Can we implement these in Stan?
Marginally specified priors for non-parametric Bayesian estimation (by David Kessler, Peter Hoff, and David Dunson):
Prior specification for non-parametric Bayesian inference involves the difficult task of quantifying prior knowledge about a parameter of high, often infinite, dimension. A statistician is unlikely to have informed opinions about all aspects of such a parameter but will have real information about functionals of the parameter, such as the population mean or variance. The paper proposes a new framework for non-parametric Bayes inference in which the prior distribution for a possibly infinite dimensional parameter is decomposed into two parts: an informative prior on a finite set of functionals, and a non-parametric conditional prior for the parameter given the functionals. . . .
Some version of this should be pretty easy in Stan, I’d think, as it’s no problem to supply probabilistic information on transformed parameters.
Adaptive Higher-order Spectral Estimators (by David Gerard and Peter Hoff):
Many applications involve estimation of a signal matrix from a noisy data matrix. In such cases, it has been observed that estimators that shrink or truncate the singular values of the data matrix perform well when the signal matrix has approximately low rank. In this article, we generalize this approach to the estimation of a tensor of parameters from noisy tensor data. We develop new classes of estimators that shrink or threshold the mode-specific singular values from the higher-order singular value decomposition. These classes of estimators are indexed by tuning parameters, which we adaptively choose from the data by minimizing Stein’s unbiased risk estimate. . . .
“Stein’s unbiased risk estimate” sounds kinda silly but it should be possible to just do this using full Bayes. In any case, the real point is to make use of this class of models rather than the messy ensembles of interactions that we are often working with now.
Hierarchical array priors for ANOVA decompositions of cross-classified data (by Alexander Volfovsky, Peter Hoff):
ANOVA decompositions are a standard method for describing and estimating heterogeneity among the means of a response variable across levels of multiple categorical factors. In such a decomposition, the complete set of main effects and interaction terms can be viewed as a collection of vectors, matrices and arrays that share various index sets defined by the factor levels. For many types of categorical factors, it is plausible that an ANOVA decomposition exhibits some consistency across orders of effects, in that the levels of a factor that have similar main-effect coefficients may also have similar coefficients in higher-order interaction terms. In such a case, estimation of the higher-order interactions should be improved by borrowing information from the main effects and lower-order interactions. To take advantage of such patterns, this article introduces a class of hierarchical prior distributions for collections of interaction arrays that can adapt to the presence of such interactions. . . .
I’d really like to work with this model, or something like it. I seem to recall talking with Volfovsky and there was some reason we can’t just fit it in Stan as is, but maybe there’s some way we could adapt the model, or the computation.