Testing hypotheses via a mixture estimation model

By K. Kamary, Kerrie L. Mengersen, Christian P. Robert, Judith Rousseau. 2014

75 citations recorded by the source corpus.

We consider a novel paradigm for Bayesian testing of hypotheses and Bayesian model comparison. Our alternative to the traditional construction of posterior probabilities that a given hypothesis is true or that the data originates from a specific model is to consider the models under comparison as components of a mixture model. We therefore replace the original testing problem with an estimation one that focus on the probability weight of a given model within a mixture model. We analyse the sensitivity on the resulting posterior distribution on the weights of various prior modelling on the weights. We stress that a major appeal in using this novel perspective is that generic improper priors are acceptable, while not putting convergence in jeopardy. Among other features, this allows for a resolution of the Lindley-Jeffreys paradox. When using a reference Beta B(a,a) prior on the mixture weights, we note that the sensitivity of the posterior estimations of the weights to the choice of a vanishes with the sample size increasing and advocate the default choice a=0.5, derived from Rousseau and Mengersen (2011). Another feature of this easily implemented alternative to the classical Bayesian solution is that the speeds of convergence of the posterior mean of the weight and of the corresponding posterior probability are quite similar.

Subfields: Artificial Intelligence, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/8497a272dcce833da99c3a35e034bd30a13fe498

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Citations in the corpus

Penalising Model Component Complexity: A Principled, Practical Approach to Constructing Priors (2014, methodology): “It is commonly (although not exclusively Bernardo, 2011; Rousseau and Robert, 2011; Kamary et al., 2014) held that we need to use different priors for testing than those used for prediction.”
Bayesian Workflow (2020, methodology): “In the extreme, model averaging can also be done so that different models can apply to different data points (Kamary et al., 2019, Pirš and Štrumbelj, 2019). In Bayesian workflow, we will fit many models that we will not be interested in including in any average; such “scaffolds” include models that”
Using Stacking to Average Bayesian Predictive Distributions (with Discussion) (2017, background): “The heterogeneity of stacking weights can also be generalized to other hierarchical data structures, and this can be seen as related to a generalization of the mixture formulation of Kamary et al. (2014).”
Abandon Statistical Significance (2017, background): “A discussion of the difficulties with this from a Bayesian perspective is provided in Kamary et al. [2014], with a proposal of setting the hypothesis problem as one of mixture estimation.”
The expected demise of the Bayes factor (2015, methodology): “…H. Jeffreys, ToP (p.13) A very relevant point made by the authors in this discussion of ToP is that Harold Jeffreys only considered embedded or nested hypotheses, a fact that allows for some common parameters between models and hence some form of reference prior, as argued in Kamary et al. (2014).”