Eliciting Dirichlet and Gaussian copula prior distributions for multinomial models

In this paper, we propose novel methods of quantifying expert opinion about prior distributions for multinomial models. Two different multivariate priors are elicited using median and quartile assessments of the multinomial probabilities. First, we start by eliciting a univariate beta distribution for the probability of each category. Then we elicit the hyperparameters of the Dirichlet distribution, as a tractable conjugate prior, from those of the univariate betas through various forms of reconciliation using least-squares techniques. However, a multivariate copula function will give a more flexible correlation structure between multinomial parameters if it is used as their multivariate prior distribution. So, second, we use beta marginal distributions to construct a Gaussian copula as a multivariate normal distribution function that binds these marginals and expresses the dependence structure between them. The proposed method elicits a positive-definite correlation matrix of this Gaussian copula. The two proposed methods are designed to be used through interactive graphical software written in Java

Elicitation of Subjective Probability Distributions

To incorporate expert opinion into a Bayesian analysis, it must be quantified as a prior distribution through an elicitation process that asks the expert meaningful questions whose answers determine this distribution. The aim of this thesis is to fill some gaps in the available techniques for eliciting prior distributions for Generalized Linear Models (GLMs) and multinomial models. A general method for quantifying opinion about GLMs was developed in Garthwaite and Al-Awadhi (2006). They model the relationship between each continuous predictor and the dependant variable as a piecewise-linear function with a regression coefficient at each of its dividing points. However, coefficients were assumed a priori independent if associated with different predictors. We relax this simplifying assumption and propose three new methods for eliciting positive-definite variance-covariance matrices of a multivariate normal prior distribution. In addition, we extend the method of Garthwaite and Dickey (1988) for eliciting an inverse chi-squared conjugate prior for the error variance in normal linear models. We also propose a novel method for eliciting a lognormal prior distribution for the scale parameter of a gamma GLM. For multinomial models, novel methods are proposed that quantify expert opinion about a conjugate Dirichlet distribution and, additionally, about three more general and flexible prior distributions. First, an elicitation method is proposed for the generalized Dirichlet distribution that was introduced by Connor and Mosimann (1969). Second, a method is developed for eliciting the Gaussian copula as a multivariate distribution with marginal beta priors. Third, a further novel method is constructed that quantifies expert opinion about the most flexible alternate prior, the logistic normal distribution (Aitchison, 1986). This third method is extended to the case of multinomial models with explanatory covariates. All proposed methods in this thesis are designed to be used with interactive Prior Elicitation Graphical Software (PEGS) that is freely available at http://statistics.open.ac.uk/elicitation

Estimating Discrete Markov Models From Various Incomplete Data Schemes

The parameters of a discrete stationary Markov model are transition probabilities between states. Traditionally, data consist in sequences of observed states for a given number of individuals over the whole observation period. In such a case, the estimation of transition probabilities is straightforwardly made by counting one-step moves from a given state to another. In many real-life problems, however, the inference is much more difficult as state sequences are not fully observed, namely the state of each individual is known only for some given values of the time variable. A review of the problem is given, focusing on Monte Carlo Markov Chain (MCMC) algorithms to perform Bayesian inference and evaluate posterior distributions of the transition probabilities in this missing-data framework. Leaning on the dependence between the rows of the transition matrix, an adaptive MCMC mechanism accelerating the classical Metropolis-Hastings algorithm is then proposed and empirically studied.Comment: 26 pages - preprint accepted in 20th February 2012 for publication in Computational Statistics and Data Analysis (please cite the journal's paper

Prior distributions for stochastic matrices

Right-stochastic matrices are used in the modelling of discrete-time Markov processes, with a property that the matrix elements are non-negative and each row sums to one. If we consider the problem of estimating these probabilities from a Bayesian standpoint, we are interested in constructing sensible probability distributions that can be used to encapsulate expert beliefs about such structures before any data is observed. Through the process of expert elicitation, this un- certainty can be represented in terms of probability distributions. In this thesis, we explore multivariate distributions on the simplex support from the view of expert elicitation. We explore properties and constraints of these distributions, and ways to elicit expert judgement about their parameters. This is interesting both mathematically and from a practical standpoint, particularly where there are many such variables to explore, which can prove cognitively challenging and tiring for the experts. Similarly, data representing proportions of a whole can be unified into the com- positional framework (Aitchison, 1986) with similar non-negativity and unit-sum properties. This thesis also explores the study of compositional data analysis, its problems and modern ways of approaching them. Application of these methods is found in exploring how high resolution imagery obtained over rural areas could be used in order to identify the distribution of tree species found in those areas where monitoring is prohibited

