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Mixtures of Zellner's g-priors have been studied extensively in linear models and have been shown to have numerous desirable properties for Bayesian variable selection and model averaging. Several extensions of g-priors to generalized linear models (GLMs) have been proposed in the literature; however, the choice of prior distribution of g and resulting properties for inference have received considerably less attention. In this article, we unify mixtures of g-priors in GLMs by assigning the truncated Compound Confluent Hypergeometric (tCCH) distribution to 1/(1 + g), which encompasses as special cases several mixtures of g-priors in the literature, such as the hyper-g, Beta-prime, truncated Gamma, incomplete inverse-Gamma, benchmark, robust, hyper-g/n, and intrinsic priors. Through an integrated Laplace approximation, the posterior distribution of 1/(1 + g) is in turn a tCCH distribution, and approximate marginal likelihoods are thus available analytically, leading to \"Compound Hypergeometric Information Criteria\" for model selection. We discuss the local geometric properties of the g-prior in GLMs and show how the desiderata for model selection proposed by Bayarri et al., such as asymptotic model selection consistency, intrinsic consistency, and measurement invariance may be used to justify the prior and specific choices of the hyper parameters. We illustrate inference using these priors and contrast them to other approaches via simulation and real data examples. The methodology is implemented in the R package BAS and freely available on CRAN. Supplementary materials for this article are available online.

Within computational neuroscience, the algorithmic and neural basis of structure learning remains poorly understood. Concept learning is one primary example, which requires both a type of internal model expansion process (adding novel hidden states that explain new observations), and a model reduction process (merging different states into one underlying cause and thus reducing model complexity via meta-learning). Although various algorithmic models of concept learning have been proposed within machine learning and cognitive science, many are limited to various degrees by an inability to generalize, the need for very large amounts of training data, and/or insufficiently established biological plausibility. Using concept learning as an example case, we introduce a novel approach for modeling structure learning – and specifically state-space expansion and reduction – within the active inference framework and its accompanying neural process theory. Our aim is to demonstrate its potential to facilitate a novel line of active inference research in this area. The approach we lay out is based on the idea that a generative model can be equipped with extra (hidden state or cause) ‘slots’ that can be engaged when an agent learns about novel concepts. This can be combined with a Bayesian model reduction process, in which any concept learning – associated with these slots – can be reset in favor of a simpler model with higher model evidence. We use simulations to illustrate this model’s ability to add new concepts to its state space (with relatively few observations) and increase the granularity of the concepts it currently possesses. We also simulate the predicted neural basis of these processes. We further show that it can accomplish a simple form of ‘one-shot’ generalization to new stimuli. Although deliberately simple, these simulation results highlight ways in which active inference could offer useful resources in developing neurocomputational models of structure learning. They provide a template for how future active inference research could apply this approach to real-world structure learning problems and assess the added utility it may offer.

This technical note introduces adiabatic dynamic causal modelling, a method for inferring slow changes in biophysical parameters that control fluctuations of fast neuronal states. The application domain we have in mind is inferring slow changes in variables (e.g., extracellular ion concentrations or synaptic efficacy) that underlie phase transitions in brain activity (e.g., paroxysmal seizure activity). The scheme is efficient and yet retains a biophysical interpretation, in virtue of being based on established neural mass models that are equipped with a slow dynamic on the parameters (such as synaptic rate constants or effective connectivity). In brief, we use an adiabatic approximation to summarise fast fluctuations in hidden neuronal states (and their expression in sensors) in terms of their second order statistics; namely, their complex cross spectra. This allows one to specify and compare models of slowly changing parameters (using Bayesian model reduction) that generate a sequence of empirical cross spectra of electrophysiological recordings. Crucially, we use the slow fluctuations in the spectral power of neuronal activity as empirical priors on changes in synaptic parameters. This introduces a circular causality, in which synaptic parameters underwrite fast neuronal activity that, in turn, induces activity-dependent plasticity in synaptic parameters. In this foundational paper, we describe the underlying model, establish its face validity using simulations and provide an illustrative application to a chemoconvulsant animal model of seizure activity.

