Improved Laplace approximation for marginal likelihoods

Ruli, Erlis; Sartori, Nicola; Ventura, Laura

doi:10.1214/16-ejs1218

Cited by 17 publications

(15 citation statements)

References 46 publications

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“…It requires computing the marginal likelihood or the normalizing constant. Numerous methods have been proposed to compute this integral, for example Laplace's approximation (Ruli, Sartori, & Ventrua, 2016), bridge sampling (Gronau et al, 2017), and Chib's MCMC approximation (Siddhartha, 1995). Further, it is common to use conjugate prior distributions that provide an analytic expression for (7).…”

Section: Bayesian Hypothesis Testingmentioning

confidence: 99%

Putting the Individual into Reliability: Bayesian Testing of Homogeneous Within-Person Variance in Hierarchical Models

Williams¹,

Martin²,

Rast³

2019

Preprint

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Measurement reliability is a fundamental concept in psychology. It is traditionally considered a stable property of a questionnaire, measurement device, or experimental task. Although intraclass correlation coefficients (ICC) are often used to assess reliability in repeated measure designs, their descriptive nature depends upon the assumption of a commonwithin-person variance.This work focuses on the presumption that each individual is adequately described by the average within-person variance in hierarchical models. And thus whether reliability generalizes to the individual level, which leads directly into the notion of individually varying ICCs. In particular, we introduce a novel approach, using the Bayes factor, wherein a researcher can directly test for homogeneous within-person variance in hierarchical models. Additionally, we introduce a membership model that allows for classifying which (and how many) individuals belong to the common variance model. The utility of our methodology is demonstrated on cognitive inhibition tasks. We find that heterogeneous within-person variance is a defining feature of these tasks, and in one case, the ratio between the largest to smallest within-person variance exceeded 20. This translates into a 10 fold difference in person-specific reliability! We also find that few individuals belong to the common variance model, and thus traditional reliability indices are potentially masking important individual variation. We discuss the implications of our findings and possible future directions. The methods are implemented in the R package vICC.

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Section: Bayesian Hypothesis Testingmentioning

confidence: 99%

Putting the Individual into Reliability: Bayesian Testing of Homogeneous Within-Person Variance in Hierarchical Models

Williams¹,

Martin²,

Rast³

2019

Preprint

View full text Add to dashboard Cite

show abstract

“…It requires computing the marginal likelihood, for which numerous approaches have been proposed, including, for example, Laplace's approximation (Ruli, Sartori, & Ventrua, 2016), bridge sampling (Gronau, Sarafoglou, et al, 2017), and various MCMC approximations (e.g., Siddhartha, 1995). Alternative strategies aim to side-step computing the marginal likelihood altogether.…”

Section: Bayesian Hypothesis Testingmentioning

confidence: 99%

Many Mixture Components, Oh My: Extending the Spike and Slab to Bayesian Hypothesis Testing with Multinoulli Indicators

Williams¹

2021

Preprint

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Bayesian analysis has become increasingly popular in the social-behavioral sciences. Because hypothesis testing has an important place on the mantel of psychological inquiry, an active area of research has been developing Bayesian analogs for commonly used frequentist tests.However, a major hurdle to this endeavour is computing the necessary ingredients, that is the marginal likelihood, resulting in the use of inflexible analytic solutions or approaches that avoid its computation altogether. In a similar spirit, I extend the spike and slab model, widely considered the gold standard for variable selection, to allow for flexible hypothesis testing. This is accomplished by employing multinoulli indicator variables, as opposed to Bernoulli, which results in a general solution for testing any number of hypotheses that correspond to components of a mixture prior distribution. In a motivating example, I first describe the qualitative relation of the proposed methodology to a popular Bayesian $t$-test, including extensions for one-sided and interval hypothesis tests. With the foundation laid, I proceed to a more complex example wherein themultinoulli spike and slab is used to model a correlation matrix, with the goal of testing joint hypotheses. This example investigated the associations among experimental effects from three cognitive inhibition tasks ($N = 121$), where the theoretical expectation is that they will be positively correlated. To the contrary, the results revealed that the null model of no associations better predicted the observed data than the positive effects model.The important topics of model selection and Bayesian model averaging are also discussed. I end with ideas to further extend the multinoulli spike and slab model. In addition, detailed {\tt R} code is provided that can serve as the building block for developing custom Bayesian models.

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“…The Laplace approximation is accurate to

O (n^{- 1})

since it takes into account only the first‐order terms of the Taylor series expansion (Ruli, Sartori, & Ventura, 2016). The application of this method in the univariate case is less cumbersome as it only requires the evaluation of a unidimensional minimization problem.…”

Section: Approximate Bayesian Inference and Predictionmentioning

confidence: 99%

A new mixed‐effects regression model for the analysis of zero‐modified hierarchical count data

et al. 2020

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Count data sets are traditionally analyzed using the ordinary Poisson distribution. However, such a model has its applicability limited as it can be somewhat restrictive to handle specific data structures. In this case, it arises the need for obtaining alternative models that accommodate, for example, (a) zero‐modification (inflation or deflation at the frequency of zeros), (b) overdispersion, and (c) individual heterogeneity arising from clustering or repeated (correlated) measurements made on the same subject. Cases (a)–(b) and (b)–(c) are often treated together in the statistical literature with several practical applications, but models supporting all at once are less common. Hence, this paper's primary goal was to jointly address these issues by deriving a mixed‐effects regression model based on the hurdle version of the Poisson–Lindley distribution. In this framework, the zero‐modification is incorporated by assuming that a binary probability model determines which outcomes are zero‐valued, and a zero‐truncated process is responsible for generating positive observations. Approximate posterior inferences for the model parameters were obtained from a fully Bayesian approach based on the Adaptive Metropolis algorithm. Intensive Monte Carlo simulation studies were performed to assess the empirical properties of the Bayesian estimators. The proposed model was considered for the analysis of a real data set, and its competitiveness regarding some well‐established mixed‐effects models for count data was evaluated. A sensitivity analysis to detect observations that may impact parameter estimates was performed based on standard divergence measures. The Bayesian ‐value and the randomized quantile residuals were considered for model diagnostics.

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Improved Laplace approximation for marginal likelihoods

Cited by 17 publications

References 46 publications

Putting the Individual into Reliability: Bayesian Testing of Homogeneous Within-Person Variance in Hierarchical Models

Putting the Individual into Reliability: Bayesian Testing of Homogeneous Within-Person Variance in Hierarchical Models

Many Mixture Components, Oh My: Extending the Spike and Slab to Bayesian Hypothesis Testing with Multinoulli Indicators

A new mixed‐effects regression model for the analysis of zero‐modified hierarchical count data

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