Variability in Encoding Precision Accounts for Visual Short-Term Memory Limitations
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Hyperprior on Symmetric Dirichlet Distribution
Hyperprior on symmetric Dirichlet distribution Jun Lu Computer Science, EPFL, Lausanne [email protected] November 5, 2018 Abstract In this article we introduce how to put vague hyperprior on Dirichlet distribution, and we update the parameter of it by adaptive rejection sampling (ARS). Finally we analyze this hyperprior in an over-fitted mixture model by some synthetic experiments. 1 Introduction It has become popular to use over-fitted mixture models in which number of cluster K is chosen as a conservative upper bound on the number of components under the expectation that only relatively few of the components K0 will be occupied by data points in the samples X . This kind of over-fitted mixture models has been successfully due to the ease in computation. Previously Rousseau & Mengersen(2011) proved that quite generally, the posterior behaviour of overfitted mixtures depends on the chosen prior on the weights, and on the number of free parameters in the emission distributions (here D, i.e. the dimension of data). Specifically, they have proved that (a) If α=min(αk; k ≤ K)>D=2 and if the number of components is larger than it should be, asymptotically two or more components in an overfitted mixture model will tend to merge with non-negligible weights. (b) −1=2 In contrast, if α=max(αk; k 6 K) < D=2, the extra components are emptied at a rate of N . Hence, if none of the components are small, it implies that K is probably not larger than K0. In the intermediate case, if min(αk; k ≤ K) ≤ D=2 ≤ max(αk; k 6 K), then the situation varies depending on the αk’s and on the difference between K and K0. -
A Dirichlet Process Mixture Model of Discrete Choice Arxiv:1801.06296V1
A Dirichlet Process Mixture Model of Discrete Choice 19 January, 2018 Rico Krueger (corresponding author) Research Centre for Integrated Transport Innovation, School of Civil and Environmental Engineering, UNSW Australia, Sydney NSW 2052, Australia [email protected] Akshay Vij Institute for Choice, University of South Australia 140 Arthur Street, North Sydney NSW 2060, Australia [email protected] Taha H. Rashidi Research Centre for Integrated Transport Innovation, School of Civil and Environmental Engineering, UNSW Australia, Sydney NSW 2052, Australia [email protected] arXiv:1801.06296v1 [stat.AP] 19 Jan 2018 1 Abstract We present a mixed multinomial logit (MNL) model, which leverages the truncated stick- breaking process representation of the Dirichlet process as a flexible nonparametric mixing distribution. The proposed model is a Dirichlet process mixture model and accommodates discrete representations of heterogeneity, like a latent class MNL model. Yet, unlike a latent class MNL model, the proposed discrete choice model does not require the analyst to fix the number of mixture components prior to estimation, as the complexity of the discrete mixing distribution is inferred from the evidence. For posterior inference in the proposed Dirichlet process mixture model of discrete choice, we derive an expectation maximisation algorithm. In a simulation study, we demonstrate that the proposed model framework can flexibly capture differently-shaped taste parameter distributions. Furthermore, we empirically validate the model framework in a case study on motorists’ route choice preferences and find that the proposed Dirichlet process mixture model of discrete choice outperforms a latent class MNL model and mixed MNL models with common parametric mixing distributions in terms of both in-sample fit and out-of-sample predictive ability. -
Part 2: Basics of Dirichlet Processes 2.1 Motivation
CS547Q Statistical Modeling with Stochastic Processes Winter 2011 Part 2: Basics of Dirichlet processes Lecturer: Alexandre Bouchard-Cˆot´e Scribe(s): Liangliang Wang Disclaimer: These notes have not been subjected to the usual scrutiny reserved for formal publications. They may be distributed outside this class only with the permission of the Instructor. Last update: May 17, 2011 2.1 Motivation To motivate the Dirichlet process, let us consider a simple density estimation problem: modeling the height of UBC students. We are going to take a Bayesian approach to this problem, considering the parameters as random variables. In one of the most basic models, one would define a mean and variance random parameter θ = (µ, σ2), and a height random variable normally distributed conditionally on θ, with parameters θ.1 Using a single normal distribution is clearly a defective approach, since for example the male/female sub- populations create a skewness in the distribution, which cannot be capture by normal distributions: The solution suggested by this figure is to use a mixture of two normal distributions, with one set of parameters θc for each sub-population or cluster c ∈ {1, 2}. Pictorially, the model can be described as follows: 1- Generate a male/female relative frequence " " ~ Beta(male prior pseudo counts, female P.C) Mean height 2- Generate the sex of each student for each i for men Mult( ) !(1) x1 x2 x3 xi | " ~ " 3- Generate the mean height of each cluster c !(2) Mean height !(c) ~ N(prior height, how confident prior) for women y1 y2 y3 4- Generate student heights for each i yi | xi, !(1), !(2) ~ N(!(xi) ,variance) 1Yes, this has many problems (heights cannot be negative, normal assumption broken, etc). -
Maximum Likelihood Estimation of Dirichlet Distribution Parameters
