A New Hyperprior Distribution for Bayesian Regression Model with Application in Genomics
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Assessing Fairness with Unlabeled Data and Bayesian Inference
Can I Trust My Fairness Metric? Assessing Fairness with Unlabeled Data and Bayesian Inference Disi Ji1 Padhraic Smyth1 Mark Steyvers2 1Department of Computer Science 2Department of Cognitive Sciences University of California, Irvine [email protected] [email protected] [email protected] Abstract We investigate the problem of reliably assessing group fairness when labeled examples are few but unlabeled examples are plentiful. We propose a general Bayesian framework that can augment labeled data with unlabeled data to produce more accurate and lower-variance estimates compared to methods based on labeled data alone. Our approach estimates calibrated scores for unlabeled examples in each group using a hierarchical latent variable model conditioned on labeled examples. This in turn allows for inference of posterior distributions with associated notions of uncertainty for a variety of group fairness metrics. We demonstrate that our approach leads to significant and consistent reductions in estimation error across multiple well-known fairness datasets, sensitive attributes, and predictive models. The results show the benefits of using both unlabeled data and Bayesian inference in terms of assessing whether a prediction model is fair or not. 1 Introduction Machine learning models are increasingly used to make important decisions about individuals. At the same time it has become apparent that these models are susceptible to producing systematically biased decisions with respect to sensitive attributes such as gender, ethnicity, and age [Angwin et al., 2017, Berk et al., 2018, Corbett-Davies and Goel, 2018, Chen et al., 2019, Beutel et al., 2019]. This has led to a significant amount of recent work in machine learning addressing these issues, including research on both (i) definitions of fairness in a machine learning context (e.g., Dwork et al. -
Bayesian Inference Chapter 4: Regression and Hierarchical Models
Bayesian Inference Chapter 4: Regression and Hierarchical Models Conchi Aus´ınand Mike Wiper Department of Statistics Universidad Carlos III de Madrid Master in Business Administration and Quantitative Methods Master in Mathematical Engineering Conchi Aus´ınand Mike Wiper Regression and hierarchical models Masters Programmes 1 / 35 Objective AFM Smith Dennis Lindley We analyze the Bayesian approach to fitting normal and generalized linear models and introduce the Bayesian hierarchical modeling approach. Also, we study the modeling and forecasting of time series. Conchi Aus´ınand Mike Wiper Regression and hierarchical models Masters Programmes 2 / 35 Contents 1 Normal linear models 1.1. ANOVA model 1.2. Simple linear regression model 2 Generalized linear models 3 Hierarchical models 4 Dynamic models Conchi Aus´ınand Mike Wiper Regression and hierarchical models Masters Programmes 3 / 35 Normal linear models A normal linear model is of the following form: y = Xθ + ; 0 where y = (y1;:::; yn) is the observed data, X is a known n × k matrix, called 0 the design matrix, θ = (θ1; : : : ; θk ) is the parameter set and follows a multivariate normal distribution. Usually, it is assumed that: 1 ∼ N 0 ; I : k φ k A simple example of normal linear model is the simple linear regression model T 1 1 ::: 1 where X = and θ = (α; β)T . x1 x2 ::: xn Conchi Aus´ınand Mike Wiper Regression and hierarchical models Masters Programmes 4 / 35 Normal linear models Consider a normal linear model, y = Xθ + . A conjugate prior distribution is a normal-gamma distribution: -
The Exponential Family 1 Definition
The Exponential Family David M. Blei Columbia University November 9, 2016 The exponential family is a class of densities (Brown, 1986). It encompasses many familiar forms of likelihoods, such as the Gaussian, Poisson, multinomial, and Bernoulli. It also encompasses their conjugate priors, such as the Gamma, Dirichlet, and beta. 1 Definition A probability density in the exponential family has this form p.x / h.x/ exp >t.x/ a./ ; (1) j D f g where is the natural parameter; t.x/ are sufficient statistics; h.x/ is the “base measure;” a./ is the log normalizer. Examples of exponential family distributions include Gaussian, gamma, Poisson, Bernoulli, multinomial, Markov models. Examples of distributions that are not in this family