of online learning in the linear contextual bandits setting, but in which there are also strong individual fairness constraints governed by an unknown similarity metric. These constraints demand that we select similar actions or individuals with approximately equal probability (arXiv:1104.3913), which may be at odds with optimizing reward, thus modeling settings where profit and social policy are in tension. We assume we learn about an unknown Mahalanobis similarity metric from only weak feedback that identifies fairness violations, but does not quantify their extent. This is intended to represent the interventions of a regulator who "knows unfairness when he sees it" but nevertheless cannot enunciate a quantitative fairness metric over individuals. Our main result is an algorithm in the adversarial context setting that has a number of fairness violations that depends only logarithmically on @math , while obtaining an optimal @math regret bound to the best fair policy. Predicated on the increasing abundance of electronic health records, we investigate the problem of inferring individualized treatment effects using observational data. Stemming from the potential outcomes model, we propose a novel multi-task learning framework in which factual and counterfactual outcomes are modeled as the outputs of a function in a vector-valued reproducing kernel Hilbert space (vvRKHS). We develop a nonparametric Bayesian method for learning the treatment effects using a multi-task Gaussian process (GP) with a linear coregionalization kernel as a prior over the vvRKHS. The Bayesian approach allows us to compute individualized measures of confidence in our estimates via pointwise credible intervals, which are crucial for realizing the full potential of precision medicine. The impact of selection bias is alleviated via a risk-based empirical Bayes method for adapting the multi-task GP prior, which jointly minimizes the empirical error in factual outcomes and the uncertainty in (unobserved) counterfactual outcomes. We conduct experiments on observational datasets for an interventional social program applied to premature infants, and a left ventricular assist device applied to cardiac patients wait-listed for a heart transplant. In both experiments, we show that our method significantly outperforms the state-of-the-art. The probability of error of classification methods based on convex combinations of simple base classifiers by boosting algorithms is investigated. The main result of the paper is that certain regularized boosting algorithms provide Bayes-risk consistent classifiers under the sole assumption that the Bayes classifier may be approximated by a convex combination of the base classifiers. Nonasymptotic distribution-free bounds are also developed which offer interesting new insight into how boosting works and help explain its success in practical classification problems. The Bayesian framework for model comparison and regularisation is demonstrated by studying interpolation and classification problems modelled with both linear and non-linear models. This framework quantitatively embodies 'Occam's razor'. Over-complex and under-regularised models are automatically inferred to be less probable, even though their flexibility allows them to fit the data better. When applied to 'neural networks', the Bayesian framework makes possible (1) objective comparison of solutions using alternative network architectures; (2) objective stopping rules for network pruning or growing procedures; (3) objective choice of type of weight decay terms (or regularisers); (4) on-line techniques for optimising weight decay (or regularisation constant) magnitude; (5) a measure of the effective number of well-determined parameters in a model; (6) quantified estimates of the error bars on network parameters and on network output. In the case of classification models, it is shown that the careful incorporation of error bar information into a classifier's predictions yields improved performance. Comparisons of the inferences of the Bayesian framework with more traditional cross-validation methods help detect poor underlying assumptions in learning models. The relationship of the Bayesian learning framework to 'active learning' is examined. Objective functions are discussed which measure the expected informativeness of candidate data measurements, in the context of both interpolation and classification problems. The concepts and methods described in this thesis are quite general and will be applicable to other data modelling problems whether they involve regression, classification or density estimation. This paper studies two-class (or binary) classification of elements X in R k that allows for a reject option. Based on n independent copies of the pair of random variables (X,Y ) with X 2 R k and Y 2 0,1 , we consider classifiers f(X) that render three possible outputs: 0, 1 and R. The option R expresses doubt and is to be used for few observations that are hard to classify in an automatic way. Chow (1970) derived the optimal rule minimizing the risk P f(X) 6 Y, f(X) 6 R + dP f(X) = R . This risk function subsumes that the cost of making a wrong decision equals 1 and that of utilizing the reject option is d. We show that the classification problem hinges on the behavior of the regression function (x) = E(Y |X = x) near d and 1 d. (Here d 2 [0,1 2] as the other cases turn out to be trivial.) Classification rules can be categorized into plug-in estimators and empirical risk minimizers. Both types are considered here and we prove that the rates of convergence of the risk of any estimate depends on P | (X) d| + P | (X) (1 d)| and on the quality of the estimate for or an appropriate measure of the size of the class of classifiers, in case of plug-in rules and empirical risk minimizers, respectively. We extend the mathematical framework even further by dierentiating between costs associated with the two possible errors: predicting f(X) = 0 whilst Y = 1 and predicting f(X) = 1 whilst Y = 0. Such situations are common in, for instance, medical studies where misclassifying a sick patient as healthy is worse than the opposite. Bayesian optimization has recently been proposed as a framework