DOCSLIB.ORG

A Non-Markov Continuous-Time Model of Topical Trends

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
A Non-Markov Continuous-Time Model of Topical Trends Topics over Time: A Non-Markov Continuous-Time Model of Topical Trends Xuerui Wang, Andrew McCallum Department of Computer Science University of Massachusetts Amherst, MA 01003 {xuerui,mccallum}@cs.umass.edu ABSTRACT patterns) [15]. In each case, graphical model structures are This paper presents an LDA-style topic model that captures carefully-designed to capture the relevant structure and co- not only the low-dimensional structure of data, but also occurrence dependencies in the data. how the structure changes over time. Unlike other recent work that relies on Markov assumptions or discretization of Many of the large data sets to which these topic models time, here each topic is associated with a continuous distri- are applied do not have static co-occurrence patterns; they bution over timestamps, and for each generated document, are instead dynamic. The data are often collected over the mixture distribution over topics is influenced by both time, and generally patterns present in the early part of word co-occurrences and the document’s timestamp. Thus, the collection are not in effect later. Topics rise and fall in the meaning of a particular topic can be relied upon as con- prominence; they split apart; they merge to form new top- stant, but the topics’ occurrence and correlations change ics; words change their correlations. For example, across 17 significantly over time. We present results on nine months years of the Neural Information Processing Systems (NIPS) of personal email, 17 years of NIPS research papers and conference, activity in “analog circuit design” has fallen off over 200 years of presidential state-of-the-union addresses, somewhat, while research in “support vector machines” has showing improved topics, better timestamp prediction, and recently risen dramatically. The topic “dynamic systems” interpretable trends. used to co-occur with “neural networks,” but now co-occurs with “graphical models.” Categories and Subject Descriptors However none of the topic models mentioned above are aware I.2.6 [Artificial Intelligence]: Learning; H.2.8 [Database of these dependencies on document timestamps. Not model- Management]: Database Applications—data mining ing time can confound co-occurrence patterns and result in unclear, sub-optimal topic discovery. For example, in topic General Terms analysis of U.S. Presidential State-of-the-Union addresses, Algorithms, experimentation LDA confounds Mexican-American War (1846-1848) with some aspects of World War I (1914-1918), because LDA is Keywords unaware of the 70-year separation between the two events. Some previous work has performed some post-hoc analysis— Graphical models, temporal analysis, topic modeling discovering topics without the use of timestamps and then projecting their occurrence counts into discretized time [4]— 1. INTRODUCTION but this misses the opportunity for time to improve topic Research in statistical models of co-occurrence has led to the discovery. development of a variety of useful topic models—mechanisms for discovering low-dimensional, multi-faceted summaries of This paper presents Topics over Time (TOT), a topic model documents or other discrete data. These include models that explicitly models time jointly with word co-occurrence of words alone, such as Latent Dirichlet Allocation (LDA) patterns. Significantly, and unlike some recent work with [2, 4], of words and research paper citations [3], of word similar goals, our model does not discretize time, and does sequences with Markov dependencies [5], of words and their not make Markov assumptions over state transitions in time. authors [11], of words in a social network of senders and Rather, TOT parameterizes a continuous distribution over recipients [9], and of words and relations (such as voting time associated with each topic, and topics are responsible for generating both observed timestamps as well as words. Parameter estimation is thus driven to discover topics that simultaneously capture word co-occurrences and locality of those patterns in time. When a strong word co-occurrence pattern appears for a brief moment in time then disappears, TOT will create a topic with a narrow time distribution. (Given enough ev- ACM SIGKDD-2006 August 20-23, 2005, Philadelphia, Pennsylvania, USA idence, arbitrarily small spans can be represented, unlike schemes based on discretizing time.) When a pattern of naturally injected into other more richly structured topic word co-occurrence remains consistent across a long time models, such as the Author-Recipient-Topic model to cap- span, TOT will create a topic with a broad time distribu- ture changes in social network roles over time [9], and the tion. In current experiments, we use a Beta distribution over Group-Topic model to capture changes in group formation a (normalized) time span covering all the data, and thus we over time [15]. can also flexibly represent various skewed shapes of rising and falling topic