Slice Sampling for General Completely Random Measures
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Markov Chain Monte Carlo 1 Importance Sampling
CS731 Spring 2011 Advanced Artificial Intelligence Markov Chain Monte Carlo Lecturer: Xiaojin Zhu [email protected] A fundamental problem in machine learning is to generate samples from a distribution: x ∼ p(x). (1) This problem has many important applications. For example, one can approximate the expectation of a function φ(x) Z µ ≡ Ep[φ(x)] = φ(x)p(x)dx (2) by the sample average over x1 ... xn ∼ p(x): n 1 X µˆ ≡ φ(x ) (3) n i i=1 This is known as the Monte Carlo method. A concrete application is in Bayesian predictive modeling where R p(x | θ)p(θ | Data)dθ. Clearly,µ ˆ is an unbiased estimator of µ. Furthermore, the variance ofµ ˆ decreases as V(φ)/n (4) R 2 where V(φ) = (φ(x) − µ) p(x)dx. Thus Monte Carlo methods depend on the sample size n, not on the dimensionality of x. 1 Often, p is only known up to a constant. That is, we assume p(x) = Z p˜(x) and we are able to evaluate p˜(x) at any point x. However, we cannot compute the normalization constant Z. For instance, in the 1 Bayesian modeling example, the posterior distribution is p(θ | Data) = Z p(θ)p(Data | θ). This makes sampling harder. All methods below work onp ˜. 1 Importance Sampling Unlike other methods discussed later, importance sampling is not a method to sample from p. Rather, it is a method to compute (3). Assume we know the unnormalizedp ˜. It turns out that we (or Matlab) only know how to directly sample from very few distributions such as uniform, Gaussian, etc.; p may not be one of them in general. -
Sampling Methods (Gatsby ML1 2017)
Probabilistic & Unsupervised Learning Sampling Methods Maneesh Sahani [email protected] Gatsby Computational Neuroscience Unit, and MSc ML/CSML, Dept Computer Science University College London Term 1, Autumn 2017 Both are often intractable. Deterministic approximations on distributions (factored variational / mean-field; BP; EP) or expectations (Bethe / Kikuchi methods) provide tractability, at the expense of a fixed approximation penalty. An alternative is to represent distributions and compute expectations using randomly generated samples. Results are consistent, often unbiased, and precision can generally be improved to an arbitrary degree by increasing the number of samples. Sampling For inference and learning we need to compute both: I Posterior distributions (on latents and/or parameters) or predictive distributions. I Expectations with respect to these distributions. Deterministic approximations on distributions (factored variational / mean-field; BP; EP) or expectations (Bethe / Kikuchi methods) provide tractability, at the expense of a fixed approximation penalty. An alternative is to represent distributions and compute expectations using randomly generated samples. Results are consistent, often unbiased, and precision can generally be improved to an arbitrary degree by increasing the number of samples. Sampling For inference and learning we need to compute both: I Posterior distributions (on latents and/or parameters) or predictive distributions. I Expectations with respect to these distributions. Both are often intractable. An alternative is to represent distributions and compute expectations using randomly generated samples. Results are consistent, often unbiased, and precision can generally be improved to an arbitrary degree by increasing the number of samples. Sampling For inference and learning we need to compute both: I Posterior distributions (on latents and/or parameters) or predictive distributions. -
Stat 451 Lecture Notes 0512 Simulating Random Variables
Stat 451 Lecture Notes 0512 Simulating Random Variables Ryan Martin UIC www.math.uic.edu/~rgmartin 1Based on Chapter 6 in Givens & Hoeting, Chapter 22 in Lange, and Chapter 2 in Robert & Casella 2Updated: March 7, 2016 1 / 47 Outline 1 Introduction 2 Direct sampling techniques 3 Fundamental theorem of simulation 4 Indirect sampling techniques 5 Sampling importance resampling 6 Summary 2 / 47 Motivation Simulation is a very powerful tool for statisticians. It allows us to investigate the performance of statistical methods before delving deep into difficult theoretical work. At a more practical level, integrals themselves are important for statisticians: p-values are nothing but integrals; Bayesians need to evaluate integrals to produce posterior probabilities, point estimates, and model selection criteria. Therefore, there is a need to understand simulation techniques and how they can be used for integral approximations. 3 / 47 Basic Monte Carlo Suppose we have a function '(x) and we'd like to compute Ef'(X )g = R '(x)f (x) dx, where f (x) is a density. There is no guarantee that the techniques we learn in calculus are sufficient to evaluate this integral analytically. Thankfully, the law of large numbers (LLN) is here to help: If X1; X2;::: are iid samples from f (x), then 1 Pn R n i=1 '(Xi ) ! '(x)f (x) dx with prob 1. Suggests that a generic approximation of the integral be obtained by sampling lots of Xi 's from f (x) and replacing integration with averaging. This is the heart of the Monte Carlo method. 4 / 47 What follows? Here we focus mostly on simulation techniques. -
A Minimax Near-Optimal Algorithm for Adaptive Rejection Sampling
