Marcio Alves Diniz

Unit Roots: Bayesian Significance Test

The unit root problem plays a central role in empirical applications in the time series econometric literature. However, significance tests developed under the frequentist tradition present various conceptual problems that jeopardize the power of these tests, especially for small samples. Bayesian alternatives, although having interesting interpretations and being precisely defined, experience problems due to the fact that that the hypothesis of interest in this case is sharp or precise. The Bayesian significance test used in this article, for the unit root hypothesis, is based solely on the posterior density function, without the need of imposing positive probabilities to sets of zero Lebesgue measure. Furthermore, it is conducted under strict observance of the likelihood principle. It was designed mainly for testing sharp null hypotheses and it is called FBST for Full Bayesian Significance Test.

Coherent predictive inference under exchangeability with imprecise probabilities

Coherent reasoning under uncertainty can be represented in a very general manner by coherent sets of desirable gambles. In a context that does not allow for indecision, this leads to an approach that is mathematically equivalent to working with coherent conditional probabilities. If we do allow for indecision, this leads to a more general foundation for coherent (imprecise-)probabilistic inference. In this framework, and for a given finite category set, coherent predictive inference under exchangeability can be represented using Bernstein coherent cones of multivariate polynomials on the simplex generated by this category set. This is a powerful generalisation of de Finettis Representation Theorem allowing for both imprecision and indecision. We define an inference system as a map that associates a Bernstein coherent cone of polynomials with every finite category set. Many inference principles encountered in the literature can then be interpreted, and represented mathematically, as restrictions on such maps. We discuss, as particular examples, two important inference principles: representation insensitivity--a strengthened version of Walleys representation invariance--and specificity. We show that there is an infinity of inference systems that satisfy these two principles, amongst which we discuss in particular the skeptically cautious inference system, the inference systems corresponding to (a modified version of) Walley and Bernards Imprecise Dirichlet Multinomial Models (IDMM), the skeptical IDMM inference systems, and the Haldane inference system. We also prove that the latter produces the same posterior inferences as would be obtained using Haldanes improper prior, implying that there is an infinity of proper priors that produce the same coherent posterior inferences as Haldanes improper one. Finally, we impose an additional inference principle that allows us to characterise uniquely the immediate predictions for the IDMM inference systems.

Representation theorems for partially exchangeable random variables

We provide representation theorems for both finite and countable sequences of finite-valued random variables that are considered to be partially exchangeable. In their most general form, our results are presented in terms of sets of desirable gambles, a very general framework for modelling uncertainty. Its key advantages are that it allows for imprecision, is more expressive than almost every other imprecise-probabilistic framework and makes conditioning on events with (lower) probability zero non-problematic. We translate our results to more conventional, although less general frameworks as well: lower previsions, linear previsions and probability measures. The usual, precise-probabilistic representation theorems for partially exchangeable random variables are obtained as special cases.

FBST for Unit Root Problems

This paper presents the Full Bayesian Significance Test for unit roots in auto-regressive time series, and compares it to other approaches on a benchmark of 14 econometric series.

A Simple Proof for the Multinomial Version of the Representation Theorem

In this work we present a demonstration for the multinomial version of de Finetti’s Representation Theorem. We use characteristic functions, following his first demonstration for binary random quantities, but simplify the argument through forward operators.

Cointegration: Bayesian Significance Test

To estimate causal relationships, time series econometricians must be aware of spurious correlation, a problem first mentioned by Yule (1926). To deal with this problem, one can work either with differenced series or multivariate models: VAR (VEC or VECM) models. These models usually include at least one cointegration relation. Although the Bayesian literature on VAR/VEC is quite advanced, Bauwens et al. (1999) highlighted that “the topic of selecting the cointegrating rank has not yet given very useful and convincing results”. The present article applies the Full Bayesian Significance Test (FBST), especially designed to deal with sharp hypotheses, to cointegration rank selection tests in VECM time series models. It shows the FBST implementation using both simulated and available (in the literature) data sets. As illustration, standard non informative priors are used.

Predictive Inference Under Exchangeability, and the Imprecise Dirichlet Multinomial Model

Coherent reasoning under uncertainty can be represented in a very general manner by coherent sets of desirable gambles. In this framework, and for a given finite category set, coherent predictive inference under exchangeability can be represented using Bernstein coherent cones of multivariate polynomials on the simplex generated by this category set. This is a powerful generalisation of de Finetti’s representation theorem allowing for both imprecision and indecision. We define an inference system as a map that associates a Bernstein coherent cone of polynomials with every finite category set. Many inference principles encountered in the literature can then be interpreted, and represented mathematically, as restrictions on such maps. We discuss two important inference principles: representation insensitivity—a strengthened version of Walley’s representation invariance—and specificity. We show that there is a infinity of inference systems that satisfy these two principles, amongst which we discuss in particular the inference systems corresponding to (a modified version of) Walley and Bernard’s imprecise Dirichlet multinomial models (IDMMs) and the Haldane inference system.

