main effect in the model. Heredity reduces the number of models from $2^{p + {p \choose 2}}$ to $\sum_{i = 0}^p {p \choose i} 2^{{i \choose 2}}$ or $ \sum_{i = 0}^p {p \choose i} 2^{p i - i (i+1)/2}$ for strong or weak heredity, respectively [@Chipman1996]. For ultra high-dimensional problems, two stage-approaches have been developed, see [@HaoFengZhang2016] and [@wang2019penalized]. However, these methods do not report uncertainties in model selection and parameter estimation, and rely on strong sparsity assumptions.

We are particularly motivated by studies of environmental health collecting data on mixtures of chemical exposures. These exposures can be moderately high-dimensional with high correlations within blocks of variables; for example, this can arise when an individual is exposed to a product having a mixture of chemicals and when chemical measurements consist of metabolites or breakdown products of a parent compound. There is a large public heath interest in studying E$\times$E, E$\times$G and G$\times$G interactions, with E = environmental exposures and G = genetic factors. However, current methods for quadratic regression are not ideal in these applications due to the level of correlation in the predictors, the fact that strong sparsity assumptions are not appropriate, and the need for uncertainty quantification. 