A minimax near-optimal algorithm for adaptive rejection sampling Juliette Achdou [email protected] Numberly (1000mercis Group) Paris, France Joseph C. Lam [email protected] Otto-von-Guericke University Magdeburg, Germany Alexandra Carpentier [email protected] Otto-von-Guericke University Magdeburg, Germany Gilles Blanchard [email protected] Potsdam University Potsdam, Germany Abstract Rejection Sampling is a fundamental Monte-Carlo method. It is used to sample from distributions admitting a probability density function which can be evaluated exactly at any given point, albeit at a high computational cost. However, without proper tuning, this technique implies a high rejection rate. Several methods have been explored to cope with this problem, based on the principle of adaptively estimating the density by a simpler function, using the information of the previous samples. Most of them either rely on strong assumptions on the form of the density, or do not offer any theoretical performance guarantee. We give the first theoretical lower bound for the problem of adaptive rejection sampling and introduce a new algorithm which guarantees a near-optimal rejection rate in a minimax sense. Keywords: Adaptive rejection sampling, Minimax rates, Monte-Carlo, Active learning. 1. Introduction The breadth of applications requiring independent sampling from a probability distribution is sizable. Numerous classical statistical results, and in particular those involved in ma- arXiv:1810.09390v1 [stat.ML] 22 Oct 2018 chine learning, rely on the independence assumption. For some densities, direct sampling may not be tractable, and the evaluation of the density at a given point may be costly. Rejection sampling (RS) is a well-known Monte-Carlo method for sampling from a density d f on R when direct sampling is not tractable (see Von Neumann, 1951, Devroye, 1986). -
On the Generalized Ratio of Uniforms As a Combination of Transformed Rejection and Extended Inverse of Density Sampling
1 On the Generalized Ratio of Uniforms as a Combination of Transformed Rejection and Extended Inverse of Density Sampling Luca Martinoy, David Luengoz, Joaqu´ın M´ıguezy yDepartment of Signal Theory and Communications, Universidad Carlos III de Madrid. Avenida de la Universidad 30, 28911 Leganes,´ Madrid, Spain. zDepartment of Circuits and Systems Engineering, Universidad Politecnica´ de Madrid. Carretera de Valencia Km. 7, 28031 Madrid, Spain. E-mail: [email protected], [email protected], [email protected] Abstract In this work we investigate the relationship among three classical sampling techniques: the inverse of density (Khintchine’s theorem), the transformed rejection (TR) and the generalized ratio of uniforms (GRoU). Given a monotonic probability density function (PDF), we show that the transformed area obtained using the generalized ratio of uniforms method can be found equivalently by applying the transformed rejection sampling approach to the inverse function of the target density. Then we provide an extension of the classical inverse of density idea, showing that it is completely equivalent to the GRoU method for monotonic densities. Although we concentrate on monotonic probability density functions (PDFs), we also discuss how the results presented here can be extended to any non-monotonic PDF that can be decomposed into a collection of intervals where it is monotonically increasing or decreasing. In this general case, we show the connections with transformations of certain random variables and the generalized inverse PDF with the GRoU technique. Finally, we also introduce a GRoU technique to arXiv:1205.0482v7 [stat.CO] 16 Jul 2013 handle unbounded target densities. Index Terms Transformed rejection sampling; inverse of density method; Khintchine’s theorem; generalized ratio of uniforms technique; vertical density representation. -
Efficient Sampling for Gaussian Linear Regression with Arbitrary Priors
Efficient sampling for Gaussian linear regression with arbitrary priors P. Richard Hahn ∗ Arizona State University and Jingyu He Booth School of Business, University of Chicago and Hedibert F. Lopes INSPER Institute of Education and Research May 18, 2018 Abstract This paper develops a slice sampler for Bayesian linear regression models with arbitrary priors. The new sampler has two advantages over current approaches. One, it is faster than many custom implementations that rely on auxiliary latent variables, if the number of regressors is large. Two, it can be used with any prior with a density function that can be evaluated up to a normalizing constant, making it ideal for investigating the properties of new shrinkage priors without having to develop custom sampling algorithms. The new sampler takes advantage of the special structure of the linear regression likelihood, allowing it to produce better effective sample size per second than common alternative approaches. Keywords: Bayesian computation, linear regression, shrinkage priors, slice sampling ∗The authors gratefully acknowledge the Booth School of Business for support. 1 1 Introduction This paper develops a computationally efficient posterior sampling algorithm for Bayesian linear regression models with Gaussian errors. Our new approach is motivated by the fact that existing software implementations for Bayesian linear regression do not readily handle problems with large number of observations (hundreds of thousands) and predictors (thou- sands). Moreover, existing sampling algorithms for popular shrinkage priors are bespoke Gibbs samplers based on case-specific latent variable representations. By contrast, the new algorithm does not rely on case-specific auxiliary variable representations, which allows for rapid prototyping of novel shrinkage priors outside the conditionally Gaussian framework. -