include student-t, mixtures, and hidden Markov models. (We are considering these families as distributions of data. The latent variables are implicitly marginalized out.) The statistic t.x/ is called sufficient because the probability as a function of only depends on x through t.x/. The exponential family has fundamental connections to the world of graphical models (Wainwright and Jordan, 2008). For our purposes, we’ll use exponential 1 families as components in directed graphical models, e.g., in the mixtures of Gaussians. The log normalizer ensures that the density integrates to 1, Z a./ log h.x/ exp >t.x/ d.x/ (2) D f g This is the negative logarithm of the normalizing constant. The function h.x/ can be a source of confusion. One way to interpret h.x/ is the (unnormalized) distribution of x when 0. It might involve statistics of x that D are not in t.x/, i.e., that do not vary with the natural parameter. -
1 Introduction to Bayesian Inference 2 Introduction to Gibbs Sampling
MCMC I: July 5, 2016 1 MCMC I 8th Summer Institute in Statistics and Modeling in Infectious Diseases Course Time Plan July 13-15, 2016 Instructors: Vladimir Minin, Kari Auranen, M. Elizabeth Halloran Course Description: This module is an introduction to Markov chain Monte Carlo methods with some simple applications in infectious disease studies. The course includes an introduction to Bayesian inference, Monte Carlo, MCMC, some background theory, and convergence diagnostics. Algorithms include Gibbs sampling and Metropolis-Hastings and combinations. Programming is in R. Familiarity with the R statistical package or other computing language is needed. Course schedule: The course is composed of 10 90-minute sessions, for a total of 15 hours of instruction. 1 Introduction to Bayesian Inference • Overview of the course. • Bayesian inference: Likelihood, prior, posterior, normalizing constant • Conjugate priors; Beta-binomial; Poisson-gamma; normal-normal • Posterior summaries, mean, mode, posterior intervals • Motivating examples: Chain binomial model (Reed-Frost), General Epidemic Model, SIS model. • Lab: – Goals: Warm-up with R for simple Bayesian computation – Example: Posterior distribution of transmission probability with a binomial sampling distribution using a conjugate beta prior distribution – Summarizing posterior inference (mean, median, posterior quantiles and intervals) – Varying the amount of prior information – Writing an R function 2 Introduction to Gibbs Sampling • Chain binomial model and data augmentation • Brief introduction -
Fast Bayesian Non-Negative Matrix Factorisation and Tri-Factorisation
Fast Bayesian Non-Negative Matrix Factorisation and Tri-Factorisation Thomas Brouwer Jes Frellsen Pietro Lio’ Computer Laboratory Department of Engineering Computer Laboratory University of Cambridge University of Cambridge University of Cambridge United Kingdom United Kingdom United Kingdom [email protected] [email protected] [email protected] Abstract We present a fast variational Bayesian algorithm for performing non-negative matrix factorisation and tri-factorisation. We show that our approach achieves faster convergence per iteration and timestep (wall-clock) than Gibbs sampling and non-probabilistic approaches, and do not require additional samples to estimate the posterior. We show that in particular for matrix tri-factorisation convergence is difficult, but our variational Bayesian approach offers a fast solution, allowing the tri-factorisation approach to be used more effectively. 1 Introduction Non-negative matrix factorisation methods Lee and Seung [1999] have been used extensively in recent years to decompose matrices into latent factors, helping us reveal hidden structure and predict missing values. In particular we decompose a given matrix into two smaller matrices so that their product approximates the original one. The non-negativity constraint makes the resulting matrices easier to interpret, and is often inherent to the problem – such as in image processing or bioinformatics (Lee and Seung [1999], Wang et al. [2013]). Some approaches approximate a maximum likelihood (ML) or maximum a posteriori (MAP) solution that minimises the difference between the observed matrix and the decomposition of this matrix. This gives a single point estimate, which can lead to overfitting more easily and neglects uncertainty. Instead, we may wish to find a full distribution over the matrices using a Bayesian approach, where we define prior distributions over the matrices and then compute their posterior after observing the actual data. -