prominence. We present experimental results with three real-world data sets. On over two centuries of U.S. Presidential State-of-the- The model’s generative storyline can be understood in two Union addresses, we show that TOT discovers topics with different ways. We fit the model parameters according to a both time-localization and word-clarity improvements over generative model in which a per-document multinomial dis- LDA. On the 17-year history of the NIPS conference, we tribution over topics is sampled from a Dirichlet, then for show clearly interpretable topical trends, as well as a two- each word occurrence we sample a topic; next a per-topic fold increase in the ability to predict time given a document. multinomial generates the word, and a per-topic Beta dis- On nine months of the second author’s email archive, we tribution generates the document’s time stamp. Here the show another example of clearly interpretable, time-localized time stamp (which in practice is always observed and con- topics, such as springtime faculty recruiting. On all three stant across the document) is associated with each word in datasets, TOT provides more distinct topics, as measured the document. We can also imagine an alternative, cor- by KL divergence. responding generative model in which the time stamp is generated once per document, conditioned directly on the 2. TOPICS OVER TIME per-document mixture over topics. In both cases, the likeli- Before introducing the Topics over Time (TOT) model, let hood contribution from the words and the contribution from us review the basic Latent Dirichlet Allocation model. Our the timestamps may need to be weighted by some factor, as notation is summarized in Table 1, and the graphical model in the balancing of acoustic models and language models representations of both LDA and our TOT models are shown in speech recognition. The later generative storyline more in Figure 1. directly corresponds to common data sets (with one times- tamp per document); the former is easier to fit, and can also Latent Dirichlet Allocation (LDA) is a Bayesian network allow some flexibility in which different parts of the docu- that generates a document using a mixture of topics [2]. ment may be discussing different time periods. In its generative process, for each document d, a multino- mial distribution θ over topics is randomly sampled from a Some previous studies have also shown that topic discov- Dirichlet with parameter α, and then to generate each word, ery can be influenced by information in addition to word a topic z is chosen from this topic distribution, and a word, co-occurrences. For example, the Group-Topic model [15] di w , is generated by randomly sampling from a topic-specific showed that the joint modeling of word co-occurrence and di multinomial distribution φ . The robustness of the model voting relations resulted in more salient, relevant topics. zdi is greatly enhanced by integrating out uncertainty about the The Mixed-Membership model [3] also showed interesting per-document topic distribution θ and the per-topic word results for research papers and their citations. distribution φ. Note that, in contrast to other work that models trajecto- In the TOT model, topic discovery is influenced not only by ries of individual topics over time, TOT topics and their word co-occurrences, but also temporal information. Rather meaning are modeled as constant over time. TOT captures than modeling a sequence of state changes with a Markov changes in the occurrence (and co-occurrence conditioned assumption on the dynamics, TOT models (normalized) ab- on time) of the topics themselves, not changes in the word solute timestamp values. This allows TOT to see long-range distribution of each topic. While choosing to model individ- dependencies in time, to predict absolute time values given ual topics as mutable could be useful, it can also be dan- an unstamped document, and to predict topic distributions gerous. Imagine a subset of documents containing strong given a timestamp. It also helps avoid a Markov model’s co-occurrence patterns across time: first between birds and risk of inappropriately dividing a topic in two when there is aerodynamics, then aerodynamics and heat, then heat and a brief gap in its appearance. quantum mechanics—this could lead to a single topic that follows this trajectory, and lead the user to inappropriately Time is intrinsically continuous. Discretization of time al- conclude that birds and quantum mechanics are time-shifted ways begs the question of selecting the slice size, and the versions of the same topic. Alternatively, consider a large size is invariably too small for some regions and too large subject like medicine, which has changed drastically over for others. time. In TOT we choose to model these shifts as changes in topic co-occurrence—a decrease in occurrence of topics Avoiding discretization, in TOT each topic is associated about blood-letting and bile, and an increase in topics about with a continuous distribution over time.
Load more

Recommended publications
  • 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.