Monte Carlo and Markov Chain Monte Carlo Methods
MONTE CARLO AND MARKOV CHAIN MONTE CARLO METHODS History: Monte Carlo (MC) and Markov Chain Monte Carlo (MCMC) have been around for a long time. Some (very) early uses of MC ideas: • Conte de Buffon (1777) dropped a needle of length L onto a grid of parallel lines spaced d > L apart to estimate P [needle intersects a line]. • Laplace (1786) used Buffon’s needle to evaluate π. • Gosset (Student, 1908) used random sampling to determine the distribution of the sample correlation coefficient. • von Neumann, Fermi, Ulam, Metropolis (1940s) used games of chance (hence MC) to study models of atomic collisions at Los Alamos during WW II. We are concerned with the use of MC and, in particular, MCMC methods to solve estimation and detection problems. As discussed in the introduction to MC methods (handout # 4), many estimation and detection problems require evaluation of integrals. EE 527, Detection and Estimation Theory, # 4b 1 Monte Carlo Integration MC Integration is essentially numerical integration and thus may be thought of as estimation — we will discuss MC estimators that estimate integrals. Although MC integration is most useful when dealing with high-dimensional problems, the basic ideas are easiest to grasp by looking at 1-D problems first. Suppose we wish to evaluate Z G = g(x) p(x) dx Ω R where p(·) is a density, i.e. p(x) ≥ 0 for x ∈ Ω and Ω p(x) dx = 1. Any integral can be written in this form if we can transform their limits of integration to Ω = (0, 1) — then choose p(·) to be uniform(0, 1). -
Concave Convex Adaptive Rejection Sampling
Concave Convex Adaptive Rejection Sampling Dilan G¨or¨ur and Yee Whye Teh Gatsby Computational Neuroscience Unit, University College London {dilan,ywteh}@gatsby.ucl.ac.uk Abstract. We describe a method for generating independent samples from arbitrary density functions using adaptive rejection sampling with- out the log-concavity requirement. The method makes use of the fact that a function can be expressed as a sum of concave and convex func- tions. Using a concave convex decomposition, we bound the log-density using piecewise linear functions for and use the upper bound as the sam- pling distribution. We use the same function decomposition approach to approximate integrals which requires only a slight change in the sampling algorithm. 1 Introduction Probabilistic graphical models have become popular tools for addressing many machine learning and statistical inference problems in recent years. This has been especially accelerated by general-purpose inference toolkits like BUGS [1], VIBES [2], and infer.NET [3], which allow users of graphical models to spec- ify the models and obtain posterior inferences given evidence without worrying about the underlying inference algorithm. These toolkits rely upon approximate inference techniques that make use of the many conditional independencies in graphical models for efficient computation. By far the most popular of these toolkits, especially in the Bayesian statistics community, is BUGS. BUGS is based on Gibbs sampling, a Markov chain Monte Carlo (MCMC) sampler where one variable is sampled at a time conditioned on its Markov blanket. When the conditional distributions are of standard form, samples are easily obtained using standard algorithms [4]. When these condi- tional distributions are not of standard form (e.g. -
Approximate Slice Sampling for Bayesian Posterior Inference
Approximate Slice Sampling for Bayesian Posterior Inference Christopher DuBois∗ Anoop Korattikara∗ Max Welling Padhraic Smyth GraphLab, Inc. Dept. of Computer Science Informatics Institute Dept. of Computer Science UC Irvine University of Amsterdam UC Irvine Abstract features, or as is now popular in the context of deep learning, dropout. We believe deep learning is so suc- cessful because it provides the means to train arbitrar- In this paper, we advance the theory of large ily complex models on very large datasets that can be scale Bayesian posterior inference by intro- effectively regularized. ducing a new approximate slice sampler that uses only small mini-batches of data in ev- Methods for Bayesian posterior inference provide a ery iteration. While this introduces a bias principled alternative to frequentist regularization in the stationary distribution, the computa- techniques. The workhorse for approximate posterior tional savings allow us to draw more samples inference has long been MCMC, which unfortunately in a given amount of time and reduce sam- is not (yet) quite up to the task of dealing with very pling variance. We empirically verify on three large datasets. The main reason is that every iter- different models that the approximate slice ation of sampling requires O(N) calculations to de- sampling algorithm can significantly outper- cide whether a proposed parameter is accepted or not. form a traditional slice sampler if we are al- Frequentist methods based on stochastic gradients are lowed only a fixed amount of computing time orders of magnitude faster because they effectively ex- for our simulations. ploit the fact that far away from convergence, the in- formation in the data about how to improve the model is highly redundant. -
Reparameterization Gradients Through Acceptance-Rejection Sampling Algorithms