We focus on a Bayesian inference framework for finite element (FE) model updating of a long-span cable-stayed bridge using long-term monitoring data collected from a wireless sensor network (WSN). A robust Bayesian inference method is proposed which marginalizes the prediction-error precisions and applies Transitional Markov Chain Monte Carlo (TMCMC) algorithm. The proposed marginalizing error precision is compared with other two treatments of prediction-error precisions, including the constant error precisions and updating error precisions through theoretical analysis and numerical investigation based on a bridge FE model. TMCMC is employed to draw samples from the posterior probability density function (PDF) of the structural model parameters and the uncertain prediction-error precision parameters if required. It is found that the proposed Bayesian inference method with prediction-error precisions marginalized as “nuisance” parameters produces an FE model with more accurate posterior uncertainty quantification and robust modal property prediction. When applying the identified modal parameters from acceleration data collected during a one-year period from the large-scale WSN on the bridge, we choose two candidate model classes using different parameter grouping based on the clustering results from a sensitivity analysis and apply Bayes’ Theorem at the model class level. By implementing the TMCMC sampler, both the posterior distributions of the structural model parameters and the plausibility of the two model classes are characterized given the real data. Computation of the posterior probabilities over the candidate model classes provides a procedure for Bayesian model class assessment, where the computation automatically implements Bayesian Ockham razor that trades off between data-fitting and model complexity, which penalizes model classes that “over-fit” the data. The results of FE model updating and assessment based on the real data using the proposed method show that the updated FE model can successfully predict modal properties of the structural system with high accuracy.

Identifying a coupled dynamical system out of many plausible candidates, each of which could serve as the underlying generator of some observed measurements, is a profoundly ill-posed problem that commonly arises when modelling real-world phenomena. In this review, we detail a set of statistical procedures for inferring the structure of nonlinear coupled dynamical systems (structure learning), which has proved useful in neuroscience research. A key focus here is the comparison of competing models of network architectures—and implicit coupling functions—in terms of their Bayesian model evidence. These methods are collectively referred to as dynamic causal modelling. We focus on a relatively new approach that is proving remarkably useful, namely Bayesian model reduction, which enables rapid evaluation and comparison of models that differ in their network architecture. We illustrate the usefulness of these techniques through modelling neurovascular coupling (cellular pathways linking neuronal and vascular systems), whose function is an active focus of research in neurobiology and the imaging of coupled neuronal systems. This article is part of the theme issue ‘Coupling functions: dynamical interaction mechanisms in the physical, biological and social sciences'.

Geochemical processes in subsurface reservoirs affected by microbial activity change the material properties of porous media. This is a complex biogeochemical process in subsurface reservoirs that currently contains strong conceptual uncertainty. This means, several modeling approaches describing the biogeochemical process are plausible and modelers face the uncertainty of choosing the most appropriate one. The considered models differ in the underlying hypotheses about the process structure. Once observation data become available, a rigorous Bayesian model selection accompanied by a Bayesian model justifiability analysis could be employed to choose the most appropriate model, i.e. the one that describes the underlying physical processes best in the light of the available data. However, biogeochemical modeling is computationally very demanding because it conceptualizes different phases, biomass dynamics, geochemistry, precipitation and dissolution in porous media. Therefore, the Bayesian framework cannot be based directly on the full computational models as this would require too many expensive model evaluations. To circumvent this problem, we suggest to perform both Bayesian model selection and justifiability analysis after constructing surrogates for the competing biogeochemical models. Here, we will use the arbitrary polynomial chaos expansion. Considering that surrogate representations are only approximations of the analyzed original models, we account for the approximation error in the Bayesian analysis by introducing novel correction factors for the resulting model weights. Thereby, we extend the Bayesian model justifiability analysis and assess model similarities for computationally expensive models. We demonstrate the method on a representative scenario for microbially induced calcite precipitation in a porous medium. Our extension of the justifiability analysis provides a suitable approach for the comparison of computationally demanding models and gives an insight on the necessary amount of data for a reliable model performance.