    [Show full text]
  • Machine Learning Conjugate Priors and Monte Carlo Methods
    Hierarchical Bayes for Non-IID Data Conjugate Priors Monte Carlo Methods CPSC 540: Machine Learning Conjugate Priors and Monte Carlo Methods Mark Schmidt University of British Columbia Winter 2016 Hierarchical Bayes for Non-IID Data Conjugate Priors Monte Carlo Methods Admin Nothing exciting? We discussed empirical Bayes, where you optimize prior using marginal likelihood, Z argmax p(xjα; β) = argmax p(xjθ)p(θjα; β)dθ: α,β α,β θ Can be used to optimize λj, polynomial degree, RBF σi, polynomial vs. RBF, etc. We also considered hierarchical Bayes, where you put a prior on the prior, p(xjα; β)p(α; βjγ) p(α; βjx; γ) = : p(xjγ) But is the hyper-prior really needed? Hierarchical Bayes for Non-IID Data Conjugate Priors Monte Carlo Methods Last Time: Bayesian Statistics In Bayesian statistics we work with posterior over parameters, p(xjθ)p(θjα; β) p(θjx; α; β) = : p(xjα; β) We also considered hierarchical Bayes, where you put a prior on the prior, p(xjα; β)p(α; βjγ) p(α; βjx; γ) = : p(xjγ) But is the hyper-prior really needed? Hierarchical Bayes for Non-IID Data Conjugate Priors Monte Carlo Methods Last Time: Bayesian Statistics In Bayesian statistics we work with posterior over parameters, p(xjθ)p(θjα; β) p(θjx; α; β) = : p(xjα; β) We discussed empirical Bayes, where you optimize prior using marginal likelihood, Z argmax p(xjα; β) = argmax p(xjθ)p(θjα; β)dθ: α,β α,β θ Can be used to optimize λj, polynomial degree, RBF σi, polynomial vs.
    [Show full text]
  • Polynomial Singular Value Decompositions of a Family of Source-Channel Models
    Polynomial Singular Value Decompositions of a Family of Source-Channel Models The MIT Faculty has made this article openly available. Please share how this access benefits you. Your story matters. Citation Makur, Anuran and Lizhong Zheng. "Polynomial Singular Value Decompositions of a Family of Source-Channel Models." IEEE Transactions on Information Theory 63, 12 (December 2017): 7716 - 7728. © 2017 IEEE As Published http://dx.doi.org/10.1109/tit.2017.2760626 Publisher Institute of Electrical and Electronics Engineers (IEEE) Version Author's final manuscript Citable link https://hdl.handle.net/1721.1/131019 Terms of Use Creative Commons Attribution-Noncommercial-Share Alike Detailed Terms http://creativecommons.org/licenses/by-nc-sa/4.0/ IEEE TRANSACTIONS ON INFORMATION THEORY 1 Polynomial Singular Value Decompositions of a Family of Source-Channel Models Anuran Makur, Student Member, IEEE, and Lizhong Zheng, Fellow, IEEE Abstract—In this paper, we show that the conditional expec- interested in a particular subclass of one-parameter exponential tation operators corresponding to a family of source-channel families known as natural exponential families with quadratic models, defined by natural exponential families with quadratic variance functions (NEFQVF). So, we define natural exponen- variance functions and their conjugate priors, have orthonor- mal polynomials as singular vectors. These models include the tial families next. Gaussian channel with Gaussian source, the Poisson channel Definition 1 (Natural Exponential Family). Given a measur- with gamma source, and the binomial channel with beta source. To derive the singular vectors of these models, we prove and able space (Y; B(Y)) with a σ-finite measure µ, where Y ⊆ R employ the equivalent condition that their conditional moments and B(Y) denotes the Borel σ-algebra on Y, the parametrized are strictly degree preserving polynomials.