Maximum Likelihood Estimation of Dirichlet Distribution Parameters Jonathan Huang Abstract. Dirichlet distributions are commonly used as priors over propor- tional data. In this paper, I will introduce this distribution, discuss why it is useful, and compare implementations of 4 different methods for estimating its parameters from observed data. 1. Introduction The Dirichlet distribution is one that has often been turned to in Bayesian statistical inference as a convenient prior distribution to place over proportional data. To properly motivate its study, we will begin with a simple coin toss example, where the task will be to find a suitable distribution P which summarizes our beliefs about the probability that the toss will result in heads, based on all prior such experiments. 16 14 12 10 8 6 4 2 0 0 0.2 0.4 0.6 0.8 1 H/(H+T) Figure 1. A distribution over possible probabilities of obtaining heads We will want to convey several things via such a distribution. First, if we have an idea of what the odds of heads are, then we will want P to reflect this. For example, if we associate P with the experiment of flipping a penny, we would hope that P gives strong probability to 50-50 odds. Second, we will want the distribution to somehow reflect confidence by expressing how many coin flips we have witnessed 1 2 JONATHAN HUANG in the past, the idea being that the more coin flips one has seen, the more confident one is about how a coin must behave. In the case where we have never seen a coin flip experiment, then P should assign uniform probability to all odds. -
A Dirichlet Process Model for Detecting Positive Selection in Protein-Coding DNA Sequences
A Dirichlet process model for detecting positive selection in protein-coding DNA sequences John P. Huelsenbeck*†, Sonia Jain‡, Simon W. D. Frost§, and Sergei L. Kosakovsky Pond§ *Section of Ecology, Behavior, and Evolution, Division of Biological Sciences, University of California at San Diego, La Jolla, CA 92093-0116; ‡Division of Biostatistics and Bioinformatics, Department of Family and Preventive Medicine, University of California at San Diego, La Jolla, CA 92093-0717; and §Antiviral Research Center, University of California at San Diego, 150 West Washington Street, La Jolla, CA 92103 Edited by Joseph Felsenstein, University of Washington, Seattle, WA, and approved March 1, 2006 (received for review September 21, 2005) Most methods for detecting Darwinian natural selection at the natural selection that is present at only a few positions in the molecular level rely on estimating the rates or numbers of non- alignment (21). In cases where such methods have been successful, synonymous and synonymous changes in an alignment of protein- many sites are typically under strong positive selection (e.g., MHC). coding DNA sequences. In some of these methods, the nonsyn- More recently, Nielsen and Yang (2) developed a method that onymous rate of substitution is allowed to vary across the allows the rate of nonsynonymous substitution to vary across the sequence, permitting the identification of single amino acid posi- sequence. The method of Nielsen and Yang has proven useful for tions that are under positive natural selection. However, it is detecting positive natural selection in sequences where only a few unclear which probability distribution should be used to describe sites are under directional selection. -
Kernel Analysis Based on Dirichlet Processes Mixture Models
entropy Article Kernel Analysis Based on Dirichlet Processes Mixture Models Jinkai Tian 1,2, Peifeng Yan 1,2 and Da Huang 1,2,* 1 State Key Laboratory of High Performance Computing, National University of Defense Technology, Changsha 410073, China 2 College of Computer, National University of Defense Technology, Changsha 410073, China * Correspondence: [email protected] Received: 22 July 2019; Accepted: 29 August 2019; Published: 2 September 2019 Abstract: Kernels play a crucial role in Gaussian process regression. Analyzing kernels from their spectral domain has attracted extensive attention in recent years. Gaussian mixture models (GMM) are used to model the spectrum of kernels. However, the number of components in a GMM is fixed. Thus, this model suffers from overfitting or underfitting. In this paper, we try to combine the spectral domain of kernels with nonparametric Bayesian models. Dirichlet processes mixture models are used to resolve this problem by changing the number of components according to the data size. Multiple experiments have been conducted on this model and it shows competitive performance. Keywords: Dirichlet processes; Dirichlet processes mixture models; Gaussian mixture models; kernel; spectral analysis 1. Introduction Probabilistic models are essential for machine learning to have a correct understanding of problems. While frequentist models are considered a traditional method for statisticians, Bayesian models are more widely used in machine learning fields. A Bayesian model is a statistical model where probability is used to represent all uncertainties in the model. Uncertainty of both input and output are under consideration in a Bayesian model. A parametric Bayesian model assumes a finite set of parameters which is not flexible because the complexity of the model is bounded while the amount of data is unbounded. -
Estimating the Mutual Information Between Two Discrete, Asymmetric Variables with Limited Samples
