## Bayes factors revisited

“Bayes factor analyses are highly sensitive to and crucially depend on prior assumptions about model parameters (…) Note that the dependency of Bayes factors on the prior goes beyond the dependency of the posterior on the prior. Importantly, for most interesting problems and models, Bayes factors cannot be computed analytically.”

Daniel J. Schad, Bruno Nicenboim, Paul-Christian Bürkner, Michael Betancourt, Shravan Vasishth have just arXived a massive document on the Bayes factor, worrying about the computation of this common tool, but also at the variability of decisions based on Bayes factors, e.g., stressing correctly that

“…we should not confuse inferences with decisions. Bayes factors provide inference on hypotheses. However, to obtain discrete decisions (…) from continuous inferences in a principled way requires utility functions. Common decision heuristics (e.g., using Bayes factor larger than 10 as a discovery threshold) do not provide a principled way to perform decisions, but are merely heuristic conventions.”

The text is long and at times meandering (at least in the sections I read), while trying a wee bit too hard to bring up the advantages of using Bayes factors versus frequentist or likelihood solutions. (The likelihood ratio being presented as a “frequentist” solution, which I think is an incorrect characterisation.) For instance, the starting point of preferring a model with a higher marginal likelihood is presented as an evidence (oops!) rather than argumented. Since this quantity depends on both the prior and the likelihood, it being high or low is impacted by both. One could then argue that using its numerical value as an absolute criterion amounts to selecting the prior a posteriori as much as checking the fit to the data! The paper also resorts to the Occam’s razor argument, which I wish we could omit, as it is a vague criterion, wide open to misappropriation. It is also qualitative, rather than quantitative, hence does not help in calibrating the Bayes factor.

Concerning the actual computation of the Bayes factor, an issue that has always been a concern and a research topic for me, the authors consider only two “very common methods”, the Savage–Dickey density ratio method and bridge sampling. We discussed the shortcomings of the Savage–Dickey density ratio method with Jean-Michel Marin about ten years ago. And while bridge sampling is an efficient approach when comparing models of the same dimension, I have reservations about this efficiency in other settings. Alternative approaches like importance nested sampling, noise contrasting estimation or SMC samplers are often performing quite efficiently as normalising constant approximations. (Not to mention our version of harmonic mean estimator with HPD support.)

Simulation-based inference is based on the notion that simulated data can be produced from the predictive distributions. Reminding me of ABC model choice to some extent. But I am uncertain this approach can be used to calibrate the decision procedure to select the most appropriate model. We thought about using this approach in our testing by mixture paper and it is favouring the more complex of the two models. This seems also to occur for the example behind Figure 5 in the paper.

Two other points: first, the paper does not consider the important issue with improper priors, which are not rigorously compatible with Bayes factors, as I discussed often in the past. And second, Bayes factors are not truly Bayesian decision procedures, since they remove the prior weights on the models, thus the mention of utility functions therein seems inappropriate unless a genuine utility function can be produced.

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