for automatically tuning the hyperparameters of machine learning models and has been shown to yield state-of-the-art performance with impressive ease and efficiency. In this paper, we explore whether it is possible to transfer the knowledge gained from previous optimizations to new tasks in order to find optimal hyperparameter settings more efficiently. Our approach is based on extending multi-task Gaussian processes to the framework of Bayesian optimization. We show that this method significantly speeds up the optimization process when compared to the standard single-task approach. We further propose a straightforward extension of our algorithm in order to jointly minimize the average error across multiple tasks and demonstrate how this can be used to greatly speed up k-fold cross-validation. Lastly, we propose an adaptation of a recently developed acquisition function, entropy search, to the cost-sensitive, multi-task setting. We demonstrate the utility of this new acquisition function by leveraging a small dataset to explore hyper-parameter settings for a large dataset. Our algorithm dynamically chooses which dataset to query in order to yield the most information per unit cost. We exhibit a poly(log k)-competitive randomized algorithm for the k-server problem on any metric space. The best previous result independent of the geometry of the underlying metric space is the 2k–1 competitive ratio established for the deterministic work function algorithm by Koutsoupias and Papadimitriou (1995). Even for the special case when the underlying metric space is the real line, the best known competitive ratio was k. Since deterministic algorithms can do no better than k on any metric space with at least k+1 points, this establishes that for every metric space on which the problem is non-trivial, randomized algorithms give an exponential improvement over deterministic algorithms. Our algorithm maintains an approximation of the underlying metric space by a distribution over HSTs. The granularity and accuracy of the approximation is adjusted dynamically according to the aggregate behavior of the HST algorithms. In short: We try to obtain more accurate approximations at the locations and scales where the gactionh is happening. Thus a crucial component of our approach is the O((log k)^2)-competitive randomized algorithm for HSTs obtained in our previous work with Bubeck, Cohen, Lee, and Ma.dry, and its "multiscale information theory" perspective. Machine learning is now being used to make crucial decisions about people's lives. For nearly all of these decisions there is a risk that individuals of a certain race, gender, sexual orientation, or any other subpopulation are unfairly discriminated against. Our recent method has demonstrated how to use techniques from counterfactual inference to make predictions fair across different subpopulations. This method requires that one provides the causal model that generated the data at hand. In general, validating all causal implications of the model is not possible without further assumptions. Hence, it is desirable to integrate competing causal models to provide counterfactually fair decisions, regardless of which causal "world" is the correct one. In this paper, we show how it is possible to make predictions that are approximately fair with respect to multiple possible causal models at once, thus mitigating the problem of exact causal specification. We frame the goal of learning a fair classifier as an optimization problem with fairness constraints entailed by competing causal explanations. We show how this optimization problem can be efficiently solved using gradient-based methods. We demonstrate the flexibility of our model on two real-world fair classification problems. We show that our model can seamlessly balance fairness in multiple worlds with prediction accuracy. This paper reports on a novel technique for literature indexing and searching in a mechanized library system. The notion of relevance is taken as the key concept in the theory of information retrieval and a comparative concept of relevance is explicated in terms of the theory of probability. The resulting technique called “Probabilistic Indexing,” allows a computing machine, given a request for information, to make a statistical inference and derive a number (called the “relevance number”) for each document, which is a measure of the probability that the document will satisfy the given request. The result of a search is an ordered list of those documents which satisfy the request ranked according to their probable relevance. The paper goes on to show that whereas in a conventional library system the cross-referencing (“see” and “see also”) is based solely on the “semantical closeness” between index terms, statistical measures of closeness between index terms can be defined and computed. Thus, given an arbitrary request consisting of one (or many) index term(s), a machine can elaborate on it to increase the probability of selecting relevant documents that would not otherwise have been selected. Finally, the paper suggests an interpretation of the whole library problem as one where the request is considered as a clue on the basis of which the library system makes a concatenated statistical inference in order to provide as an output an ordered list of those documents which most probably satisfy the information needs of the user. We introduce a hierarchical Bayesian approach to tackle the challenging problem of size recommendation in e-commerce fashion. Our approach jointly models a size purchased by a customer, and its possible return event: 1. no return, 2. returned too small 3. returned too big. Those events are drawn following a multinomial distribution parameterized on the joint probability of each event, built following a hierarchy combining priors. Such a model allows us to incorporate extended domain expertise and article characteristics as prior knowledge, which in turn makes it possible for the underlying parameters to emerge thanks to sufficient data. Experiments are presented on real (anonymized) data from millions of customers along with a detailed discussion on the efficiency of such an approach within a large scale production system.