entropy Article Estimating the Mutual Information between Two Discrete, Asymmetric Variables with Limited Samples Damián G. Hernández * and Inés Samengo Department of Medical Physics, Centro Atómico Bariloche and Instituto Balseiro, 8400 San Carlos de Bariloche, Argentina; [email protected] * Correspondence: [email protected] Received: 3 May 2019; Accepted: 13 June 2019; Published: 25 June 2019 Abstract: Determining the strength of nonlinear, statistical dependencies between two variables is a crucial matter in many research fields. The established measure for quantifying such relations is the mutual information. However, estimating mutual information from limited samples is a challenging task. Since the mutual information is the difference of two entropies, the existing Bayesian estimators of entropy may be used to estimate information. This procedure, however, is still biased in the severely under-sampled regime. Here, we propose an alternative estimator that is applicable to those cases in which the marginal distribution of one of the two variables—the one with minimal entropy—is well sampled. The other variable, as well as the joint and conditional distributions, can be severely undersampled. We obtain a consistent estimator that presents very low bias, outperforming previous methods even when the sampled data contain few coincidences. As with other Bayesian estimators, our proposal focuses on the strength of the interaction between the two variables, without seeking to model the specific way in which they are related. A distinctive property of our method is that the main data statistics determining the amount of mutual information is the inhomogeneity of the conditional distribution of the low-entropy variable in those states in which the large-entropy variable registers coincidences. -
Nonparametric Density Estimation for Learning Noise Distributions in Mobile Robotics
Nonparametric Density Estimation for Learning Noise Distributions in Mobile Robotics David M. Rosen and John J. Leonard Histogram of observations Abstract—By admitting an explicit representation of the uncer- 350 tainty associated with sensing and actuation in the real world, the probabilistic robotics paradigm has led to remarkable improve- 300 ments in the robustness and performance of autonomous systems. However, this approach generally requires that the sensing and 250 actuation noise acting upon an autonomous system be specified 200 in the form of probability density functions, thus necessitating Count that these densities themselves be estimated. To that end, in this 150 paper we present a general framework for directly estimating the probability density function of a noise distribution given only a 100 set of observations sampled from that distribution. Our approach is based upon Dirichlet process mixture models (DPMMs), a 50 class of infinite mixture models commonly used in Bayesian 0 −10 −5 0 5 10 15 20 25 nonparametric statistics. These models are very expressive and Observation enjoy good posterior consistency properties when used in density estimation, but the density estimates that they produce are often too computationally complex to be used in mobile robotics Fig. 1. A complicated noise distribution: This figure shows a histogram of applications, where computational resources are limited and 3000 observations sampled from a complicated noise distribution in simulation speed is crucial. We derive a straightforward yet principled (cf. equations (37)–(40)). The a priori identification of a parametric family capable of accurately characterizing complex or poorly-understood noise approximation method for simplifying the densities learned by distributions such as the one shown here is a challenging problem, which DPMMs in order to produce computationally tractable density renders parametric density estimation approaches vulnerable to model mis- estimates. -
Dirichlet Process Gaussian Mixture Models: Choice of the Base Distribution
GÄorÄurD, Rasmussen CE. Dirichlet process Gaussian mixture models: Choice of the base distribution. JOURNAL OF COMPUTER SCIENCE AND TECHNOLOGY 25(4): 615{626 July 2010/DOI 10.1007/s11390-010-1051-1 Dirichlet Process Gaussian Mixture Models: Choice of the Base Distribution Dilan GÄorÄur1 and Carl Edward Rasmussen2;3 1Gatsby Computational Neuroscience Unit, University College London, London WC1N 3AR, U.K. 2Department of Engineering, University of Cambridge, Cambridge CB2 1PZ, U.K. 3Max Planck Institute for Biological Cybernetics, D-72076 TÄubingen,Germany E-mail: [email protected]; [email protected] Received May 2, 2009; revised February 10, 2010. Abstract In the Bayesian mixture modeling framework it is possible to infer the necessary number of components to model the data and therefore it is unnecessary to explicitly restrict the number of components. Nonparametric mixture models sidestep the problem of ¯nding the \correct" number of mixture components by assuming in¯nitely many components. In this paper Dirichlet process mixture (DPM) models are cast as in¯nite mixture models and inference using Markov chain Monte Carlo is described. The speci¯cation of the priors on the model parameters is often guided by mathematical and practical convenience. The primary goal of this paper is to compare the choice of conjugate and non-conjugate base distributions on a particular class of DPM models which is widely used in applications, the Dirichlet process Gaussian mixture model (DPGMM). We compare computational e±ciency and modeling performance of DPGMM de¯ned using a conjugate and a conditionally conjugate base distribution. We show that better density models can result from using a wider class of priors with no or only a modest increase in computational e®ort. -
