### Rob Tibshirani - Recent Advances in Post-Selection Statistical Inference

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Dear Rob, regarding the polyhedral result for stepwise model selection: A data set that stepwise procedure adds variable 1 and then 2 and removes any others and a set that adds 2 and then 1 and selects no other give the same model at the end, but the selection region is the union of polyhedrons, but this union is not itself a polyhedron. Is this a problem?

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It can be a problem depending on how large the selected model is. In the (fixed-lambda) lasso case, for example, if the selected model has r variables then the region is a union of 2^r polyhedra. So it is a problem if the selected model is not sufficiently sparse. However, we can still proceed by redefining a model to include not just the set of selected variables but also their signs. Then the model selection region reduces to a single polyhedron. Or equivalently, instead of changing the definition of a model we can just say we're conducting the conditional test, conditional not only on the selected model but also the observed signs. Either way, this makes computation feasible again at the cost of losing power.