Bayes factor

Résumé

The Bayes factor is a ratio of two competing statistical models represented by their evidence, and is used to quantify the support for one model over the other. The models in questions can have a common set of parameters, such as a null hypothesis and an alternative, but this is not necessary; for instance, it could also be a non-linear model compared to its linear approximation. The Bayes factor can be thought of as a Bayesian analog to the likelihood-ratio test, although it uses the (integrated) marginal likelihood rather than the maximized likelihood. As such, both quantities only coincide under simple hypotheses (e.g., two specific parameter values). Also, in contrast with null hypothesis significance testing, Bayes factors support evaluation of evidence in favor of a null hypothesis, rather than only allowing the null to be rejected or not rejected. Although conceptually simple, the computation of the Bayes factor can be challenging depending on the complexity of the model and t

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https://en.wikipedia.org/wiki/Bayes_factor

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Objective Bayesian Model Selection

Thomas Lugrin

Frequentist and Bayesian approaches to statistics have long been seen as incompatible, but recent work has been done to try and unify them (Bayarri and Berger, 2004; Efron, 2005). Empirical Bayes, approximate Bayesian analysis, and the matching prior approach are examples of methods where prior elicitation is driven by a frequentist interpretation of the data. In this report, we review some standard frequentist and Bayesian model selection techniques and describe how objective Bayes theory can help in this decision-oriented framework, typically by enabling consideration of uncertainty in the model-building process. Objective Bayes mehtods are then used and compared with standard model selection methods on data from the financial sector in Switzerland, Greece, and the United States during 1999 to 2013; results show broad agreement between the methods, but conclusions are less clear-cut in the objective Bayes framework.

2015

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A Bayesian Alternative to Gain Adaptation in Autoregressive Hidden Markov Models

Bertrand Mesot

Models dealing directly with the raw acoustic speech signal are an alternative to conventional feature-based HMMs. A popular way to model the raw speech signal is by means of an autoregressive (AR) process. Being too simple to cope with the nonlinearity of the speech signal, the AR process is generally embedded into a more elaborate model, such as the switching autoregressive HMM (SAR-HMM). A fundamental issue faced by models based on AR processes is that they are very sensitive to variations in the amplitude of the signal. One way to overcome this limitation is to use Gain Adaptation to adjust the amplitude by maximising the likelihood of the observed signal. However, adjusting model parameters by maximising test likelihoods is fundamentally outside the framework of standard statistical approaches to machine learning, since this may lead to overfitting when the models are sufficiently flexible. We propose a statistically principled alternative based on an exact Bayesian procedure in which priors are explicitly defined on the parameters of the AR process. Explicitly, we present the Bayesian SAR-HMM and compare the performance of this model against the standard Gain-Adapted SAR-HMM on a single digit recognition task, showing the effectiveness of the approach and suggesting thereby a principled and straightforward solution to the issue of Gain Adaptation.

IDIAP2006

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