    [Show full text]
  • A Compendium of Conjugate Priors
    A Compendium of Conjugate Priors Daniel Fink Environmental Statistics Group Department of Biology Montana State Univeristy Bozeman, MT 59717 May 1997 Abstract This report reviews conjugate priors and priors closed under sampling for a variety of data generating processes where the prior distributions are univariate, bivariate, and multivariate. The effects of transformations on conjugate prior relationships are considered and cases where conjugate prior relationships can be applied under transformations are identified. Univariate and bivariate prior relationships are verified using Monte Carlo methods. Contents 1 Introduction Experimenters are often in the position of having had collected some data from which they desire to make inferences about the process that produced that data. Bayes' theorem provides an appealing approach to solving such inference problems. Bayes theorem, π(θ) L(θ x ; : : : ; x ) g(θ x ; : : : ; x ) = j 1 n (1) j 1 n π(θ) L(θ x ; : : : ; x )dθ j 1 n is commonly interpreted in the following wayR. We want to make some sort of inference on the unknown parameter(s), θ, based on our prior knowledge of θ and the data collected, x1; : : : ; xn . Our prior knowledge is encapsulated by the probability distribution on θ; π(θ). The data that has been collected is combined with our prior through the likelihood function, L(θ x ; : : : ; x ) . The j 1 n normalized product of these two components yields a probability distribution of θ conditional on the data. This distribution, g(θ x ; : : : ; x ) , is known as the posterior distribution of θ. Bayes' j 1 n theorem is easily extended to cases where is θ multivariate, a vector of parameters.
    [Show full text]
  • Bayesian Filtering: from Kalman Filters to Particle Filters, and Beyond ZHE CHEN
    MANUSCRIPT 1 Bayesian Filtering: From Kalman Filters to Particle Filters, and Beyond ZHE CHEN Abstract— In this self-contained survey/review paper, we system- IV Bayesian Optimal Filtering 9 atically investigate the roots of Bayesian filtering as well as its rich IV-AOptimalFiltering..................... 10 leaves in the literature. Stochastic filtering theory is briefly reviewed IV-BKalmanFiltering..................... 11 with emphasis on nonlinear and non-Gaussian filtering. Following IV-COptimumNonlinearFiltering.............. 13 the Bayesian statistics, different Bayesian filtering techniques are de- IV-C.1Finite-dimensionalFilters............ 13 veloped given different scenarios. Under linear quadratic Gaussian circumstance, the celebrated Kalman filter can be derived within the Bayesian framework. Optimal/suboptimal nonlinear filtering tech- V Numerical Approximation Methods 14 niques are extensively investigated. In particular, we focus our at- V-A Gaussian/Laplace Approximation ............ 14 tention on the Bayesian filtering approach based on sequential Monte V-BIterativeQuadrature................... 14 Carlo sampling, the so-called particle filters. Many variants of the V-C Mulitgrid Method and Point-Mass Approximation . 14 particle filter as well as their features (strengths and weaknesses) are V-D Moment Approximation ................. 15 discussed. Related theoretical and practical issues are addressed in V-E Gaussian Sum Approximation . ............. 16 detail. In addition, some other (new) directions on Bayesian filtering V-F Deterministic
    [Show full text]
  • 36-463/663: Hierarchical Linear Models
    36-463/663: Hierarchical Linear Models Taste of MCMC / Bayes for 3 or more “levels” Brian Junker 132E Baker Hall [email protected] 11/3/2016 1 Outline Practical Bayes Mastery Learning Example A brief taste of JAGS and RUBE Hierarchical Form of Bayesian Models Extending the “Levels” and the “Slogan” Mastery Learning – Distribution of “mastery” (to be continued…) 11/3/2016 2 Practical Bayesian Statistics (posterior) ∝ (likelihood) ×(prior) We typically want quantities like point estimate: Posterior mean, median, mode uncertainty: SE, IQR, or other measure of ‘spread’ credible interval (CI) (^θ - 2SE, ^θ + 2SE) (θ. , θ. ) Other aspects of the “shape” of the posterior distribution Aside: If (prior) ∝ 1, then (posterior) ∝ (likelihood) 1/2 posterior mode = mle, posterior SE = 1/I( θ) , etc. 11/3/2016 3 Obtaining posterior point estimates, credible intervals Easy if we recognize the posterior distribution and we have formulae for means, variances, etc. Whether or not we have such formulae, we can get similar information by simulating from the posterior distribution. Key idea: where θ, θ, …, θM is a sample from f( θ|data). 11/3/2016 4 Example: Mastery Learning Some computer-based tutoring systems declare that you have “mastered” a skill if you can perform it successfully r times. The number of times x that you erroneously perform the skill before the r th success is a measure of how likely you are to perform the skill correctly (how well you know the skill). The distribution for the number of failures x before the r th success is the negative binomial distribution.