Summary The Joint United Nations Programme on HIV/AIDS (UNAIDS) has decided to use Bayesian melding as the basis for its probabilistic projections of HIV prevalence in countries with generalized epidemics. This combines a mechanistic epidemiological model, prevalence data, and expert opinion. Initially, the posterior distribution was approximated by sampling-importance-resampling, which is simple to implement, easy to interpret, transparent to users, and gave acceptable results for most countries. For some countries, however, this is not computationally efficient because the posterior distribution tends to be concentrated around nonlinear ridges and can also be multimodal. We propose instead incremental mixture importance sampling (IMIS), which iteratively builds up a better importance sampling function. This retains the simplicity and transparency of sampling importance resampling, but is much more efficient computationally. It also leads to a simple estimator of the integrated likelihood that is the basis for Bayesian model comparison and model averaging. In simulation experiments and on real data, it outperformed both sampling importance resampling and three publicly available generic Markov chain Monte Carlo algorithms for this kind of problem.

In this paper, we propose a Bayesian generative model that can form multiple categories based on each sensory-channel and can associate words with any of the four sensory-channels (action, position, object, and color). This paper focuses on cross-situational learning using the co-occurrence between words and information of sensory-channels in complex situations rather than conventional situations of cross-situational learning. We conducted a learning scenario using a simulator and a real humanoid iCub robot. In the scenario, a human tutor provided a sentence that describes an object of visual attention and an accompanying action to the robot. The scenario was set as follows: the number of words per sensory-channel was three or four, and the number of trials for learning was 20 and 40 for the simulator and 25 and 40 for the real robot. The experimental results showed that the proposed method was able to estimate the multiple categorizations and to learn the relationships between multiple sensory-channels and words accurately. In addition, we conducted an action generation task and an action description task based on word meanings learned in the cross-situational learning scenario. The experimental results showed that the robot could successfully use the word meanings learned by using the proposed method.

Multinomial processing trees (MPTs) are a popular class of cognitive models for categorical data. Typically, researchers compare several MPTs, each equipped with many parameters, especially when the models are implemented in a hierarchical framework. A Bayesian solution is to compute posterior model probabilities and Bayes factors. Both quantities, however, rely on the marginal likelihood, a high-dimensional integral that cannot be evaluated analytically. In this case study, we show how Warp-III bridge sampling can be used to compute the marginal likelihood for hierarchical MPTs. We illustrate the procedure with two published data sets and demonstrate how Warp-III facilitates Bayesian model averaging.

The Bayesian approach is being a fundamental tool in forensic and legal field where inferences and decisions are made. In this study, a full Bayesian calibration model was developed to make probabilistic inferences about age estimation in a reference sample of 891 periapical X-rays of upper and lower canines. These teeth belonged to both deceased and living adult subjects, aged between 20 and 86 years, coming from five different countries (Turkey, Italy, Portugal, Japan and Mexico). For this purpose, the narrowing of pulp chamber due to the apposition of secondary dentine was analysed by means of the pulp/tooth area ratio. To determine the agreement of the method, intra- and inter-observer differences for measuring process were calculated by means of the intraclass correlation coefficient (ICC) analysis. Observer error tests showed excellent agreement between observers and between repeated assessments. According to the results of the ANCOVA, neither nationality nor sex was associated to the secondary dentine apposition while it is associated with individual’s age. The results of the present study indicated that the concept of probability is intrinsically linked to the assessment of age in a forensic context, and the Bayesian approach could be considered a robust tool to overtake the bias generated by traditional regression models, thus helping the decision-making process in a legal framework.