The Bayesian Lasso
The Bayesian Lasso Trevor Park and George Casella y University of Florida, Gainesville, Florida, USA Summary. The Lasso estimate for linear regression parameters can be interpreted as a Bayesian posterior mode estimate when the priors on the regression parameters are indepen- dent double-exponential (Laplace) distributions. This posterior can also be accessed through a Gibbs sampler using conjugate normal priors for the regression parameters, with indepen- dent exponential hyperpriors on their variances. This leads to tractable full conditional distri- butions through a connection with the inverse Gaussian distribution. Although the Bayesian Lasso does not automatically perform variable selection, it does provide standard errors and Bayesian credible intervals that can guide variable selection. Moreover, the structure of the hierarchical model provides both Bayesian and likelihood methods for selecting the Lasso pa- rameter. The methods described here can also be extended to other Lasso-related estimation methods like bridge regression and robust variants. Keywords: Gibbs sampler, inverse Gaussian, linear regression, empirical Bayes, penalised regression, hierarchical models, scale mixture of normals 1. Introduction The Lasso of Tibshirani (1996) is a method for simultaneous shrinkage and model selection in regression problems. It is most commonly applied to the linear regression model y = µ1n + Xβ + ; where y is the n 1 vector of responses, µ is the overall mean, X is the n p matrix × T × of standardised regressors, β = (β1; : : : ; βp) is the vector of regression coefficients to be estimated, and is the n 1 vector of independent and identically distributed normal errors with mean 0 and unknown× variance σ2. The estimate of µ is taken as the average y¯ of the responses, and the Lasso estimate β minimises the sum of the squared residuals, subject to a given bound t on its L1 norm. -
Posterior Propriety and Admissibility of Hyperpriors in Normal
The Annals of Statistics 2005, Vol. 33, No. 2, 606–646 DOI: 10.1214/009053605000000075 c Institute of Mathematical Statistics, 2005 POSTERIOR PROPRIETY AND ADMISSIBILITY OF HYPERPRIORS IN NORMAL HIERARCHICAL MODELS1 By James O. Berger, William Strawderman and Dejun Tang Duke University and SAMSI, Rutgers University and Novartis Pharmaceuticals Hierarchical modeling is wonderful and here to stay, but hyper- parameter priors are often chosen in a casual fashion. Unfortunately, as the number of hyperparameters grows, the effects of casual choices can multiply, leading to considerably inferior performance. As an ex- treme, but not uncommon, example use of the wrong hyperparameter priors can even lead to impropriety of the posterior. For exchangeable hierarchical multivariate normal models, we first determine when a standard class of hierarchical priors results in proper or improper posteriors. We next determine which elements of this class lead to admissible estimators of the mean under quadratic loss; such considerations provide one useful guideline for choice among hierarchical priors. Finally, computational issues with the resulting posterior distributions are addressed. 1. Introduction. 1.1. The model and the problems. Consider the block multivariate nor- mal situation (sometimes called the “matrix of means problem”) specified by the following hierarchical Bayesian model: (1) X ∼ Np(θ, I), θ ∼ Np(B, Σπ), where X1 θ1 X2 θ2 Xp×1 = . , θp×1 = . , arXiv:math/0505605v1 [math.ST] 27 May 2005 . X θ m m Received February 2004; revised July 2004. 1Supported by NSF Grants DMS-98-02261 and DMS-01-03265. AMS 2000 subject classifications. Primary 62C15; secondary 62F15. Key words and phrases. Covariance matrix, quadratic loss, frequentist risk, posterior impropriety, objective priors, Markov chain Monte Carlo. -
Bayesian Inference in the Normal Linear Regression Model