Preface: XI Brazilian Meeting on Bayesian Statistics

Related Articles Editorial: Thank you J. Renewable Sustainable Energy 4, 050401 (2012) Preface to Special Topic: Low-Carbon Society for a Green Economy J. Renewable Sustainable Energy 4, 041301 (2012) Announcement: New Format for Physics of Fluids Phys. Fluids 24, 010202 (2012) Editorial: Women in technology for renewable energies J. Renewable Sustainable Energy 3, 030401 (2011) Preface: Proceedings of the 55th Annual Conference on Magnetism and Magnetic Materials, Atlanta, Georgia, November 2010 J. Appl. Phys. 109, 07A101 (2011)

Comparing probabilistic predictive models applied to football

We propose two Bayesian multinomial-Dirichlet models to predict the final outcome of football (soccer) matches and compare them to three well-known models regarding their predictive power. All the models predicted the full-time results of 1710 matches of the first division of the Brazilian football championship and the comparison used three proper scoring rules, the proportion of errors and a calibration assessment. We also provide a goodness of fit measure. Our results show that multinomial-Dirichlet models are not only competitive with standard approaches, but they are also well calibrated and present reasonable goodness of fit.

Coherent Predictive Inference under Exchangeability with Imprecise Probabilities (Extended Abstract)

Coherent reasoning under uncertainty can be represented in a very general manner by coherent sets of desirable gambles. This leads to a more general foundation for coherent (imprecise-)probabilistic inference that allows for indecision. In this framework, and for a given finite category set, coherent predictive inference under exchangeability can be represented using Bernstein coherent cones of multivariate polynomials on the simplex generated by this category set. We define an inference system as a map that associates a Bernstein coherent cone of polynomials with every finite category set. Inference principles can then be represented mathematically as restrictions on such maps, which allows us to develop a notion of conservative inference under such inference principles. We discuss, as particular examples, representation insensitivity and specificity, and show that there is an infinity of inference systems that satisfy these two principles. 1 The Setting: Predictive Inference We deal with predictive inference for categorical variables, and are therefore concerned with a (possibly infinite) sequence of variables Xn that assume values in some finite set of categories A. After having observed a number ?̌? of them, and having found that, say X1 = x1, X2 = x2, . . . , X?̌? = x?̌?, we consider some subject’s belief model for the next ?̂? ≥ 1 variables X?̌?+1, . . . X?̌?+?̂?. In the probabilistic tradition that we build on in the paper, this belief is modelled by a conditional predictive probability mass function p(·|x1, . . . , x?̌?) on the set A of their possible values. These probability mass functions can be used for prediction or estimation, for statistical inferences, and in decision making. In this sense, predictive inference lies at the heart of statistics, and more generally, of learning under uncertainty. For this reason, it is also of crucial importance for dealing with uncertainty in Artificial Intelligence. We refer to the synthesis by Geisser [1993] and the collection of essays by Zabell [2005] for introductions to predictive inference and the underlying issues that the paper is also concerned with. *This paper is an extended abstract of an article in the Journal of Artificial Intelligence Research [De Cooman et al., 2015]. The predictive probability mass functions for various values of ?̌?, ?̂? and (x1, . . . , x?̌?) are connected by requirements of time consistency and coherence. The former requires that when n1 ≤ n2, p1(·|x1, . . . , x?̌?) can be obtained from p2(·|x1, . . . , x?̌?) through marginalisation; the latter essentially demands that these conditional probability mass functions should be connected with time-consistent unconditional probability mass functions through Bayes’s Rule. A common assumption about the variables Xn is that they are exchangeable, meaning roughly that the subject believes that the order in which they are observed, or present themselves, has no influence on the decisions and inferences she will make regarding these variables. This assumption, and the analysis of its consequences, goes back to de Finetti [1937]. His famous Representation Theorem states, in essence, that the time-consistent and coherent conditional and unconditional predictive probability mass functions associated with a countably infinite exchangeable sequence of variables in A are characterised by, and characterise, a unique probability measure on the Borel sets of the simplex of all probability mass functions on A, called their representation.1 2 The Central Problem of Predictive Inference This leads us to the central problem of predictive inference: since there is an infinity of such probability measures on the simplex, which one does a subject choose in a particular context, and how can a given choice be motivated and justified? The subjectivists of de Finetti’s persuasion might answer that this question needs no answer: a subject’s personal predictive probabilities are entirely hers, and time consistency and coherence are the only requirements she should heed. Earlier scholars, like Laplace and Bayes, whom we would now also call subjectivists, invoked the Principle of Indifference To clarify the connection with our argumentation in the paper, the essence of de Finetti’s argument is that the representation is a coherent prevision on the set of all multinomial polynomials on this simplex [De Cooman et al., 2009b]. As a (finitely additive) coherent prevision, it can be extended uniquely only so far as to the set of all lower semicontinuous functions, but it does determine a unique (countably additive) probability measure on the Borel sets of that simplex, through the F. Riesz Representation Theorem [De Cooman and Miranda, 2008; Troffaes and De Cooman, 2014]. Proceedings of the Twenty-Sixth International Joint Conference on Artificial Intelligence (IJCAI-17)