Lecture Notes Bayesian Statistics, Spring 2011
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Overall Objective Priors
Bayesian Analysis (2015) 10, Number 1, pp. 189–221 Overall Objective Priors∗ James O. Berger†, Jose M. Bernardo‡, and Dongchu Sun§ Abstract. In multi-parameter models, reference priors typically depend on the parameter or quantity of interest, and it is well known that this is necessary to produce objective posterior distributions with optimal properties. There are, however, many situations where one is simultaneously interested in all the param- eters of the model or, more realistically, in functions of them that include aspects such as prediction, and it would then be useful to have a single objective prior that could safely be used to produce reasonable posterior inferences for all the quantities of interest. In this paper, we consider three methods for selecting a sin- gle objective prior and study, in a variety of problems including the multinomial problem, whether or not the resulting prior is a reasonable overall prior. Keywords: Joint Reference Prior, Logarithmic Divergence, Multinomial Model, Objective Priors, Reference Analysis. 1 Introduction 1.1 The problem Objective Bayesian methods, where the formal prior distribution is derived from the assumed model rather than assessed from expert opinions, have a long history (see e.g., Bernardo and Smith, 1994; Kass and Wasserman, 1996, and references therein). Reference priors (Bernardo, 1979, 2005; Berger and Bernardo, 1989, 1992a,b, Berger, Bernardo and Sun, 2009, 2012) are a popular choice of objective prior. Other interesting developments involving objective priors include Clarke and Barron (1994), Clarke and Yuan (2004), Consonni, Veronese and Guti´errez-Pe˜na (2004), De Santis et al. (2001), De Santis (2006), Datta and Ghosh (1995a; 1995b), Datta and Ghosh (1996), Datta et al. -
Minimum Description Length Model Selection
Minimum Description Length Model Selection Problems and Extensions Steven de Rooij Minimum Description Length Model Selection Problems and Extensions ILLC Dissertation Series DS-2008-07 For further information about ILLC-publications, please contact Institute for Logic, Language and Computation Universiteit van Amsterdam Plantage Muidergracht 24 1018 TV Amsterdam phone: +31-20-525 6051 fax: +31-20-525 5206 e-mail: [email protected] homepage: http://www.illc.uva.nl/ Minimum Description Length Model Selection Problems and Extensions Academisch Proefschrift ter verkrijging van de graad van doctor aan de Universiteit van Amsterdam op gezag van de Rector Magnificus prof.dr. D. C. van den Boom ten overstaan van een door het college voor promoties ingestelde commissie, in het openbaar te verdedigen in de Agnietenkapel op woensdag 10 september 2008, te 12.00 uur door Steven de Rooij geboren te Amersfoort. Promotiecommissie: Promotor: prof.dr.ir. P.M.B. Vit´anyi Co-promotor: dr. P.D. Grunwald¨ Overige leden: prof.dr. P.W. Adriaans prof.dr. A.P. Dawid prof.dr. C.A.J. Klaassen prof.dr.ir. R.J.H. Scha dr. M.W. van Someren prof.dr. V. Vovk dr.ir. F.M.J. Willems Faculteit der Natuurwetenschappen, Wiskunde en Informatica The investigations were funded by the Netherlands Organization for Scientific Research (NWO), project 612.052.004 on Universal Learning, and were supported in part by the IST Programme of the European Community, under the PASCAL Network of Excellence, IST-2002-506778. Copyright c 2008 by Steven de Rooij Cover inspired by Randall Munroe’s xkcd webcomic (www.xkcd.org) Printed and bound by PrintPartners Ipskamp (www.ppi.nl) ISBN: 978-90-5776-181-2 Parts of this thesis are based on material contained in the following papers: An Empirical Study of Minimum Description Length Model Selection with Infinite • Parametric Complexity Steven de Rooij and Peter Grunwald¨ In: Journal of Mathematical Psychology, special issue on model selection. -
Part II Bayesian Statistics
