Bayesian model comparison confirmed that delay-choices were best predicted by a hyperbolic model, with the largest reward devaluations occurring at shorter delays.

There is a high degree of correlation between findings from cross-validation and Bayesian model comparison (see for example [98]).

However, the Bayesian model evidence penalizes models in proportion to how far the posterior is from the prior (as quantified by the KL-divergence).

This property renders the Bayesian model evidence a better model comparison criterion than AIC or BIC [95].

By using a robust Bayesian modeling approach (Experiment 1), and by directly measuring participants’ indifference points for different effort levels (Experiment 2), our results instead indicate that effort discounting is best characterized by a sigmoidal two-parameter model that allows initially concave discounting shapes.

Comparison between the resulting fits was conducted using Bayesian model comparison (see Materials and Methods).

Importantly, in either task version, the hyperbolic model outperformed the sigmoidal model ( Bayesian model comparison for delay task involving words: xp 2 0.98; mp = 0.75), consistent with a large literature on delay-based choices.

Our Bayesian Model Comparison accounts for model complexity by using the Kullback-Leibler divergence between prior and posterior densities over parameters.

AB, Bayesian Model Comparison comparing all four (A) or five (B) behavioral models: in both comparisons, the hyperbolic model provides the best explanation for choices on the delay task.

These values determine the results of the Bayesian model comparison shown in Fig.

We have therefore chosen a Bayesian model structure where To is informed largely by the within projection variability, T1, 12,. .

However, if policymakers believe that some modeling assumptions are more reliable in terms of capturing the variability of outcomes, we envisage that the Bayesian model structure can be altered to include this.

We argue that the framework we introduce here has great potential, and foresee that many of the questions addressed in epidemiological modeling would require further developments of the Bayesian model , structured to fit with the specific problem.

We use Markov Chain Monte Carlo (MCMC) techniques to obtain samples from the full posterior distribution of the proposed Bayesian models (NHW, SHW and IHW).

To our knowledge, no ensemble study has implemented weights based exclusively on expert opinion, but Bayesian model averaging can incorporate expert opinion as a subjective prior on model probabilities [38].

This could be done in different ways, but here we offer a straightforward multi-quantity extension of the Bayesian model for the single epidemic quantity, based on the supposition that the relative weights are equal for all quantities.

We expand the Bayesian model by defining where xi, q and yi, q are the mean projections of modeling assumption 1' for epidemic quantity q for the implemented and alternative control action, respectively, and x0, q is the corresponding observed value.

As for the single epidemic quantity example, yqand anre the eXpected values of quantity q, and because we cannot eXpect to have the same correlation between control actions for all quantities, fiq is included as unique for each q. Parameters 9% q and 01,, q scales the precision of models between actions and quantities and the parameters of the Bayesian model are identifiable by defining 9H, 1 = 1.

[47] however points out that while it is certainly possible to construct a Bayesian model that takes this uncertainty into account, the effect is minimal if the number of replicates is large.

We analyzed human brain activity patterns from 148 studies of emotion categories (2159 total participants) using a novel hierarchical Bayesian model .

The above procedure for a Bayesian model can be very computationally expensive since it involves multiple posterior simulations.

From a meta-analytic database of nearly 400 neuroimaging studies (6,827 participants) on affect and emotion, we used a subset of studies (148 studies) focused on the five emotion categories mentioned above to develop an integrated, hierarchical Bayesian model of the functional brain patterns underlying them.

Once the Bayesian model is estimated, it can be inverted in a straightforward manner to estimate the posterior probability of each emotion category given a set of brain activation coordinates (see Methods).

Intensity maps reflect the distribution of study activation centers (Level 2 in the Bayesian model ) across brain space.

The edges (lines) reflect co-activation between pairs of regions or networks, assessed based on the joint distribution of activation intensity in the Bayesian model at P <.

Top: Average correlation in regional intensity across 10,000 MCMC samples in the Bayesian model .

This flexibility is a hallmark of generative Bayesian models that provides advantages in allowing researchers to test new metrics, features, and patterns, rather than being limited to a fixed set of features such as pairwise correlations.