    [Show full text]
  • A Geometric View of Conjugate Priors
    Proceedings of the Twenty-Second International Joint Conference on Artificial Intelligence A Geometric View of Conjugate Priors Arvind Agarwal Hal Daume´ III Department of Computer Science Department of Computer Science University of Maryland University of Maryland College Park, Maryland USA College Park, Maryland USA [email protected] [email protected] Abstract Using the same geometry also gives the closed-form solution for the maximum-a-posteriori (MAP) problem. We then ana- In Bayesian machine learning, conjugate priors are lyze the prior using concepts borrowed from the information popular, mostly due to mathematical convenience. geometry. We show that this geometry induces the Fisher In this paper, we show that there are deeper reasons information metric and 1-connection, which are respectively, for choosing a conjugate prior. Specifically, we for- the natural metric and connection for the exponential family mulate the conjugate prior in the form of Bregman (Section 5). One important outcome of this analysis is that it divergence and show that it is the inherent geome- allows us to treat the hyperparameters of the conjugate prior try of conjugate priors that makes them appropriate as the effective sample points drawn from the distribution un- and intuitive. This geometric interpretation allows der consideration. We finally extend this geometric interpre- one to view the hyperparameters of conjugate pri- tation of conjugate priors to analyze the hybrid model given ors as the effective sample points, thus providing by [7] in a purely geometric setting, and justify the argument additional intuition. We use this geometric under- presented in [1] (i.e. a coupling prior should be conjugate) standing of conjugate priors to derive the hyperpa- using a much simpler analysis (Section 6).
    [Show full text]
  • 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].
    [Show full text]
  • Sparsity Enforcing Priors in Inverse Problems Via Normal Variance
    Sparsity enforcing priors in inverse problems via Normal variance mixtures: model selection, algorithms and applications Mircea Dumitru1 1Laboratoire des signaux et systemes` (L2S), CNRS – CentraleSupelec´ – Universite´ Paris-Sud, CentraleSupelec,´ Plateau de Moulon, 91192 Gif-sur-Yvette, France Abstract The sparse structure of the solution for an inverse problem can be modelled using different sparsity enforcing pri- ors when the Bayesian approach is considered. Analytical expression for the unknowns of the model can be obtained by building hierarchical models based on sparsity enforcing distributions expressed via conjugate priors. We consider heavy tailed distributions with this property: the Student-t distribution, which is expressed as a Normal scale mixture, with the mixing distribution the Inverse Gamma distribution, the Laplace distribution, which can also be expressed as a Normal scale mixture, with the mixing distribution the Exponential distribution or can be expressed as a Normal inverse scale mixture, with the mixing distribution the Inverse Gamma distribution, the Hyperbolic distribution, the Variance-Gamma distribution, the Normal-Inverse Gaussian distribution, all three expressed via conjugate distribu- tions using the Generalized Hyperbolic distribution. For all distributions iterative algorithms are derived based on hierarchical models that account for the uncertainties of the forward model. For estimation, Maximum A Posterior (MAP) and Posterior Mean (PM) via variational Bayesian approximation (VBA) are used. The performances of re- sulting algorithm are compared in applications in 3D computed tomography (3D-CT) and chronobiology. Finally, a theoretical study is developed for comparison between sparsity enforcing algorithms obtained via the Bayesian ap- proach and the sparsity enforcing algorithms issued from regularization techniques, like LASSO and some others.