Part II Bayesian Statistics 1 Chapter 6 Bayesian Statistics: Background In the frequency interpretation of probability, the probability of an event is limiting proportion of times the event occurs in an infinite sequence of independent repetitions of the experiment. This interpretation assumes that an experiment can be repeated! Problems with this interpretation: • Independence is defined in terms of probabilities; if probabilities are defined in terms of independent events, this leads to a circular defini- tion. • How can we check whether experiments were independent, without doing more experiments? • In practice we have only ever a finite number of experiments. 6.1 Subjective probability Let P (A) denote your personal probability of an event A; this is a numerical measure of the strength of your degree of belief that A will occur, in the light of available information. 2 Your personal probabilities may be associated with a much wider class of events than those to which the frequency interpretation pertains. For example: - non-repeatable experiments (e.g. that England will win the World Cup next time); - propositions about nature (e.g. that this surgical procedure results in increased life expectancy). All subjective probabilities are conditional, and may be revised in the light of additional information. Subjective probabilities are assessments in the light of incomplete information; they may even refer to events in the past. 6.2 Axiomatic development Coherence states that a system of beliefs should avoid internal inconsisten- cies. Basically, a quantitative, coherent belief system must behave as if it was governed by a subjective probability distribution. In particular this assumes that all events of interest can be compared. -
Conjugate Priors, Uninformative Priors
Conjugate Priors, Uninformative Priors Nasim Zolaktaf UBC Machine Learning Reading Group January 2016 Outline Exponential Families Conjugacy Conjugate priors Mixture of conjugate prior Uninformative priors Jeffreys prior = h(X) exp[θT φ(X) − A(θ)] (2) where Z Z(θ) = h(X) exp[θT φ(X)]dx (3) X m A(θ) = log Z(θ) (4) Z(θ) is called the partition function, A(θ) is called the log partition function or cumulant function, and h(X) is a scaling constant. Equation 2 can be generalized by writing p(Xjθ) = h(X) exp[η(θ)T φ(X) − A(η(θ))] (5) The Exponential Family A probability mass function (pmf) or probability distribution m d function (pdf) p(Xjθ), for X = (X1; :::; Xm) 2 X and θ ⊆ R , is said to be in the exponential family if it is the form: 1 p(Xjθ) = h(X) exp[θT φ(X)] (1) Z(θ) where Z Z(θ) = h(X) exp[θT φ(X)]dx (3) X m A(θ) = log Z(θ) (4) Z(θ) is called the partition function, A(θ) is called the log partition function or cumulant function, and h(X) is a scaling constant. Equation 2 can be generalized by writing p(Xjθ) = h(X) exp[η(θ)T φ(X) − A(η(θ))] (5) The Exponential Family A probability mass function (pmf) or probability distribution m d function (pdf) p(Xjθ), for X = (X1; :::; Xm) 2 X and θ ⊆ R , is said to be in the exponential family if it is the form: 1 p(Xjθ) = h(X) exp[θT φ(X)] (1) Z(θ) = h(X) exp[θT φ(X) − A(θ)] (2) Z(θ) is called the partition function, A(θ) is called the log partition function or cumulant function, and h(X) is a scaling constant. -
Prior Distributions
5 Prior distributions The prior distribution plays a dening role in Bayesian analysis. In view of the controversy surrounding its use it may be tempting to treat it almost as an embarrassment and to emphasise its lack of importance in particular appli- cations, but we feel it is a vital ingredient and needs to be squarely addressed. In this chapter we introduce basic ideas by focusing on single parameters, and in subsequent chapters consider multi-parameter situations and hierarchical models. Our emphasis is on understanding what is being used and being aware of its (possibly unintentional) inuence. 5.1 Different purposes of priors A basic division can be made between so-called “non-informative” (also known as “reference” or “objective”) and “informative” priors. The former are in- tended for use in situations where scientic objectivity is at a premium, for example, when presenting results to a regulator or in a scientic journal, and essentially means the Bayesian apparatus is being used as a convenient way of dealing with complex multi-dimensional models. The term “non-informative” is misleading, since all priors contain some information, so such priors are generally better referred to as “vague” or “diffuse.” In contrast, the use of in- formative prior distributions explicitly acknowledges that the analysis is based on more than the immediate data in hand whose relevance to the parameters of interest is modelled through the likelihood, and also includes a considered judgement concerning plausible values of the parameters based on external information. In fact the division between these two options is not so clear-cut — in par- ticular, we would claim that any “objective” Bayesian analysis is a lot more “subjective” than it may wish to appear.