On quantifying expert opinion about multinomial models that contain covariates

This paper addresses the task of forming a prior distribution to represent expert opinion about a multinomial model that contains covariates. The task has not previously been addressed. We suppose the sampling model is a multinomial logistic regression and represent expert opinion about the regression coefficients by a multivariate normal distribution. This logistic-normal model gives a flexible prior distribution that can capture a broad variety of expert opinion. The challenge is to (i) find meaningful assessment tasks that an expert can perform and which should yield appropriate information to determine the values of parameters in the prior distribution, and (ii) develop theory for determining the parameter values from the assessments. A method is proposed that meets this challenge. The method is implemented in interactive user-friendly software that is freely available. It provides a graphical interface that the expert uses to assess quartiles of sets of proportions and the method determines a mean vector and a positive-definite covariance matrix to represent the expert's opinions. The chosen assessment tasks yield parameter values that satisfy the usual laws of probability without the expert being aware of the constraints this imposes. Special attention is given to feedback that encourages the expert to consider his/her opinions from a different perspective. The method is illustrated in an example that shows its viability and usefulness

Prior knowledge elicitation: The past, present, and future

Specification of the prior distribution for a Bayesian model is a central part of the Bayesian workflow for data analysis, but it is often difficult even for statistical experts. Prior elicitation transforms domain knowledge of various kinds into well-defined prior distributions, and offers a solution to the prior specification problem, in principle. In practice, however, we are still fairly far from having usable prior elicitation tools that could significantly influence the way we build probabilistic models in academia and industry. We lack elicitation methods that integrate well into the Bayesian workflow and perform elicitation efficiently in terms of costs of time and effort. We even lack a comprehensive theoretical framework for understanding different facets of the prior elicitation problem.Why are we not widely using prior elicitation? We analyze the state of the art by identifying a range of key aspects of prior knowledge elicitation, from properties of the modelling task and the nature of the priors to the form of interaction with the expert. The existing prior elicitation literature is reviewed and categorized in these terms. This allows recognizing under-studied directions in prior elicitation research, finally leading to a proposal of several new avenues to improve prior elicitation methodology.Fil: Mikkola, Petrus. Aalto University; FinlandiaFil: Martín, Osvaldo Antonio. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - San Luis. Instituto de Matemática Aplicada de San Luis "Prof. Ezio Marchi". Universidad Nacional de San Luis. Facultad de Ciencias Físico, Matemáticas y Naturales. Instituto de Matemática Aplicada de San Luis "Prof. Ezio Marchi"; Argentina. Aalto University; FinlandiaFil: Chandramoul, Suyog. Aalto University; FinlandiaFil: Hartmann, Marcelo. University of Helsinki; FinlandiaFil: Abril Pla, Oriol. University of Helsinki; FinlandiaFil: Thomas, Owen. University of Oslo; NoruegaFil: Pesonen, Henri. University of Oslo; NoruegaFil: Corander, Jukka. University of Oslo; NoruegaFil: Vehtari, Aki. Aalto University; FinlandiaFil: Kaski, Samuel. Aalto University; FinlandiaFil: Bürkner, Paul Christian. University Of Stuttgart; AlemaniaFil: Klami, Arto. University of Helsinki; Finlandi