Bayesian Inference in the Normal Linear Regression Model () Bayesian Methods for Regression 1 / 53 Bayesian Analysis of the Normal Linear Regression Model Now see how general Bayesian theory of overview lecture works in familiar regression model Reading: textbook chapters 2, 3 and 6 Chapter 2 presents theory for simple regression model (no matrix algebra) Chapter 3 does multiple regression In lecture, I will go straight to multiple regression Begin with regression model under classical assumptions (independent errors, homoskedasticity, etc.) Chapter 6 frees up classical assumptions in several ways Lecture will cover one way: Bayesian treatment of a particular type of heteroskedasticity () Bayesian Methods for Regression 2 / 53 The Regression Model Assume k explanatory variables, xi1,..,xik for i = 1, .., N and regression model: yi = b1 + b2xi2 + ... + bk xik + #i . Note xi1 is implicitly set to 1 to allow for an intercept. Matrix notation: y1 y2 y = 2 . 3 6 . 7 6 7 6 y 7 6 N 7 4 5 # is N 1 vector stacked in same way as y () Bayesian Methods for Regression 3 / 53 b is k 1 vector X is N k matrix 1 x12 .. x1k 1 x22 .. x2k X = 2 ..... 3 6 ..... 7 6 7 6 1 x .. x 7 6 N2 Nk 7 4 5 Regression model can be written as: y = X b + #. () Bayesian Methods for Regression 4 / 53 The Likelihood Function Likelihood can be derived under the classical assumptions: 1 2 # is N(0N , h IN ) where h = s . All elements of X are either fixed (i.e. not random variables). Exercise 10.1, Bayesian Econometric Methods shows that likelihood function can be written in -
Empirical Bayes Methods for Combining Likelihoods Bradley EFRON
Empirical Bayes Methods for Combining Likelihoods Bradley EFRON Supposethat several independent experiments are observed,each one yieldinga likelihoodLk (0k) for a real-valuedparameter of interestOk. For example,Ok might be the log-oddsratio for a 2 x 2 table relatingto the kth populationin a series of medical experiments.This articleconcerns the followingempirical Bayes question:How can we combineall of the likelihoodsLk to get an intervalestimate for any one of the Ok'S, say 01? The resultsare presented in the formof a realisticcomputational scheme that allows model buildingand model checkingin the spiritof a regressionanalysis. No specialmathematical forms are requiredfor the priorsor the likelihoods.This schemeis designedto take advantageof recentmethods that produceapproximate numerical likelihoodsLk(6k) even in very complicatedsituations, with all nuisanceparameters eliminated. The empiricalBayes likelihood theoryis extendedto situationswhere the Ok'S have a regressionstructure as well as an empiricalBayes relationship.Most of the discussionis presentedin termsof a hierarchicalBayes model and concernshow such a model can be implementedwithout requiringlarge amountsof Bayesianinput. Frequentist approaches, such as bias correctionand robustness,play a centralrole in the methodology. KEY WORDS: ABC method;Confidence expectation; Generalized linear mixed models;Hierarchical Bayes; Meta-analysis for likelihoods;Relevance; Special exponential families. 1. INTRODUCTION for 0k, also appearing in Table 1, is A typical statistical analysis blends data from indepen- dent experimental units into a single combined inference for a parameter of interest 0. Empirical Bayes, or hierar- SDk ={ ak +.S + bb + .5 +k + -5 dk + .5 chical or meta-analytic analyses, involve a second level of SD13 = .61, for example. data acquisition. Several independent experiments are ob- The statistic 0k is an estimate of the true log-odds ratio served, each involving many units, but each perhaps having 0k in the kth experimental population, a different value of the parameter 0. -
An Introduction to Markov Chain Monte Carlo Methods and Their Actuarial Applications
AN INTRODUCTION TO MARKOV CHAIN MONTE CARLO METHODS AND THEIR ACTUARIAL APPLICATIONS DAVID P. M. SCOLLNIK Department of Mathematics and Statistics University of Calgary Abstract This paper introduces the readers of the Proceed- ings to an important class of computer based simula- tion techniques known as Markov chain Monte Carlo (MCMC) methods. General properties characterizing these methods will be discussed, but the main empha- sis will be placed on one MCMC method known as the Gibbs sampler. The Gibbs sampler permits one to simu- late realizations from complicated stochastic models in high dimensions by making use of the model’s associated full conditional distributions, which will generally have a much simpler and more manageable form. In its most extreme version, the Gibbs sampler reduces the analy- sis of a complicated multivariate stochastic model to the consideration of that model’s associated univariate full conditional distributions. In this paper, the Gibbs sampler will be illustrated with four examples. The first three of these examples serve as rather elementary yet instructive applications of the Gibbs sampler. The fourth example describes a reasonably sophisticated application of the Gibbs sam- pler in the important arena of credibility for classifica- tion ratemaking via hierarchical models, and involves the Bayesian prediction of frequency counts in workers compensation insurance. 114 AN INTRODUCTION TO MARKOV CHAIN MONTE CARLO METHODS 115 1. INTRODUCTION The purpose of this paper is to acquaint the readership of the Proceedings with a class of simulation techniques known as Markov chain Monte Carlo (MCMC) methods. These methods permit a practitioner to simulate a dependent sequence of ran- dom draws from very complicated stochastic models. -
