Hearing Aid HA algorithms need to be tuned “fitted” to match the impairment of each specific patient. The lack of a fundamental HA fitting theory is a strong contributing factor to an unsatisfying sound experience for about 20 of HA patients. This paper proposes a probabilistic modeling approach to the design of HA algorithms. The proposed method relies on a generative probabilistic model for the hearing loss problem and provides for automated inference of the corresponding 1 signal processing algorithm, 2 the fitting solution as well as 3 a principled performance evaluation metric. All three tasks are realized as message passing algorithms in a factor graph representation of the generative model, which in principle allows for fast implementation on HA or mobile device hardware. The methods are theoretically worked out and simulated with a custom-built factor graph toolbox for a specific hearing loss model.
This paper deals with Bayesian selection of models that can be specified using inequality constraints among the model parameters. The concept of encompassing priors is introduced, that is, a prior distribution for an unconstrained model from which the prior distributions of the constrained models can be derived. It is shown that the Bayes factor for the encompassing and a constrained model has a very nice interpretation: it is the ratio of the proportion of the prior and posterior distribution of the encompassing model in agreement with the constrained model. It is also shown that, for a specific class of models, selection based on encompassing priors will render a virtually objective selection procedure. The paper concludes with three illustrative examples: an analysis of variance with ordered means; a contingency table analysis with ordered odds-ratios; and a multilevel model with ordered slopes. 1 Inequality constrained statistical models Researchers often have one or more (competing) theories about their field of research. Consider, for example, theories about the effect of behavioral therapy versus medication for children with an attention deficit disorder (ADD). Some researchers in this area believe medication is the only effective treatment for ADD, some believe strongly in behavioral therapy, and others may expect an additive effect of both therapies. To test or compare the plausibility of these theories they need to be translated into statistical models. Subsequently, empirical data can be used to determine which model is best. Inequality constraints on model parameters can be useful in the specification of statistical models. This paper deals with competing models that have the same parameter vector, but in one or more of the models parameters are subjected to inequality constraints. To continue the example, consider an experiment where children with ADD are randomly assigned to one of four conditions: no treatment (1), behavioral therapy (2), medication (3), and behavioral therapy plus medication (4). Let the outcome Abstract Constrained parameter problems arise in a wide variety of applications, including bioassay, actuarial graduation, ordinal categorical data, response surfaces, reliability development testing, and variance component models. Truncated data problems arise naturally in survival and failure time studies, ordinal data models, and categorical data studies aimed at uncovering underlying continuous distributions. In many applications both parameter constraints and data truncation are present. The statistical literature on such problems is very extensive, reflecting both the problems’ widespread occurrence in applications and the methodological challenges that they pose. However, it is striking that so little of this applied and theoretical literature involves a parametric Bayesian perspective. From a technical viewpoint, this perhaps is not difficult to understand. The fundamental tool for Bayesian calculations in typical realistic models is (multidimensional) numerical integration, which often is problem... The Bayes factor is a useful tool for evaluating sets of inequality and about equality constrained models. In the approach described, the Bayes factor for a constrained model with the encompassing model reduces to the ratio of two proportions, namely the proportion of, respectively, the encompassing prior and posterior in agreement with the constraints. This enables easy and straightforward estimation of the Bayes factor and its Monte Carlo Error. In this set-up, the issue of sensitivity to model specific prior distributions reduces to sensitivity to one prior distribution, that is, the prior for the encompassing model. It is shown that for specific classes of inequality constrained models, the Bayes factors for the constrained with the unconstrained model is virtually independent of the encompassing prior, that is, model selection is virtually objective. An encompassing prior (EP) approach to facilitate Bayesian model selection for nested models with inequality constraints has been previously proposed. In this approach, samples are drawn from the prior and posterior distributions of an encompassing model that contains an inequality restricted version as a special case. The Bayes factor in favor of the inequality restriction then simplifies to the ratio of the proportions of posterior and prior samples consistent with the inequality restriction. This formalism has been applied almost exclusively to models with inequality or ''about equality'' constraints. It is shown that the EP approach naturally extends to exact equality constraints by considering the ratio of the heights for the posterior and prior distributions at the point that is subject to test (i.e., the Savage-Dickey density ratio). The EP approach generalizes the Savage-Dickey ratio method, and can accommodate both inequality and exact equality constraints. The general EP approach is found to be a computationally efficient procedure to calculate Bayes factors for nested models. However, the EP approach to exact equality constraints is vulnerable to the Borel-Kolmogorov paradox, the consequences of which warrant careful consideration.