    [Show full text]
  • Jeffreys Priors 1 Priors for the Multivariate Gaussian
    Stat260: Bayesian Modeling and Inference Lecture Date: February 10th, 2010 Jeffreys priors Lecturer: Michael I. Jordan Scribe: Timothy Hunter 1 Priors for the multivariate Gaussian Consider a multivariate Gaussian variable X of size p. Its probability density function can be parametrized by a mean vector µ Rp and a covariance matrix Σ +: ∈ ∈ Sp 1 1 1 p (X µ, Σ) = exp (X µ)T Σ−1 (X µ) (1) | (2π)p/2 Σ −2 − − | | We will consider three cases of conjugate priors:p the case when the covariance is fixed, the case when the mean is fixed and the general case. 1.1 The case of fixed variance The conjugate prior is a multivariate Gaussian of mean µ0 and covariance matrix Σ0. The derivations are the same as in the univariate case. 1.2 The case of fixed mean The conjugate prior is the inverse Wishart distribution. One can see this using the trace trick: (X µ)T Σ−1 (X µ) = Tr Σ−1 (X µ) (X µ)T (2) − − − − This is an inner product in the matrix space between the matrices Σ−1 and (X µ) (X µ)T . This partic- ular inner product corresponds to summing all the pairwise products of elements− of each− matrix. We will derive here the Wishart distribution, and you can derive the inverse Wishart by a change of variable. The computations are similar for the inverse Wishart distribution. The distribution over the precision matrix is the Wishart distribution with one degree of freedom is: 1 −1 M p/2 e− 2 Tr(MV ) p (M V ) = | | (3) | 2p/2 V 1/2 Γ 1 | | p 2 where Γp is the generalized (multivariate) Gamma function, and M,V are positive definite matrices.
    [Show full text]
  • Bayesian Inference for the Multivariate Normal
    Bayesian Inference for the Multivariate Normal Will Penny Wellcome Trust Centre for Neuroimaging, University College, London WC1N 3BG, UK. November 28, 2014 Abstract Bayesian inference for the multivariate Normal is most simply instanti- ated using a Normal-Wishart prior over the mean and covariance. Predic- tive densities then correspond to multivariate T distributions, and the moments from the marginal densities are provided analytically or via Monte-Carlo sampling. We show how this textbook approach is applied to a simple two-dimensional example. 1 Introduction This report addresses the following question. Given samples of multidimensional vectors drawn from a multivariate Normal density with mean m and precision Λ, what are the likely values of m and Λ ? Here, the precision is simply the inverse of the covariance i.e. Λ = C−1. Whilst it is simple to compute the sample mean and covariance, for inference we need the probability distributions over these quantities. In a Bayesian conception of this problem we place prior distributions over all quantities of interest and use Bayes rule to compute the posterior. We follow the formulation in Bernardo and Smith [1] (tabularised on page 441). 2 Preliminaries 2.1 Multivariate T distribution The multivariate T distribution over a d-dimensional random variable x is p(x) = T (x; µ, Λ; v) (1) with parameters µ, Λ and v. The mean and covariance are given by E(x) = µ (2) v V ar(x) = Λ−1 v − 2 The multivariate T approaches a multivariate Normal for large degrees of free- dom, v, as shown in Figure 1.
    [Show full text]
  • Prior Distributions for Variance Parameters in Hierarchical Models
    Bayesian Analysis (2006) 1, Number 3, pp. 515–533 Prior distributions for variance parameters in hierarchical models Andrew Gelman Department of Statistics and Department of Political Science Columbia University Abstract. Various noninformative prior distributions have been suggested for scale parameters in hierarchical models. We construct a new folded-noncentral-t family of conditionally conjugate priors for hierarchical standard deviation pa- rameters, and then consider noninformative and weakly informative priors in this family. We use an example to illustrate serious problems with the inverse-gamma family of “noninformative” prior distributions. We suggest instead to use a uni- form prior on the hierarchical standard deviation, using the half-t family when the number of groups is small and in other settings where a weakly informative prior is desired. We also illustrate the use of the half-t family for hierarchical modeling of multiple variance parameters such as arise in the analysis of variance. Keywords: Bayesian inference, conditional conjugacy, folded-noncentral-t distri- bution, half-t distribution, hierarchical model, multilevel model, noninformative prior distribution, weakly informative prior distribution 1 Introduction Fully-Bayesian analyses of hierarchical linear models have been considered for at least forty years (Hill, 1965, Tiao and Tan, 1965, and Stone and Springer, 1965) and have remained a topic of theoretical and applied interest (see, e.g., Portnoy, 1971, Box and Tiao, 1973, Gelman et al., 2003, Carlin and Louis, 1996, and Meng and van Dyk, 2001). Browne and Draper (2005) review much of the extensive literature in the course of comparing Bayesian and non-Bayesian inference for hierarchical models. As part of their article, Browne and Draper consider some different prior distributions for variance parameters; here, we explore the principles of hierarchical prior distributions in the context of a specific class of models.
    [Show full text]