26,257 research outputs found
Robust Bayesian inference via coarsening
The standard approach to Bayesian inference is based on the assumption that
the distribution of the data belongs to the chosen model class. However, even a
small violation of this assumption can have a large impact on the outcome of a
Bayesian procedure. We introduce a simple, coherent approach to Bayesian
inference that improves robustness to perturbations from the model: rather than
condition on the data exactly, one conditions on a neighborhood of the
empirical distribution. When using neighborhoods based on relative entropy
estimates, the resulting "coarsened" posterior can be approximated by simply
tempering the likelihood---that is, by raising it to a fractional power---thus,
inference is often easily implemented with standard methods, and one can even
obtain analytical solutions when using conjugate priors. Some theoretical
properties are derived, and we illustrate the approach with real and simulated
data, using mixture models, autoregressive models of unknown order, and
variable selection in linear regression
More "normal" than normal: scaling distributions and complex systems
One feature of many naturally occurring or engineered complex systems is tremendous variability in event sizes. To account for it, the behavior of these systems is often described using power law relationships or scaling distributions, which tend to be viewed as "exotic" because of their unusual properties (e.g., infinite moments). An alternate view is based on mathematical, statistical, and data-analytic arguments and suggests that scaling distributions should be viewed as "more normal than normal". In support of this latter view that has been advocated by Mandelbrot for the last 40 years, we review in this paper some relevant results from probability theory and illustrate a powerful statistical approach for deciding whether the variability associated with observed event sizes is consistent with an underlying Gaussian-type (finite variance) or scaling-type (infinite variance) distribution. We contrast this approach with traditional model fitting techniques and discuss its implications for future modeling of complex systems
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