Robust Estimation of Risks from Small Samples
Robust estimation of risks from small samples Simon H. Tindemans∗ and Goran Strbac Department of Electrical and Electronic Engineering, Imperial College London, London SW7 2AZ, UK Data-driven risk analysis involves the inference of probability distributions from measured or simulated data. In the case of a highly reliable system, such as the electricity grid, the amount of relevant data is often exceedingly limited, but the impact of estimation errors may be very large. This paper presents a robust nonparametric Bayesian method to infer possible underlying distributions. The method obtains rigorous error bounds even for small samples taken from ill- behaved distributions. The approach taken has a natural interpretation in terms of the intervals between ordered observations, where allocation of probability mass across intervals is well-specified, but the location of that mass within each interval is unconstrained. This formulation gives rise to a straightforward computational resampling method: Bayesian Interval Sampling. In a comparison with common alternative approaches, it is shown to satisfy strict error bounds even for ill-behaved distributions. Keywords: Rare event analysis, Bayesian inference, nonparametric methods, Dirichlet process, imprecise probabilities, resampling methods I. MOTIVATION: RISK ANALYSIS FOR GRIDS Managing a critical infrastructure such as the electricity grid requires constant vigilance on the part of its operator. The system operator identifies threats, determines the risks associated with these threats and, where necessary, takes action to mitigate them. In the case of the electricity grid, these threats include sudden outages of lines and generators, misprediction of renewable generating capacity, and long-term investment shortages. Quantitative risk assessments often amounts to data analysis. -
Paper: a Tutorial on Dirichlet Process Mixture Modeling
Journal of Mathematical Psychology 91 (2019) 128–144 Contents lists available at ScienceDirect Journal of Mathematical Psychology journal homepage: www.elsevier.com/locate/jmp Tutorial A tutorial on Dirichlet process mixture modeling ∗ Yuelin Li a,b, , Elizabeth Schofield a, Mithat Gönen b a Department of Psychiatry & Behavioral Sciences, Memorial Sloan Kettering Cancer Center, New York, NY 10022, USA b Department of Epidemiology & Biostatistics, Memorial Sloan Kettering Cancer Center, New York, NY 10017, USA h i g h l i g h t s • Mathematical derivations essential to the development of Dirichlet Process are provided. • An accessible pedagogy for complex and abstract mathematics in DP modeling. • A publicly accessible computer program in R is explained line-by-line. article info a b s t r a c t Article history: Bayesian nonparametric (BNP) models are becoming increasingly important in psychology, both as Received 14 June 2018 theoretical models of cognition and as analytic tools. However, existing tutorials tend to be at a level of Received in revised form 21 March 2019 abstraction largely impenetrable by non-technicians. This tutorial aims to help beginners understand Available online xxxx key concepts by working through important but often omitted derivations carefully and explicitly, Keywords: with a focus on linking the mathematics with a practical computation solution for a Dirichlet Process Bayesian nonparametric Mixture Model (DPMM)—one of the most widely used BNP methods. Abstract concepts are made Dirichlet process explicit and concrete to non-technical readers by working through the theory that gives rise to them. Mixture model A publicly accessible computer program written in the statistical language R is explained line-by-line Gibbs sampling to help readers understand the computation algorithm. -
Variational Inference for Beta-Bernoulli Dirichlet Process Mixture Models
Variational Inference for Beta-Bernoulli Dirichlet Process Mixture Models Mengrui Ni, Erik B. Sudderth (Advisor), Mike Hughes (Advisor) Department of Computer Science, Brown University [email protected], [email protected], [email protected] Abstract A commonly used paradigm in diverse application areas is to assume that an observed set of individual binary features is generated from a Bernoulli distribution with probabilities varying according to a Beta distribution. In this paper, we present our nonparametric variational inference algorithm for the Beta-Bernoulli observation model. Our primary focus is clustering discrete binary data using the Dirichlet process (DP) mixture model. 1 Introduction In many biological and vision studies, the presences and absences of a certain set of engineered attributes create binary outcomes which can be used to describe and distinguish individual objects. Observations of such discrete binary feature data are usually assumed to be generated from mixtures of Bernoulli distributions with probability parameters µ governed by Beta distributions. Such a model, also known as latent class analysis [6], is practically important in its own right. Our discussion will focus on clustering discrete binary data via the Dirichlet process (DP) mixture model. 2 Beta-Bernoulli Observation Model The Beta-Bernoulli model is one of the simplest Bayesian models with Bernoulli likelihood p(X | µ) and its conjugate Beta prior p(µ). 2.1 Bernoulli distribution Consider a binary dataset with N observations and D attributes: x11 x12 ··· x1D x x ··· x 21 22 2D X = . , . .. xN1 xN2 ··· xND where X1,X2,...,XN ∼ i.i.d. Bernoulli(X | µ). Each Xi is a vector of 0’s and 1’s.