Prior knowledge elicitation: The past, present, and future

Specification of the prior distribution for a Bayesian model is a central part of the Bayesian workflow for data analysis, but it is often difficult even for statistical experts. Prior elicitation transforms domain knowledge of various kinds into well-defined prior distributions, and offers a solution to the prior specification problem, in principle. In practice, however, we are still fairly far from having usable prior elicitation tools that could significantly influence the way we build probabilistic models in academia and industry. We lack elicitation methods that integrate well into the Bayesian workflow and perform elicitation efficiently in terms of costs of time and effort. We even lack a comprehensive theoretical framework for understanding different facets of the prior elicitation problem.Why are we not widely using prior elicitation? We analyze the state of the art by identifying a range of key aspects of prior knowledge elicitation, from properties of the modelling task and the nature of the priors to the form of interaction with the expert. The existing prior elicitation literature is reviewed and categorized in these terms. This allows recognizing under-studied directions in prior elicitation research, finally leading to a proposal of several new avenues to improve prior elicitation methodology.Fil: Mikkola, Petrus. Aalto University; FinlandiaFil: Martín, Osvaldo Antonio. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - San Luis. Instituto de Matemática Aplicada de San Luis "Prof. Ezio Marchi". Universidad Nacional de San Luis. Facultad de Ciencias Físico, Matemáticas y Naturales. Instituto de Matemática Aplicada de San Luis "Prof. Ezio Marchi"; Argentina. Aalto University; FinlandiaFil: Chandramoul, Suyog. Aalto University; FinlandiaFil: Hartmann, Marcelo. University of Helsinki; FinlandiaFil: Abril Pla, Oriol. University of Helsinki; FinlandiaFil: Thomas, Owen. University of Oslo; NoruegaFil: Pesonen, Henri. University of Oslo; NoruegaFil: Corander, Jukka. University of Oslo; NoruegaFil: Vehtari, Aki. Aalto University; FinlandiaFil: Kaski, Samuel. Aalto University; FinlandiaFil: Bürkner, Paul Christian. University Of Stuttgart; AlemaniaFil: Klami, Arto. University of Helsinki; Finlandi

Prior Elicitation for Generalised Linear Models and Extensions

A statistical method for the elicitation of priors in Bayesian generalised linear models (GLMs) and extensions is proposed. Probabilistic predictions are elicited from the expert to parametrise a multivariate t prior distribution for the unknown linear coefficients of the GLM and an inverse gamma prior for the dispersion parameter, if unknown. The elicited predictions condition on defined elicitation scenarios. Dependencies among scenarios are then elicited from the expert by additionally conditioning on hypothetical experiments. Elicited conditional medians efficiently parametrise a canonical vine copula model of dependence that may be truncated for efficiency. The statistical elicitation method permits prior parametrisation of GLMs with alternative choices of design matrices or observation models from the same elicitation session. Extensions of the method apply to multivariate data, data with bounded support, semi-continuous data with point mass at zero, and count data with overdispersion or zero-inflation. A case study elicits a prior for an extended GLM embedded in a statistical model of overdispersed counts described by a binomial-simplex mixture distribution. The elicited canonical vine model of dependence is found to incorporate substantial information into the prior. The procedures of the statistical elicitation method are implemented in the R package eglm

Expert judgement for dependence in probabilistic modelling : a systematic literature review and future research directions

Many applications in decision making under uncertainty and probabilistic risk assessment require the assessment of mul- tiple, dependent uncertain quantities, so that in addition to marginal distributions, interdependence needs to be modelled in order to properly understand the overall risk. Nevertheless, relevant historical data on dependence information are often not available or simply too costly to obtain. In this case, the only sensible option is to elicit this uncertainty through the use of expert judgements. In expert judgement studies, a structured approach to eliciting variables of interest is desirable so that their assessment is methodologically robust. One of the key decisions during the elicitation process is the form in which the uncertainties are elicited. This choice is subject to various, potentially con icting, desiderata related to e.g. modelling convenience, coherence between elicitation parameters and the model, combining judgements, and the assessment burden for the experts. While extensive and systematic guidance to address these considerations exists for single variable uncertainty elicitation, for higher dimensions very little such guidance is available. Therefore this paper o ers a systematic review of the current literature on eliciting dependence. The literature on the elicitation of dependence parameters such as correlations is presented alongside commonly used dependence models and experience from case studies. From this, guidance about the strategy for dependence assessment is given and gaps in the existing research are identi ed to determine future directions for structured methods to elicit dependence