Accelerating Bayesian Inference on Structured Graphs Using Parallel Gibbs Sampling
Accelerating Bayesian Inference on Structured Graphs Using Parallel Gibbs Sampling Glenn G. Ko∗, Yuji Chai∗, Rob A. Rutenbary, David Brooks∗, Gu-Yeon Wei∗ ∗Harvard University, Cambridge, USA yUniversity of Pittsburgh, Pittsburgh, USA [email protected], [email protected], [email protected], fdbrooks, [email protected] Abstract—Bayesian models and inference is a class of machine with substantial uncertainty and noise is normally reserved for learning that is useful for solving problems where the amount Bayesian models and inference. of data is scarce and prior knowledge about the application Bayesian models pose challenges that are different from allows you to draw better conclusions. However, Bayesian models often requires computing high-dimensional integrals and finding deep learning. They tend to have more complex structure, the posterior distribution can be intractable. One of the most that may be hierarchical with dependencies between different commonly used approximate methods for Bayesian inference distributions Without an easily-defined common computational is Gibbs sampling, which is a Markov chain Monte Carlo kernel and hardware accelerators like GPU and TPU [5], and (MCMC) technique to estimate target stationary distribution. software frameworks such as Tensorflow [6] and PyTorch [7], The idea in Gibbs sampling is to generate posterior samples by iterating through each of the variables to sample from its it is hard to build and run models fast and efficiently. This conditional given all the other variables fixed. While Gibbs work explores hardware accelerators for Bayesian models and sampling is a popular method for probabilistic graphical models inference which has growing interest and demand. such as Markov Random Field (MRF), the plain algorithm is Probabilistic graphical models are Bayesian models described slow as it goes through each of the variables sequentially. -
Gibbs Sampling, Exponential Families and Orthogonal Polynomials
Statistical Science 2008, Vol. 23, No. 2, 151–178 DOI: 10.1214/07-STS252 c Institute of Mathematical Statistics, 2008 Gibbs Sampling, Exponential Families and Orthogonal Polynomials1 Persi Diaconis, Kshitij Khare and Laurent Saloff-Coste Abstract. We give families of examples where sharp rates of conver- gence to stationarity of the widely used Gibbs sampler are available. The examples involve standard exponential families and their conjugate priors. In each case, the transition operator is explicitly diagonalizable with classical orthogonal polynomials as eigenfunctions. Key words and phrases: Gibbs sampler, running time analyses, expo- nential families, conjugate priors, location families, orthogonal polyno- mials, singular value decomposition. 1. INTRODUCTION which has f as stationary density under mild con- ditions discussed in [4, 102]. The Gibbs sampler, also known as Glauber dy- The algorithm was introduced in 1963 by Glauber namics or the heat-bath algorithm, is a mainstay of [49] to do simulations for Ising models, and inde- scientific computing. It provides a way to draw sam- pendently by Turcin [103]. It is still a standard tool ples from a multivariate probability density f(x1,x2, of statistical physics, both for practical simulation ...,xp), perhaps only known up to a normalizing (e.g., [88]) and as a natural dynamics (e.g., [10]). constant, by a sequence of one-dimensional sampling The basic Dobrushin uniqueness theorem showing problems. From (X1,...,Xp) proceed to (X1′ , X2,..., existence of Gibbs measures was proved based on Xp), then (X1′ , X2′ , X3,...,Xp),..., (X1′ , X2′ ,...,Xp′ ) this dynamics (e.g., [54]). It was introduced as a where at the ith stage, the coordinate is sampled base for image analysis by Geman and Geman [46].