Reparameterization Gradients through Acceptance-Rejection Sampling Algorithms Christian A. Naessethyz Francisco J. R. Ruizzx Scott W. Lindermanz David M. Bleiz yLink¨opingUniversity zColumbia University xUniversity of Cambridge Abstract 2014] and text [Hoffman et al., 2013]. By definition, the success of variational approaches depends on our Variational inference using the reparameteri- ability to (i) formulate a flexible parametric family zation trick has enabled large-scale approx- of distributions; and (ii) optimize the parameters to imate Bayesian inference in complex pro- find the member of this family that most closely ap- babilistic models, leveraging stochastic op- proximates the true posterior. These two criteria are timization to sidestep intractable expecta- at odds|the more flexible the family, the more chal- tions. The reparameterization trick is appli- lenging the optimization problem. In this paper, we cable when we can simulate a random vari- present a novel method that enables more efficient op- able by applying a differentiable determinis- timization for a large class of variational distributions, tic function on an auxiliary random variable namely, for distributions that we can efficiently sim- whose distribution is fixed. For many dis- ulate by acceptance-rejection sampling, or rejection tributions of interest (such as the gamma or sampling for short. Dirichlet), simulation of random variables re- For complex models, the variational parameters can lies on acceptance-rejection sampling. The be optimized by stochastic gradient ascent on the evi- discontinuity introduced by the accept{reject dence lower bound (elbo), a lower bound on the step means that standard reparameterization marginal likelihood of the data. There are two pri- tricks are not applicable. -
Two Adaptive Rejection Sampling Schemes for Probability Density Functions Log-Convex Tails
1 Two adaptive rejection sampling schemes for probability density functions log-convex tails Luca Martino and Joaqu´ın M´ıguez Department of Signal Theory and Communications, Universidad Carlos III de Madrid. Avenida de la Universidad 30, 28911 Leganes,´ Madrid, Spain. E-mail: [email protected], [email protected] Abstract Monte Carlo methods are often necessary for the implementation of optimal Bayesian estimators. A fundamental technique that can be used to generate samples from virtually any target probability distribution is the so-called rejection sampling method, which generates candidate samples from a proposal distribution and then accepts them or not by testing the ratio of the target and proposal densities. The class of adaptive rejection sampling (ARS) algorithms is particularly interesting because they can achieve high acceptance rates. However, the standard ARS method can only be used with log-concave target densities. For this reason, many generalizations have been proposed. In this work, we investigate two different adaptive schemes that can be used to draw exactly from a large family of univariate probability density functions (pdf’s), not necessarily log-concave, possibly multimodal and with tails of arbitrary concavity. These techniques are adaptive in the sense that every time a candidate sample is rejected, the acceptance rate is improved. The two proposed algorithms can work properly when the target pdf is multimodal, with first and second derivatives analytically intractable, and when the tails are log-convex in a infinite domain. Therefore, they can be applied in a number of scenarios in which the other generalizations of the standard ARS fail. -
Random Number Generation for the Generalized Normal Distribution Using the Modified Adaptive Rejection Method
INFORMATION ISSN 1343-4500 Volume 8, Number 6, pp. 829-836 @2005 International Information Institute Random number generation for the generalized normal distribution using the modified adaptive rejection method Hideo Hirose, Akihiro Todoroki Department of systems Innovation & Informatics Kyushu Institute of Technology Iizuka, Fukuoka 820-8502 Japan Abstract When we want to grasp the characteristics of the time series signals emitted massively from electric power apparatuses or electroencephalogram, and want to decide some diagnoses about the apparatuses or human brains, we may use some statistical distribution functions. In such cases, the generalized normal distribution is frequently used in pattern analysts. In assessing the correctness of the estimates of the shape of the distribution function accurately, we often use a Monte Carlo simulation study; thus, a fast and efficient random number generation method for the distribution function is needed. However, the method for generating the random numbers of the distribution seems not easy and not yet to have been developed. In this paper, we propose a random number generation method for the distribution function using the the rejection method. A newly developed modified adaptive rejection method works well in the case of log-convex density functions. Keywords: modified adaptive rejection method, exponential distribution, generalized normal distri bution, envelop function, log-convex, log-concave 1 Introduction vVhen we want to grasp the characteristics of the time series signals emitted massively from electric power apparatuses or electroencephalogram, and want to decide some diagnoses about the apparatuses or human brains, we may use some statistical distribution functions. For example, we take a distribution pattern in certain time zone by making average in the zone, and we see the differences among those zones by discriminating the distribution patterns zone by zone.