Density Ratio Class Research Articles

Agreeing to disagree and dilation

We consider Aumann's famous result on “agreeing to disagree” in the context of imprecise probabilities. Our primary aim is to reveal a connection between the possibility of agreeing to disagree and the interesting and anomalous phenomenon known as dilation. For such a purpose it is convenient to use Geanakoplos and Polemarchakis' communication setting, where agents repeatedly announce and update credences until no new information is conveyed by the announcements. We show that for agents who share the same set of priors and update by conditioning on every prior, once the procedure of communicating credences stops, it is impossible to agree to disagree on the lower or upper probability of a hypothesis unless a certain dilation occurs. With some common topological assumptions, the result entails that it is impossible to agree not to have the same set of posterior probability values unless dilation is present. This result may be used to generate sufficient conditions for guaranteed full agreement for some important models of imprecise priors, and we illustrate the potential with an agreement result involving density ratio classes. We also provide a formulation of our results in terms of “dilation-averse” agents who ignore information about the value of a dilating partition but otherwise update by full Bayesian conditioning.

Robustness in Bayesian nonparametrics

We consider Bayesian robustness in the context of Bayesian Nonparametrics, and specifically for the Dirichlet Process prior. We show how to find an optimal procedure, based on C-minimax posterior regret (CMPR) for a class of priors C. We consider regret based on squared error loss. The neighborhood classes considered are the density ratio (DR) class and the epsilon-contamination class.

The effect of ambiguous prior knowledge on Bayesian model parameter inference and prediction

Environmental modeling often requires combining prior knowledge with information obtained from data. The robust Bayesian approach makes it possible to consider ambiguity in this prior knowledge. Describing such ambiguity using sets of probability distributions defined by the Density Ratio Class has important conceptual advantages over alternative robust formulations. Earlier studies showed that the Density Ratio Class is invariant under Bayesian inference and marginalization. We prove that (i) the Density Ratio Class is also invariant under propagation through deterministic models, whereas (ii) predictions of a stochastic model with parameters defined by a Density Ratio Class are embedded in a Density Ratio Class. These invariance properties make it possible to describe sequential learning and prediction under a unified framework. We developed numerical algorithms to minimize the additional computational burden relative to the use of single priors. Practical feasibility of these methods is demonstrated by their application to a simple ecological model.

Density-ratio robustness in dynamic state estimation

The filtering problem is addressed by taking into account imprecision in the knowledge about the probabilistic relationships involved. Imprecision is modelled in this paper by a particular closed convex set of probabilities that is known with the name of density ratio class or constant odds-ratio (COR) model. The contributions of this paper are the following. First, we shall define an optimality criterion based on the squared-loss function for the estimates derived from a general closed convex set of distributions. Second, after revising the properties of the density ratio class in the context of parametric estimation, we shall extend these properties to state estimation accounting for system dynamics. Furthermore, for the case in which the nominal density of the COR model is a multivariate Gaussian, we shall derive closed-form solutions for the set of optimal estimates and for the credible region. Third, we discuss how to perform Monte Carlo integrations to compute lower and upper expectations from a COR set of densities. Then we shall derive a procedure that, employing Monte Carlo sampling techniques, allows us to propagate in time both the lower and upper state expectation functionals and, thus, to derive an efficient solution of the filtering problem. Finally, we empirically compare the proposed estimator with the Kalman filter. This shows that our solution is more robust to the presence of modelling errors in the system and that, hence, appears to be a more realistic approach than the Kalman filter in such a case.

Bridging uncertain and ambiguous knowledge with imprecise probabilities

Model-based environmental decision support requires that uncertainty be rigorously evaluated. Whether uncertainty is aleatory or epistemic, we argue that probability is the natural mathematical construct for describing uncertainty in predictions used for decision-making. If expert knowledge is elicited using stated preferences between lotteries, and the experts are rational in the sense of avoiding sure loss, then the resulting knowledge quantifications will be consistent with the axiomatic foundation of probability theory. This idea can be extended to the description of intersubjective knowledge when the intent is to characterize the state of knowledge of the scientific community. Many methods for probability elicitation have been reported, but there is nearly always some degree of ambiguity in translating elicited quantities into probabilistic description. This would include: any lack of fit of a particular distributional form to elicited data; incertitude in the elicited data themselves; and/or disagreement in the elicited data across multiple experts. By replacing a precise probability distribution by a set of distributions, the mathematical concept of imprecise probabilities provides a means for representing this ambiguity. In this way, imprecise probabilities can form a bridge between total ignorance and precisely characterized risk by allowing for a continuous degree of imprecision to represent ambiguity. We introduce three metrics to describe the relative ambiguity of important attributes of probability distributions, namely their width, shape, and mode. These metrics are applicable to sets of distributions characterized by using any available method, and we derive the specific forms of these metrics for the Density Ratio Class, which we have found to have many desirable properties. Based on these metrics and on elicitation data from the literature, we use three examples to demonstrate the wide variety of ambiguity that can be present in elicited knowledge. Imprecise probabilities allow us to quantify this ambiguity and consider it in environmental decision-making. Our examples were implemented using a package we recently developed and made freely available for the R statistical programming environment.

