Sprecher
Beschreibung
Classical convergence results show that Bayesian agents who entertain the true hypothesis H as one of their alternatives will become certain of H’ in the limit. If H is not in the set of alternatives, on the other hand, we may still converge on the 'best' alternative (e.g. in terms of minimal KL-divergence, see Barron 1998). However, it has also been demonstrated how this can fail, if certain structural (in particular, convexity-) properties are violated (Grünwald & van Ommen 2017). If we are careful to set up the model space properly, we may be able to ensure convergence on the best alternative. However, if the parameter space is very complex, doing so may leave the model intractable. In this contribution, I want to examine the prospects for setting up the hypothesis space such as to enable convergence on the best alternative as far as possible, while keeping the model tractable and allowing for efficient data collection.
References:
Barron, A. R. (1998). “Information-Theoretic Characterization of Bayes Performance and the Choice of Priors in Parametric and Nonparametric Problems.” In Bernardo, J. M., Berger, J. O., Dawid, A. P., and Smith, A. F. M. (eds.), Bayesian Statistics, volume 6, 27–52. Oxford: Oxford University Press.
Grünwald, P. and van Ommen, T. (2017). Inconsistency of Bayesian inference for misspecified linear models, and a proposal for repairing it. Bayesian Analysis 9.