Eliciting density ratio classes

The probability distributions of uncertain quantities needed for predictive modelling and decision support are frequently elicited from subject matter experts. However, experts are often uncertain about quantifying their beliefs using precise probability distributions. Therefore, it seems natural to describe their uncertain beliefs using sets of probability distributions. There are various possible structures, or classes, for defining set membership of continuous random variables. The Density Ratio Class has desirable properties, but there is no established procedure for eliciting this class. Thus, we propose a method for constructing Density Ratio Classes that builds on conventional quantile or probability elicitation, but allows the expert to state intervals for these quantities. Parametric shape functions, ideally also suggested by the expert, are then used to bound the nonparametric set of shapes of densities that belong to the class and are compatible with the stated intervals. This leads to a natural metric for the size of the class based on the ratio of the total areas under upper and lower bounding shape functions. This ratio will be determined by the characteristics of the shape functions, the scatter of the elicited values, and the explicit expert imprecision, as characterized by the width of the stated intervals. We provide some examples, both didactic and real, and conclude with recommendations for the further development and application of the Density Ratio Class.

Optimally robust credible sets with a class of priors

We determine a credible set A that is the “best” with respect to the variation of the prior distribution in a neighborhood Γ of the starting prior π 0 ( θ ) . Among the class of sets with credibility γ under π 0 , the “optimally robust” set will be the one which maximizes the minimum probability of including θ as the prior varies over Γ . This procedure is also Γ -minimax with respect to the risk function, probability of non-inclusion. We find the optimally robust credible set for three neighborhood classes Γ , the ε -contamination class, the density ratio class and the density bounded class. A consequence of this investigation is that the maximum likelihood set is seen to be an optimal credible set from a robustness perspective.

On Smoothness Constraints with Shape Constraints in a Robust Bayesian Analysis

We examine the role of the likelihood in Bayesian robustness with the density ratio class (DeRobertis and Hartigan, 1981. Ann. Stat.) and show how to impose smoothness on the density ratio class after imposing shape constraints. We discuss how to impose shape constraints on the density bounded class (Lavine, 1991. JASA)

On the Imposition of Shape Constraints in a Robust Bayesian Analysis

Abstract We consider the problem of imposing shape constraints on a neighborhood class -the density ratio class (DeRobertis and Hartigan, 1981). Bose (1994) used mixture distributions to impose shape and smoothness constraints simultaneously. We discuss how one may impose either or both unimodality and symmetry without requiring simultaneous imposition of a smoothness constraint.

Asymptotics in Bayesian decision theory with applications to global robustness

We provide the rate of convergence of the Bayes action derived from non smooth loss functions involved in Bayesian robustness. Such loss functions are typically not twice differentiable but admit right and left second derivatives. The asymptotic limit of three measures of global robustness is given. These measures are the range of the Bayes actions set associated with a class of loss functions, the maximum regret of using a particular loss when the subjective loss belongs to a given class and the range of the posterior expected loss when the loss ranges over a given class. An application to prior robustness with density ratio classes is provided.

Robustness of the posterior mean in normal hierarchical models

We consider the problem of robustness in hierarchical Bayes models. Let X = (X1,X2, … ,Xp)τ be a random vector, the X1 being independently distributed as N(θ1,σ2) random variables (σ2 known), while the θ1 are thought to be exchangeable, modelled as i.i.d, N(μ,τ2). The hyperparameter µ is given a noninformative prior distribution π(μ) = 1 and τ2 is assumed to be independent of µ having a distribution g(τ2) lying in a certain class of distributions g. For several g's, including e-contaminations classes and density ratio classes we determine the range of the posterior mean of θ1 as g ranges over g.

An Approach to Robust Bayesian Analysis for Multidimensional Parameter Spaces

Abstract In some multiparameter Bayesian problems the prior is more naturally described as a joint density for all of the parameters and in others it is more naturally described as a product of marginal and conditional densities. Although ordinary quantile, e-contamination, and density-ratio classes of priors may be sensible choices for robust Bayesian analyses in the former case, they may be less satisfactory in the latter. This article shows how to construct product classes of priors for use in the second case, using one class of priors for the marginal and another for each conditional. Then the article introduces density-bounded classes as an alternative to the more familiar classes for representing uncertainty about the prior. It is often of interest to find tight bounds on posterior expectations as the prior ranges over the class of priors—an apparently difficult nonlinear optimization problem regardless of which class of priors is used. This article describes a technique for turning the problem into...